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  SOUND

  BY

  JOHN TYNDALL, D.C.L., LL.D., F.R.S.

  [Illustration]

  NEW YORK P. F. COLLIER & SON MCMII 7

  SCIENCE

  TO THE MEMORY

  OF

  MY FRIEND RICHARD DAWES

  LATE DEAN OF HEREFORD

  THIS BOOK IS DEDICATED

  J. T.




CONTENTS


CHAPTER I

  The Nerves and Sensation—Production and Propagation of
  Sonorous Motion—Experiments on Sounding Bodies placed
  in Vacuo—Deadening of Sound by Hydrogen—Action of
  Hydrogen on the Voice—Propagation of Sound through Air
  of Varying Density—Reflection of Sound—Echoes—Refraction
  of Sound—Diffraction of Sound; Case of Erith Village and
  Church—Influence of Temperature on Velocity—Influence
  of Density on Elasticity—Newton’s Calculation of
  Velocity—Thermal Changes Produced by the Sonorous
  Wave—Laplace’s Correction of Newton’s Formula—Ratio of
  Specific Heats at Constant Pressure and at Constant
  Volume deduced from Velocities of Sound—Mechanical
  Equivalent of Heat deduced from this Ratio—Inference that
  Atmospheric Air Possesses no Sensible Power to Radiate
  Heat—Velocity of Sound in Different Gases—Velocity in
  Liquids and Solids—Influence of Molecular Structure on
  the Velocity of Sound.                                       31

  SUMMARY OF CHAPTER I                                         77


CHAPTER II

  Physical Distinction between Noise and Music—A
  Musical Tone Produced by Periodic, Noise Produced
  by Unperiodic, Impulses—Production of Musical
  Sounds by Taps—Production of Musical Sounds by
  Puffs—Definition of Pitch in Music—Vibrations of a
  Tuning-Fork; their Graphic Representation on Smoked
  Glass—Optical Expression of the Vibrations of a
  Tuning-Fork—Description of the Siren—Limits of the
  Ear; Highest and Deepest Tones—Rapidity of Vibration
  Determined by the Siren—Determination of the Lengths
  of Sonorous Waves—Wave-Lengths of the Voice in Man and
  Woman—Transmission of Musical Sounds through Liquids and
  Solids.                                                      82

  SUMMARY OF CHAPTER II                                       117


CHAPTER III

  Vibration of Strings—How employed in Music—Influence of
  Sound-Boards—Laws of Vibrating String—Combination of
  Direct and Reflected Pulses—Stationary and Progressive
  Waves—Nodes and Ventral Segments—Application of Results
  to the Vibrations of Musical Strings—Experiments of
  Melde—Springs set in Vibration by Tuning-Forks—Laws
  of Vibration thus demonstrated—Harmonic Tones of
  Strings—Definitions of Timbre or Quality, or Overtones
  and Clang—Abolition of Special Harmonies—Conditions
  which affect the Intensity of the Harmonic Tones—Optical
  Examination of the Vibrations of a Piano-Wire               120

  SUMMARY OF CHAPTER III                                      161


CHAPTER IV

  Vibrations of a Rod fixed at Both Ends: its Subdivisions
  and Corresponding Overtones—Vibrations of a Rod fixed
  at One End—The Kaleidophone—The Iron Fiddle and Musical
  Box—Vibrations of a Rod free at Both Ends—The Claque-bois
  and Glass Harmonica—Vibrations of a Tuning-Fork:
  its Subdivisions and Overtones—Vibrations of Square
  Plates—Chladni’s Discoveries—Wheatstone’s Analysis of
  the Vibrations of Plates—Chladni’s Figures—Vibrations of
  Disks and Bells—Experiments of Faraday and Strehlke.        165

  SUMMARY OF CHAPTER IV                                       196


CHAPTER V

  Longitudinal Vibrations of a Wire—Relative Velocities of
  Sound in Brass and Iron—Longitudinal Vibrations of Rods
  fixed at One End—Of Rods free at Both Ends—Divisions and
  Overtones of Rods vibrating longitudinally—Examination
  of Vibrating Bars by Polarized Light—Determination of
  Velocity of Sound in Solids—Resonance—Vibrations of
  Stopped Pipes: their Divisions and Overtones—Relation
  of the Tones of Stopped Pipes to those of Open
  Pipes—Condition of Column of Air within a Sounding
  Organ-Pipe—Reeds and Reed-Pipes—The Voice—Overtones of
  the Vocal Chords—The Vowel Sounds—Kundt’s Experiments—New
  Methods of determining the Velocity of Sound.               200

  SUMMARY OF CHAPTER V                                        254


CHAPTER VI

  Singing Flames—Influence of the Tube surrounding
  the Flame—Influence of Size of Flame—Harmonic Notes
  of Flames—Effect of Unisonant Notes on Singing
  Flames—Action of Sound on Naked Flames—Experiments with
  Fish-Tail and Bat’s-Wing Burners—Experiments on Tall
  Flames—Extraordinary Delicacy of Flames as Acoustic
  Reagents—The Vowel-Flame—Action of Conversational
  Tones upon Flames—Action of Musical Sounds on
  Smoke-Jets—Constitution of Water-Jets—Plateau’s Theory
  of the Resolution of a Liquid Vein into Drops—Action of
  Musical Sounds on Water-Jets—A Liquid Vein may compete in
  Point of Delicacy with the Ear                              260

  SUMMARY OF CHAPTER VI                                       301


CHAPTER VII

_PART I_

  RESEARCHES ON THE ACOUSTIC TRANSPARENCY OF THE ATMOSPHERE
  IN RELATION TO THE QUESTION OF FOG-SIGNALLING

  Introduction—Instruments and Observations—Contradictory
  Results from the 19th of May to the 1st of July
  inclusive—Solution of Contradictions—Aërial Reflection
  and its Causes—Aërial Echoes—Acoustic Clouds—Experimental
  Demonstration of Stoppage of Sound by Aërial Reflection
                                                              305

_PART II_

  INVESTIGATION OF THE CAUSES WHICH HAVE HITHERTO BEEN
  SUPPOSED EFFECTIVE IN PREVENTING THE TRANSMISSION OF
  SOUND THROUGH THE ATMOSPHERE

  Action of Hail and Rain—Action of Snow—Action of Fog;
  Observations in London—Experiments on Artificial
  Fogs—Observations on Fogs at the South Foreland—Action of
  Wind—Atmospheric Selection—Influence of Sound-Shadow        341

  SUMMARY OF CHAPTER VII                                      374


CHAPTER VIII

  Law of Vibratory Motions in Water and
  Air—Superposition of Vibrations—Interference of
  Sonorous Waves—Destruction of Sound by Sound—Combined
  Action of Two Sounds nearly in Unison with each
  other—Theory of Beats—Optical Illustration of the
  Principle of Interference—Augmentation of Intensity
  by Partial Extinction of Vibrations—Resultant
  Tones—Conditions of their Production—Experimental
  Illustrations—Difference-Tones and
  Summation-Tones—Theories of Young and Helmholtz             377

  SUMMARY OF CHAPTER VIII                                     407


CHAPTER IX

  Combination of Musical Sounds—The smaller the Two
  Numbers which express the Ratio of their Rates of
  Vibration, the more perfect is the Harmony of Two
  Sounds—Notions of the Pythagoreans regarding Musical
  Consonance—Euler’s Theory of Consonance—Theory of
  Helmholtz—Dissonance due to Beats—Interference of Primary
  Tones and of Overtones—Mechanism of Hearing—Schultze’s
  Bristles—The Otoliths—Corti’s Fibres—Graphic
  Representation of Consonance and Dissonance—Musical
  Chords—The Diatonic Scale—Optical Illustration of
  Musical Intervals—Lissajous’s Figures—Sympathetic
  Vibrations—Various Modes of illustrating the Composition
  of Vibrations                                               410

  Summary of Chapter IX                                       450


  APPENDIX I

  ON THE INFLUENCE OF MUSICAL SOUNDS ON THE FLAME OF A JET OF
  COAL-GAS. BY JOHN LE CONTE, M.D.                            454


  APPENDIX II

  ON ACOUSTIC REVERSIBILITY                                   461


  INDEX                                                       471

  ILLUSTRATION—FOG-SIREN                            _Frontispiece_


[Illustration: FOG-SIREN]




PREFACE TO THE THIRD EDITION


In preparing this new edition of “Sound,” I have carefully gone over
the last one; amended, as far as possible, its defects of style
and matter, and paid at the same time respectful attention to the
criticisms and suggestions which the former editions called forth.

The cases are few in which I have been content to reproduce what I
have _read_ of the works of acousticians. I have sought to make myself
experimentally familiar with the ground occupied; trying, in all cases,
to present the illustrations in the form and connection most suitable
for educational purposes.

Though bearing, it may be, an undue share of the imperfection which
cleaves to all human effort, the work has already found its way into
the literature of various nations of diverse intellectual standing.
Last year, for example, a new German edition was published “under the
special supervision” of Helmholtz and Wiedemann. That men so eminent,
and so overladen with official duties, should add to these the labor of
examining and correcting every proof-sheet of a work like this, shows
that they consider it to be what it was meant to be—a serious attempt
to improve the public knowledge of science. It is especially gratifying
to me to be thus assured that not in England alone has the book met a
public want, but also in that learned land to which I owe my scientific
education.

Before me, on the other hand, lie two volumes of foolscap size,
curiously stitched, and printed in characters the meaning of which I
am incompetent to penetrate. Here and there, however, I notice the
familiar figures of the former editions of “Sound.” For these volumes
I am indebted to Mr. John Fryer, of Shanghai, who, along with them,
favored me, a few weeks ago, with a letter from which the following is
an extract: “One day,” writes Mr. Fryer, “soon after the first copy
of your work on Sound reached Shanghai, I was reading it in my study,
when an intelligent official, named Hsii-chung-hu, noticed some of the
engravings and asked me to explain them to him. He became so deeply
interested in the subject of Acoustics that nothing would satisfy him
but to make a translation. Since, however, engineering and other works
were then considered to be of more practical importance by the higher
authorities, we agreed to translate your work during our leisure time
every evening, and publish it separately ourselves. Our translation,
however, when completed, and shown to the higher officials, so much
interested them, and pleased them, that they at once ordered it to be
published at the expense of the Government, and sold at cost price. The
price is four hundred and eighty copper cash per copy, or about one
shilling and eightpence. This will give you an idea of the cheapness of
native printing.”

Mr. Fryer adds that his Chinese friend had no difficulty in grasping
every idea in the book.


The new matter of greatest importance which has been introduced into
this edition is an account of an investigation which, during the past
two years, I have had the honor of conducting in connection with the
Elder Brethren of the Trinity House. Under the title “Researches on the
Acoustic Transparency of the Atmosphere, in Relation to the Question of
Fog-signalling,” the subject is treated in Chapter VII. of this volume.
It was only by Governmental appliances that such an investigation
could have been made; and it gives me pleasure to believe that not
only have the practical objects of the inquiry been secured, but that
a crowd of scientific errors, which for more than a century and a half
have surrounded this subject, have been removed, their place being
now taken by the sure and certain truth of Nature. In drawing up the
account of this laborious inquiry, I aimed at linking the observations
so together that they alone should offer a substantial demonstration of
the principles involved. Further labors enabled me to bring the whole
inquiry within the firm grasp of _experiment_; and thus to give it a
certainty which, without this final guarantee, it could scarcely have
enjoyed.

Immediately after the publication of the first brief abstract of the
investigation, it was subjected to criticism. To this I did not deem it
necessary to reply, believing that the grounds of it would disappear
in presence of the full account. The only opinion to which I thought
it right to defer was to some extent a private one, communicated to
me by Prof. Stokes. He considered that I had, in some cases, ascribed
too exclusive an influence to the mixed currents of aqueous vapor and
air, to the neglect of differences of temperature. That differences
of temperature, when they come into play, are an efficient cause of
acoustic opacity, I never doubted. In fact, aërial reflection arising
from this cause is, in the present inquiry, for the first time made
the subject of experimental demonstration. What the relative potency
of differences of temperature and differences due to aqueous vapor, in
the cases under consideration, may be, I do not venture to state; but
as both are active, I have, in Chapter VII., referred to them jointly
as concerned in the production of those “acoustic clouds” to which the
stoppage of sound in the atmosphere is for the most part due.


Subsequently, however, to the publication of the full investigation
another criticism appeared, to which, in consideration of its source,
I would willingly pay all respect and attention. In this criticism,
which reached me first through the columns of an American newspaper,
differences in the amounts of aqueous vapor, and differences of
temperature, are alike denied efficiency as causes of acoustic
opacity. At a meeting of the Philosophical Society of Washington the
emphatic opinion had, it was stated, been expressed that I was wrong
in ascribing the opacity of the atmosphere to its flocculence, the
really efficient cause being _refraction_. This view appeared to me so
obviously mistaken that I assumed, for a time, the incorrectness of the
newspaper account.

Recently, however, I have been favored with the “Report of the United
States Lighthouse Board for 1874,” in which the account just referred
to is corroborated. A brief reference to the Report will here suffice.
Major Elliott, the accomplished officer and gentleman referred to at
page 261, had published a record of his visit of inspection to this
country, in which he spoke, with a perfectly enlightened appreciation
of the facts, of the differences between our system of lighthouse
illumination and that of the United States. He also embodied in his
Report some account of the investigation on fog-signals, the initiation
of which he had witnessed, and indeed aided, at the South Foreland.

On this able Report of their own officer the Lighthouse Board at
Washington make the following remark: “Although this account is
interesting in itself and to the public generally, yet, being addressed
to the Lighthouse Board of the United States, it would tend to convey
the idea that the facts which it states were new to the Board, and that
the latter had obtained no results of a similar kind; while a reference
to the appendix to this Report[1] will show that the researches of our
Lighthouse Board have been much more extensive on this subject than
those of the Trinity House, and that the latter has established no
facts of practical importance which had not been previously observed
and used by the former.”

The “appendix” here referred to is from the pen of the venerable Prof.
Joseph Henry, chairman of the Lighthouse Board at Washington. To his
credit be it recorded that at a very early period in the history of
fog-signalling Prof. Henry reported in favor of Daboll’s trumpet,
though he was opposed by one of his colleagues on the ground that
“fog-signals were of little importance, since the mariner should know
his place by the character of his soundings.” In the appendix, he
records the various efforts made in the United States with a view to
the establishment of fog-signals. He describes experiments on bells,
and on the employment of reflectors to reinforce their sound. These,
though effectual close at hand, were found to be of no use at a
distance. He corrects current errors regarding steam-whistles, which
by some inventors were thought to act like ringing bells. He cites the
opinion of the Rev. Peter Ferguson, that sound is better heard in fog
than in clear air. This opinion is founded on observations of the noise
of locomotives; in reference to which it may be said that others have
drawn from similar experiments diametrically opposite conclusions. On
the authority of Captain Keeney he cites an occurrence, “in the first
part of which the captain was led to suppose that fog had a marked
influence in deadening sound, though in a subsequent part he came to
an opposite conclusion.” Prof. Henry also describes an experiment made
during a fog at Washington, in which he employed “a small bell rung by
clockwork, the apparatus being the part of a moderator lamp, intended
to give warning to the keepers when the supply of oil ceased. The
result of the experiment was, he affirms, contrary to the supposition
of absorption of the sound by the fog.” This conclusion is not founded
on comparative experiments, but on observations made in the fog alone;
for, adds Prof. Henry, “the change in the condition of the atmosphere,
as to temperature and the motion of the air, before the experiment
could be repeated in clear weather, rendered the result not entirely
satisfactory.”

This, I may say, is the only experiment on fog which I have found
recorded in the appendix.

In 1867 the steam-siren was mounted at Sandy Hook, and examined by
Prof. Henry. He compared its action with that of a Daboll trumpet,
employing for this purpose a stretched membrane covered with sand, and
placed at the small end of a tapering tube which concentrated the
sonorous motion upon the membrane. The siren proved most powerful. “At
a distance of 50, the trumpet produced a decided motion of the sand,
while the siren gave a similar result at a distance of 58.” Prof. Henry
also varied the pitch of the siren, and found that in association with
its trumpet 400 impulses per second yielded the maximum sound; while
the best result with the unaided siren was obtained when the impulses
were 360 a second. Experiments were also made on the influence of
pressure; from which it appeared that when the pressure varied from
100 lbs. to 20 lbs., the distance reached by the sound (as determined
by the vibrating membrane) varied only in the ratio of 61 to 51. Prof.
Henry also showed the sound of the fog-trumpet to be independent of
the material employed in its construction; and he furthermore observed
the decay of the sound when the angular distance from the axis of the
instrument was increased. Further observations were made by Prof. Henry
and his colleagues in August, 1873, and in August, and September,
1874. In the brief but interesting account of these experiments a
hypothetical element appears, which is absent from the record of the
earlier observations.

It is quite evident from the foregoing that, in regard to the question
of fog-signalling, the Lighthouse Board of Washington have not been
idle. Add to this the fact that their eminent chairman gives his
services gratuitously, conducting without fee or reward experiments and
observations of the character here revealed, and I think it will be
conceded that he not only deserves well of his own country, but also
sets his younger scientific contemporaries, both in his country and
ours, an example of high-minded devotion.

I was quite aware, in a general way, that labors like those now for the
first time made public had been conducted in the United States, and
this knowledge was not without influence upon my conduct. The first
instruments mounted at the South Foreland were of English manufacture;
and I, on various accounts, entertained a strong sympathy for their
able constructor, Mr. Holmes. From the outset, however, I resolved to
suppress such feelings, as well as all other extraneous considerations,
individual or national; and to aim at obtaining the best instruments,
irrespective of the country which produced them. In reporting,
accordingly, on the observations of May 19 and 20, 1873 (our first two
days at the South Foreland), these were my words to the Elder Brethren
of the Trinity House:

“In view of the reported performance of horns and whistles in other
places, the question arises whether those mounted at the South
Foreland, and to which the foregoing remarks refer, are of the best
possible description.... I think our first duty is to make ourselves
acquainted with the best instruments hitherto made, no matter where
made; and then, if home genius can transcend them, to give it all
encouragement. Great and unnecessary expense may be incurred, through
our not availing ourselves of the results of existing experience.

“I have always sympathized, and I shall always sympathize, with the
desire of the Elder Brethren to encourage the inventor who first
made the magneto-electric light available for lighthouse purposes. I
regard his aid and counsel as, in many respects, invaluable to the
corporation. But, however original he may be, our duty is to demand
that his genius shall be expended in making advances on that which
has been already achieved elsewhere. If the whistles and horns that
we heard on the 19th and 20th be the very best hitherto constructed,
my views have been already complied with; but if they be not—and I
am strongly inclined to think that they are not—then I would submit
that it behooves us to have the best, and to aim at making the South
Foreland, both as regards light and sound, a station not excelled by
any other in the world.”

On this score it gives me pleasure to say that I never had a difficulty
with the Elder Brethren. They agreed with me; and two powerful
steam-whistles, the one from Canada, the other from the United States,
together with a steam-siren—also an American instrument—were in due
time mounted at the South Foreland. It will be seen in Chapter VII.
that my strongest recommendation applies to an instrument for which we
are indebted to the United States.

In presence of these facts, it will hardly be assumed that I wish
to withhold from the Lighthouse Board of Washington any credit that
they may fairly claim. My desire is to be strictly just; and this
desire compels me to express the opinion that their Report fails to
establish the inordinate claim made in its first paragraph. It contains
observations, but contradictory observations; while as regards the
establishment of any principle which should reconcile the conflicting
results, it leaves our condition unimproved.

But I willingly turn aside from the discussion of “claims” to the
discussion of science. Inserted, as a kind of intrusive element,
into the Report of Prof. Henry, is a second Report by General Duane,
founded on an extensive series of observations made by him in 1870 and
1871. After stating with distinctness the points requiring decision,
the General makes the following remarks:

“Before giving the results of these experiments, some facts will be
stated which will explain the difficulties of determining the power of
a fog-signal.

“There are six steam fog-whistles on the coast of Maine: these have
been frequently heard at a distance of twenty miles, and as frequently
cannot be heard at the distance of two miles, and this with no
perceptible difference in the state of the atmosphere.

“The signal is often heard at a great distance in one direction, while
in another it will be scarcely audible at the distance of a mile. This
is not the effect of wind, as the signal is frequently heard much
further against the wind than with it.[2] For example, the whistle on
Cape Elizabeth can always be distinctly heard in Portland, a distance
of nine miles, during a heavy northeast snowstorm, the wind blowing a
gale directly from Portland toward the whistle.[3]

“The most perplexing difficulties, however, arise from the fact that
the signal often appears to be surrounded by a belt, varying in radius
from one mile to one mile and a half, from which the sound appears to
be entirely absent. Thus, in moving directly from a station the sound
is audible for the distance of a mile, is then lost for about the
same distance, after which it is again distinctly heard for a long
time. This action is common to all ear-signals, and has been at times
observed at all the stations, at one of which the signal is situated on
a bare rock twenty miles from the mainland, with no surrounding objects
to affect the sound.”

It is not necessary to assume here the existence of a “belt,” at some
distance from the station. The passage of an acoustic cloud over the
station itself would produce the observed phenomenon.

Passing over the record of many other valuable observations in the
Report of General Duane, I come to a few very important remarks which
have a direct bearing upon the present question:

“From an attentive observation,” writes the General, “during three
years, of the fog-signals on this coast, and from the reports received
from the captains and pilots of coasting vessels, I am convinced that,
in some conditions of the atmosphere, the most powerful signals will be
at times unreliable.[4]

“Now it frequently occurs that a signal which, under ordinary
circumstances, would be audible at the distance of fifteen miles,
cannot be heard from a vessel at the distance of a single mile. This is
probably due to the reflection mentioned by Humboldt.

“The temperature of the air over the land where the fog-signal is
located being very different from that over the sea, the sound, in
passing from the former to the latter, undergoes reflection at their
surface of contact. The correctness of this view is rendered more
probable by the fact that, when the sound is thus impeded in the
direction of the sea, it has been observed to be much stronger inland.

“Experiments and observation lead to the conclusion that these
anomalies in the penetration and direction of sound from fog-signals
are to be attributed mainly to the want of uniformity in the
surrounding atmosphere, and that snow, rain, and fog, and the direction
of the wind, have much less influence than has been generally supposed.”

The Report of General Duane is marked throughout by fidelity to facts,
rare sagacity, and soberness of speculation. The last three of the
paragraphs just quoted exhibit, in my opinion, the only approach to a
true explanation of the phenomena which the Washington Report reveals.
At this point, however, the eminent Chairman of the Lighthouse Board
strikes in with the following criticism:

“In the foregoing I differ entirely in opinion from General Duane as
to the cause of extinction of powerful sounds being due to the unequal
density of the atmosphere. The velocity of sound is not at all affected
by barometric pressure; but if the difference in pressure is caused by
a difference in heat, or by the expansive power of vapor mingled with
the air, a slight degree of obstruction of sound may be observed. But
this effect we think is entirely too minute to produce the results
noted by General Duane and Dr. Tyndall, while we shall find in the
action of currents above and below a true and efficient cause.”

I have already cited the remarkable observation of General Duane,
that with a snowstorm from the northeast blowing against the sound,
the signal at Cape Elizabeth is always heard at Portland, a distance
of nine miles. The observations at the South Foreland, where the
sound has-been proved to reach a distance of more than twelve miles
against the wind, backed by decisive experiments, reduce to certainty
the surmises of General Duane. It has, for example, been proved that
a couple of gas-flames placed in a chamber can, in a minute or two,
render its air so non-homogeneous as to cut a sound practically off;
while the same sound passes without sensible impediment through showers
of paper-scraps, seeds, bran, raindrops, and through fumes and fogs of
the densest description. The sound also passes through thick layers of
calico, silk, serge, flannel, baize, close felt, and through pads of
cotton-net impervious to the strongest light.

As long, indeed, as the air on which snow, hail, rain or fog is
suspended is homogeneous, so long will sound pass through the air,
sensibly heedless of the suspended matter.[5] This point is illustrated
upon a large scale by my own observations on the Mer de Glace, and by
those of General Duane, at Portland, which prove the snow-laden air
from the northeast to be a highly homogeneous medium. Prof. Henry thus
accounts for the fact that the northeast snow-wind renders the sound
of Cape Elizabeth audible at Portland: In the higher regions of the
atmosphere he places an ideal wind, blowing in a direction opposed to
the real one, which _always_ accompanies the latter, and which more
than neutralizes its action. In speculating thus he bases himself on
the reasoning of Prof. Stokes, according to which a sound-wave moving
against the wind is tilted upward. The upper, and opposing wind, is
invented for the purpose of tilting again the already lifted sound-wave
downward. Prof. Henry does not explain how the sound-wave recrosses
the hostile lower current, nor does he give any definite notion of the
conditions under which it can be shown that it will reach the observer.

This, so far as I know, is the only theoretic gleam cast by the
Washington Report on the conflicting results which have hitherto
rendered experiments on fog-signals so bewildering. I fear it is an
_ignis fatuus_, instead of a safe guiding light. Prof. Henry, however,
boldly applies the hypothesis in a variety of instances. But he dwells
with particular emphasis upon a case of non-reciprocity which he
considers absolutely fatal to my views regarding the flocculence of
the atmosphere. The observation was made on board the steamer “City
of Richmond,” during a thick fog in a night of 1872. “The vessel was
approaching Whitehead from the southwestward, when, at a distance of
about six miles from the station, the fog-signal, which is a 10-inch
steam-whistle, was distinctly perceived, and continued to be heard with
increasing intensity of sound until within about three miles, when
the sound suddenly ceased to be heard, and was not perceived again
until the vessel approached within a quarter of a mile of the station,
although from conclusive evidence, furnished by the keeper, it was
shown that the signal had been sounding during the whole time.”

But while the 10-inch shore-signal thus failed to make itself heard
at sea, a 6-inch whistle on board the steamer made itself heard on
shore. Prof. Henry thus turns this fact against me. “It is evident,”
he writes, “that this result could not be due to any mottled condition
or want of acoustic transparency in the atmosphere, since this would
absorb the sound equally in both directions.” Had the observation been
made in a still atmosphere, this argument would, at one time, have had
great force. But the atmosphere was not still, and a sufficient reason
for the observed non-reciprocity is to be found in the recorded fact
that the wind was blowing against the shore-signal, and in favor of the
ship-signal.

But the argument of Prof. Henry, on which he places his main reliance,
would be untenable, even had the air been still. By the very aërial
reflection which he practically ignores, reciprocity may be destroyed
in a calm atmosphere. In proof of this assertion I would refer him
to a short paper on “Acoustic Reversibility,” printed at the end of
this volume.[6] The most remarkable case of non-reciprocity on record,
and which, prior to the demonstration of the existence and power of
acoustic clouds, remained an insoluble enigma, is there shown to be
capable of satisfactory solution. These clouds explain perfectly the
“abnormal phenomena” of Prof. Henry. Aware of their existence, the
falling off and subsequent recovery of a signal-sound, as noticed by
him and General Duane, is no more a mystery than the interception of
the solar light by a common cloud, and its restoration after the cloud
has moved or melted away.

The clew to all the difficulties and anomalies of this question is to
be found in the aërial echoes, the significance of which has been
overlooked by General Duane, and misinterpreted by Prof. Henry. And
here a word might be said with regard to the injurious influence still
exercised by authority in science. The affirmations of the highest
authorities, that from clear air no sensible echo ever comes, were
so distinct that my mind for a time refused to entertain the idea.
Authority caused me for weeks to depart from the truth, and to seek
counsel among delusions. On the day our observations at the South
Foreland began I heard the echoes. They perplexed me. I heard them
again and again, and listened to the explanations offered by some
ingenious persons at the Foreland. They were an “ocean-echo”: this
is the very phraseology now used by Prof. Henry. They were echoes
“from the crests and slopes of the waves”: these are the words of the
hypothesis which he now espouses. Through a portion of the month of
May, through the whole of June, and through nearly the whole of July,
1873, I was occupied with these echoes; one of the phases of thought
then passed through, one of the solutions then weighed in the balance
and found wanting, being identical with that which Prof. Henry now
offers for acceptation.

But though it thus deflected me from the proper track, shall I say that
authority in science is injurious? Not without some qualification. It
is not only injurious, but deadly, when it cows the intellect into fear
of questioning it. But the authority which so merits our respect as to
compel us to test and overthrow all its supports, before accepting a
conclusion opposed to it, is not wholly noxious. On the contrary, the
disciplines it imposes may be in the highest degree salutary, though
they may end, as in the present case, in the ruin of authority. The
truth thus established is rendered firmer by our struggles to reach it.
I groped day after day, carrying this problem of aërial echoes in my
mind; to the weariness, I fear, of some of my colleagues who did not
know my object. The ships and boats afloat, the “slopes and crests of
the waves,” the visible clouds, the cliffs, the adjacent lighthouses,
the objects landward, were all in turn taken into account, and all in
turn rejected.

With regard to the particular notion which now finds favor with Prof.
Henry, it suggests the thought that his observations, notwithstanding
their apparent variety and extent, were really limited as regards the
weather. For did they, like ours, embrace weather of all kinds, it is
not likely that he would have ascribed to the sea-waves an action which
often reaches its maximum intensity when waves are entirely absent.
I will not multiply instances, but confine myself to the definite
statement that the echoes have often manifested an astonishing strength
when the sea was of glassy smoothness. On days when the echoes were
powerful, I have seen the southern cumuli mirrored in the waveless
ocean, in forms almost as definite as the clouds themselves. By no
possible application of the law of incidence and reflection could the
echoes from such a sea return to the shore; and if we accept for a
moment a statement which Prof. Henry seems to indorse, that sound-waves
of great intensity, when they impinge upon a solid or liquid surface,
do not obey the law of incidence and reflection, but “roll along the
surface like a cloud of smoke,” it only increases the difficulty. Such
a “cloud,” instead of returning to the coast of England, would, in our
case, have rolled toward the coast of France. Nothing that I could say
in addition could strengthen the case here presented. I will only add
one further remark. When the sun shines uniformly on a smooth sea, thus
producing a practically uniform distribution of the aërial currents to
which the echoes are due, the direction in which the trumpet-echoes
reach the shore is always that in which the axis of the instrument is
pointed. At Dungeness this was proved to be the case throughout an
arc of 210°—an impossible result, if the direction of reflection were
determined by that of the ocean waves.

Rightly interpreted and followed out, these aërial echoes lead to a
solution which penetrates and reconciles the phenomena from beginning
to end. On this point I would stake the issue of the whole inquiry,
and to this point I would, with special earnestness, direct the
attention of the Lighthouse Board of Washington. Let them prolong
their observations into calm weather: if their atmosphere resembles
ours—which I cannot doubt—then I affirm that they will infallibly
find the echoes strong on days when all thought of reflection “from
the crests and slopes of the waves” must be discarded. The echoes
afford the easiest access to the core of this question, and it is for
this reason that I dwell upon them thus emphatically. It requires
no refined skill or profound knowledge to master the conditions of
their production; and these once mastered, the Lighthouse Board of
Washington will find themselves in the real current of the phenomena,
outside of which—I say it with respect—they are now vainly speculating.
The acoustic deportment of the atmosphere in haze, fog, sleet, snow,
rain, and hail will be no longer a mystery; even those “abnormal
phenomena” which are now referred to an imaginary cause, or reserved
for future investigation, will be found to fall naturally into place,
as illustrations of a principle as simple as it is universal.


“With the instruments now at our disposal wisely established along
our coasts, I venture to think that the saving of property, in ten
years, will be an exceedingly large multiple of the outlay necessary
for the establishment of such signals. The saving of life appeals to
the higher motives of humanity.” Such were the words with which I
wound up my Report on Fog-Signals.[7] One year after their utterance,
the “Schiller” goes to pieces on the Scilly rocks. A single calamity
covers the predicted multiple, while the sea receives three hundred
and thirty-three victims. As regards the establishment of fog-signals,
energy has been hitherto paralyzed by their reputed uncertainty. We
now know both the reason and the range of their variations; and such
knowledge places it within our power to prevent disasters like the
recent one. The inefficiency of bells, which caused their exclusion
from our inquiry, was sadly illustrated in the case of the “Schiller.”

  JOHN TYNDALL.

  ROYAL INSTITUTION, _June, 1875_.




PREFACE TO THE FIRST EDITION


In the following pages I have tried to render the science of Acoustics
interesting to all intelligent persons, including those who do not
possess any special scientific culture.

The subject is treated experimentally throughout, and I have endeavored
so to place each experiment before the reader that he should realize it
as an actual operation. My desire, indeed, has been to give distinct
images of the various phenomena of acoustics, and to cause them to be
seen mentally in their true relations.

I have been indebted to the kindness of some of my English friends for
a more or less complete examination of the proof-sheets of this work.
To my celebrated German friend Clausius, who has given himself the
trouble of reading the proofs from beginning to end, my especial thanks
are due and tendered.

There is a growing desire for scientific culture throughout the
civilized world. The feeling is natural, and, under the circumstances,
inevitable. For a power which influences so mightily the intellectual
and material action of the age could not fail to arrest attention and
challenge examination. In our schools and universities a movement
in favor of science has begun which, no doubt, will end in the
recognition of its claims, both as a source of knowledge and as a
means of discipline. If by showing, however inadequately, the methods
and results of physical science to men of influence, who derive their
culture from another source, this book should indirectly aid in
promoting the movement referred to, it will not have been written in
vain.




SOUND




CHAPTER I

  The Nerves and Sensation—Production and Propagation of
  Sonorous Motion—Experiments on Sounding Bodies placed
  in Vacuo—Deadening of Sound by Hydrogen—Action of
  Hydrogen on the Voice—Propagation of Sound through Air
  of Varying Density—Reflection of Sound—Echoes—Refraction
  of Sound—Diffraction of Sound; Case of Erith Village and
  Church—Influence of Temperature on Velocity—Influence of
  Density on Elasticity—Newton’s Calculation of Velocity—Thermal
  Changes Produced by the Sonorous Wave—Laplace’s Correction
  of Newton’s Formula—Ratio of Specific Heats at Constant
  Pressure and at Constant Volume deduced from Velocities
  of Sound—Mechanical Equivalent of Heat deduced from this
  Ratio—Inference that Atmospheric Air Possesses no Sensible
  Power to Radiate Heat—Velocity of Sound in Different
  Gases—Velocity in Liquids and Solids—Influence of Molecular
  Structure on the Velocity of Sound


§ 1. _Introduction: Character of Sonorous Motion. Experimental
Illustrations_

The various nerves of the human body have their origin in the brain,
which is the seat of sensation. When the finger is wounded, the
sensor nerves convey to the brain intelligence of the injury, and if
these nerves be severed, however serious the hurt may be, no pain is
experienced. We have the strongest reason for believing that what
the nerves convey to the brain is in all cases _motion_. The motion
here meant is not, however, that of the nerve as a whole, but of its
molecules or smallest particles.[8]

Different nerves are appropriated to the transmission of different
kinds of molecular motion. The nerves of taste, for example, are not
competent to transmit the tremors of light, nor is the optic nerve
competent to transmit sonorous vibrations. For these a special nerve
is necessary, which passes from the brain into one of the cavities of
the ear, and there divides into a multitude of filaments. It is the
motion imparted to this, the _auditory nerve_, which, in the brain, is
translated into sound.

Applying a flame to a small collodion balloon which contains a mixture
of oxygen and hydrogen, the gases explode, and every ear in this
room is conscious of a shock, which we name a sound. How was this
shock transmitted from the balloon to our organs of hearing? Have the
exploding gases shot the air-particles against the auditory nerve as a
gun shoots a ball against a target? No doubt, in the neighborhood of
the balloon, there is to some extent a propulsion of particles; but no
particle of air from the vicinity of the balloon reached the ear of any
person here present. The process was this: When the flame touched the
mixed gases they combined chemically, and their union was accompanied
by the development of intense heat. The heated air expanded suddenly,
forcing the surrounding air violently away on all sides. This motion
of the air close to the balloon was rapidly imparted to that a little
further off, the air first set in motion coming at the same time to
rest. The air, at a little distance, passed its motion on to the air
at a greater distance, and came also in its turn to rest. Thus each
shell of air, if I may use the term, surrounding the balloon took up
the motion of the shell next preceding, and transmitted it to the next
succeeding shell, the motion being thus propagated as a _pulse_ or
_wave_ through the air.

The motion of the pulse must not be confounded with the motion of the
particles which at any moment constitute the pulse. For while the wave
moves forward through considerable distances, each particular particle
of air makes only a small excursion to and fro.

[Illustration: FIG. 1.]

The process may be rudely represented by the propagation of motion
through a row of glass balls, such as are employed in the game of
_solitaire_. Placing the balls along a groove thus, Fig. 1, each of
them touching its neighbor, and urging one of them against the end of
the row: the motion thus imparted to the first ball is delivered up to
the second, the motion of the second is delivered up to the third, the
motion of the third is imparted to the fourth; each ball, after having
given up its motion, returning itself to rest. The last ball only of
the row flies away. In a similar way is sound conveyed from particle
to particle through the air. The particles which fill the cavity of
the ear are finally driven against _the tympanic membrane_, which is
stretched across the passage leading from the external ear toward the
brain. This membrane, which closes outwardly the “drum” of the ear, is
thrown into vibration, its motion is transmitted to the ends of the
auditory nerve, and afterward along that nerve to the brain, where the
vibrations are translated into sound. How it is that the motion of the
nervous matter can thus excite the consciousness of sound is a mystery
which the human mind cannot fathom.

[Illustration: FIG. 2.]

The propagation of sound may be illustrated by another homely but
useful illustration. I have here five young assistants, A, B, C, D,
and E, Fig. 2, placed in a row, one behind the other, each boy’s hands
resting against the back of the boy in front of him. E is now foremost,
and A finishes the row behind. I suddenly push A, A pushes B, and
regains his upright position; B pushes C; C pushes D; D pushes E; each
boy, after the transmission of the push, becoming himself erect. E,
having nobody in front, is thrown forward. Had he been standing on the
edge of a precipice, he would have fallen over; had he stood in contact
with a window, he would have broken the glass; had he been close to a
drumhead, he would have shaken the drum. “We could thus transmit a
push through a row of a hundred boys, each particular boy, however,
only swaying to and fro. Thus, also, we send sound through the air,
and shake the drum of a distant ear, while each particular particle of
the air concerned in the transmission of the pulse makes only a small
oscillation.

But we have not yet extracted from our row of boys all that they can
teach us. When A is pushed he may yield languidly, and thus tardily
deliver up the motion to his neighbor B. B may do the same to C, C
to D, and D to E. In this way the motion might be transmitted with
comparative slowness along the line. But A, when pushed, may, by a
sharp muscular effort and sudden recoil, deliver up promptly his motion
to B, and come himself to rest; B may do the same to C, C to D, and
D to E, the motion being thus transmitted rapidly along the line.
Now this sharp muscular effort and sudden recoil is analogous to the
_elasticity_ of the air in the case of sound. In a wave of sound, a
lamina of air, when urged against its neighbor lamina, delivers up its
motion and recoils, in virtue of the elastic force exerted between
them; and the more rapid this delivery and recoil, or in other words
the greater the elasticity of the air, the greater is the velocity of
the sound.

[Illustration: FIG. 3.]

A very instructive mode of illustrating the transmission of a
sound-pulse is furnished by the apparatus represented in Fig. 3,
devised by my assistant, Mr. Cottrell. It consists of a series of
wooden balls separated from each other by spiral springs. On striking
the knob A, a rod attached to it impinges upon the first ball B, which
transmits its motion to C, thence it passes to E, and so on through
the entire series. The arrival at D is announced by the shock of the
terminal ball against the wood, or, if we wish, by the ringing of a
bell. Here the elasticity of the air is represented by that of the
springs. The pulse may be rendered slow enough to be followed by the
eye.

Scientific education ought to teach us to see the invisible as well
as the visible in nature, to picture with the vision of the mind
those operations which entirely elude bodily vision; to look at the
very atoms of matter in motion and at rest, and to follow them forth,
without ever once losing sight of them, into the world of the senses,
and see them there integrating themselves in natural phenomena. With
regard to the point now under consideration, we must endeavor to form
a definite image of a wave of sound. We ought to see mentally the
air-particles, when urged outward by the explosion of our balloon,
crowding closely together; but immediately behind this condensation we
ought to see the particles separated more widely apart. We must, in
short, to be able to seize the conception that a sonorous wave consists
of two portions, in the one of which the air is more dense, and in
the other of which it is less dense than usual. A condensation and a
rarefaction, then, are the two constituents of a wave of sound. This
conception shall be rendered more complete in our next lecture.


§ 2. _Experiments in Vacuo, in Hydrogen, and on Mountains_

[Illustration: FIG. 4.]

That air is thus necessary to the propagation of sound was proved by a
celebrated experiment made before the Royal Society, by a philosopher
named Hawksbee, in 1705.[9] He so fixed a bell within the receiver
of an air-pump that he could ring the bell when the receiver was
exhausted. Before the air was withdrawn the sound of the bell was heard
within the receiver; after the air was withdrawn the sound became
so faint as to be hardly perceptible. An arrangement is before you
which enables us to repeat in a very perfect manner the experiment of
Hawksbee. Within this jar, G G′, Fig. 4, resting on the plate of an
air-pump is a bell, B, associated with clockwork.[10] After the jar
has been exhausted as perfectly as possible, I release, by means of a
rod, _r r′_, which passes air-tight through the top of the vessel, the
detent which holds the hammer. It strikes, and you see it striking,
but only those close to the bell can hear the sound. Hydrogen gas,
which you know is fourteen times lighter than air, is now allowed
to enter the vessel. The sound of the bell is not augmented by the
presence of this attenuated gas, though the receiver is now full of
it. By working the pump, the atmosphere round the bell is rendered
still more attenuated. In this way we obtain a vacuum more perfect than
that of Hawksbee, and this is important, for it is the last traces of
air that are chiefly effective in this experiment. You now see the
hammer pounding the bell, but you hear no sound. Even when the ear is
placed against the exhausted receiver not the faintest tinkle is heard.
Observe also that the bell is suspended by strings, for if it were
allowed to rest upon the plate of the air-pump the vibrations would be
communicated to the plate, and thence transmitted to the air outside.
Permitting the air to enter the jar with as little noise as possible,
you immediately hear a feeble sound, which grows louder as the air
becomes more dense, until finally every person in this large assembly
distinctly hears the ringing of the bell.[11]

Sir John Leslie found hydrogen singularly incompetent to act as the
vehicle of the sound of a bell rung in the gas. More than this, he
emptied a receiver like that before you of half its air, and plainly
heard the ringing of the bell. On permitting hydrogen to enter the
half-filled receiver until it was wholly filled, the sound sank
until it was scarcely audible. This result remained an enigma until
it received a simple and satisfactory explanation at the hands of
Prof. Stokes. When a common pendulum oscillates it tends to form a
condensation in front and a rarefaction behind. But it is only _a
tendency_; the motion is so slow, and the air is so elastic, that
it moves away in front before it is sensibly condensed, and fills
the space behind before it can become sensibly dilated. Hence waves
or pulses are not generated by the pendulum. It requires a certain
sharpness of shock to produce the condensation and rarefaction which
constitute a wave of sound in air.

The more elastic and mobile the gas, the more able will it be to move
away in front and to fill the space behind, and thus to oppose the
formation of rarefactions and condensations by a vibrating body. Now
hydrogen is much more mobile than air; and hence the production of
sonorous waves in it is attended with greater difficulty than in air. A
rate of vibration quite competent to produce sound-waves in the one may
be wholly incompetent to produce them in the other. Both calculation
and observation prove the correctness of this explanation, to which we
shall again refer.

At great elevations in the atmosphere sound is sensibly diminished in
loudness. De Saussure thought the explosion of a pistol at the summit
of Mont Blanc to be about equal to that of a common cracker below.
I have several times repeated this experiment; first, in default of
anything better, with a little tin cannon, the torn remnants of which
are now before you, and afterward with pistols. What struck me was the
absence of that density and sharpness in the sound which characterize
it at lower elevations. The pistol-shot resembled the explosion of a
champagne bottle, but it was still loud. The withdrawal of half an
atmosphere does not very materially affect our ringing bell, and air
of the density found at the top of Mont Blanc is still capable of
powerfully affecting the auditory nerve. That highly attenuated air
is able to convey sound of great intensity is forcibly illustrated
by the explosion of meteorites at elevations where the tenuity of
the atmosphere must be almost infinite. Here, however, the initial
disturbance must be exceedingly great.

The motion of sound, like all other motion, is enfeebled by its
transference from a light body to a heavy one. When the receiver
which has hitherto covered our bell is removed you hear how much
more loudly it rings in the open air. When the bell was covered the
aërial vibrations were first communicated to the heavy glass jar,
and afterward by the jar to the air outside; a great diminution of
intensity being the consequence. The action of hydrogen gas upon the
voice is an illustration of the same kind. The voice is formed by
urging air from the lungs through an organ called the larynx, where
it is thrown into vibration by the _vocal chords_ which thus generate
sound. But when the lungs are filled with hydrogen, the vocal chords
on speaking produce a vibratory motion in the hydrogen, which then
transfers the motion to the outer air. By this transference from a
light gas to a heavy one the voice is so weakened as to become a mere
squeak.[12]

The intensity of a sound depends on the density of the air in which
the sound is generated, and not on that of the air in which it is
heard.[13] Supposing the summit of Mont Blanc to be equally distant
from the top of the Aiguille Verte and the bridge at Chamouni; and
supposing two observers stationed, the one upon the bridge and the
other upon the Aiguille: the report of a cannon fired on Mont Blanc
would reach both observers with the same intensity, though in the one
case the sound would pursue its way through the rare air above, while
in the other it would descend though the denser air below. Again, let a
straight line equal to that from the bridge at Chamouni to the summit
of Mont Blanc be measured along the earth’s surface in the valley of
Chamouni, and let two observers be stationed, the one on the summit and
the other at the end of the line: the report of a cannon fired on the
bridge would reach both observers with the same intensity, though in
the one case the sound would be propagated through the dense air of the
valley, and in the other case would ascend through the rarer air of the
mountain. Finally, charge two cannon equally, and fire one of them at
Chamouni and the other at the top of Mont Blanc: the one fired in the
heavy air below may be heard above, while the one fired in the light
air above is unheard below.


§ 3. _Intensity of Sound. Law of Inverse Squares_

In the case of our exploding balloon the wave of sound expands on all
sides, the motion produced by the explosion being thus diffused over a
continually augmenting mass of air. It is perfectly manifest that this
cannot occur without an enfeeblement of the motion. Take the case of a
thin shell of air with a radius of one foot, reckoned from the centre
of explosion. A shell of air of the same thickness, but of two feet
radius, will contain four times the quantity of matter; if its radius
be three feet, it will contain nine times the quantity of matter; if
four feet, it will contain sixteen times the quantity of matter, and so
on. Thus the quantity of matter set in motion _augments_ as the square
of the distance from the centre of explosion. The intensity or loudness
of sound _diminishes_ in the same proportion. We express this law by
saying that the intensity of the sound _varies inversely as the square
of the distance_.

Let us look at the matter in another light. The mechanical effect of a
ball striking a target depends on two things—the weight of the ball,
and the velocity with which it moves. The effect is proportional to the
weight simply; but it is proportional to the square of the velocity.
The proof of this is easy, but it belongs to ordinary mechanics rather
than to our present subject. Now what is true of the cannon-ball
striking a target is also true of an air-particle striking the tympanum
of the ear. Fix your attention upon a particle of air as the sound-wave
passes over it; it is urged from its position of rest toward a neighbor
particle, first with an accelerated motion, and then with a retarded
one. The force which first urges it is opposed by the resistance of
the air, which finally stops the particle and causes it to recoil. At
a certain point of its excursion the velocity of the particle is its
maximum. _The intensity of the sound is proportional to the square of
this maximum velocity._

The distance through which the air-particle moves to and fro, when the
sound-wave passes it, is called the _amplitude_ of the vibration. The
intensity of the sound is proportional to the square of the amplitude.


§ 4. _Confinement of Sound-waves in Tubes_

This weakening of the sound, according to the law of inverse squares,
would not take place if the sound-wave were so confined as to prevent
its lateral diffusion. By sending it through a tube with a smooth
interior surface we accomplish this, and the wave thus confined
may be transmitted to great distances with very little diminution
of intensity. Into one end of this tin tube, fifteen feet long, I
whisper in a manner quite inaudible to the people nearest to me,
but a listener at the other end hears me distinctly. If a watch be
placed at one end of the tube, a person at the other end hears the
ticks, though nobody else does. At the distant end of the tube is
now placed a lighted candle, _c_, Fig. 5. When the hands are clapped
at this end, the flame instantly ducks down at the other. It is not
quite extinguished, but it is forcibly depressed. When two books, B
B′, Fig. 5, are clapped together, the candle is blown out.[14] You may
here observe, in a rough way, the speed with which the sound-wave is
propagated. The instant the clap is heard the flame is extinguished. I
do not say that the time required by the sound to travel this tube is
immeasurably short, but simply that the interval is too short for your
senses to appreciate it.

[Illustration: FIG. 5.]

That it is a _pulse_ and not a _puff_ of air is proved by filling one
end of the tube with the smoke of brown paper. On clapping the books
together no trace of this smoke is ejected from the other end. The
pulse has passed through both smoke and air without carrying either of
them along with it.

An effective mode of throwing the propagation of a pulse through air
has been devised by my assistant. The two ends of a tin tube fifteen
feet long are stopped by sheet India-rubber stretched across them.
At one end, _e_, a hammer with a spring handle rests against the
India-rubber; at the other end is an arrangement for the striking
of a bell, _c_. Drawing back the hammer _e_ to a distance measured
on the graduated circle and liberating it, the generated pulse is
propagated through the tube, strikes the other end, drives away the
cork termination _a_ of the lever _a b_, and causes the hammer _b_ to
strike the bell. The rapidity of propagation is well illustrated here.
When hydrogen (sent through the India-rubber tube H) is substituted for
air the bell does not ring.

[Illustration: FIG. 6.]

The celebrated French philosopher, Biot, observed the transmission
of sound through the empty water-pipes of Paris, and found that he
could hold a conversation in a low voice through an iron tube 3,120
feet in length. The lowest possible whisper, indeed, could be heard at
this distance, while the firing of a pistol into one end of the tube
quenched a lighted candle at the other.

§ 5. _The Reflection of Sound. Resemblances to Light_

The action of sound thus illustrated is exactly the same as that of
light and radiant heat. They, like sound, are wave-motions. Like
sound they diffuse themselves in space, diminishing in intensity
according to the same law. Like sound also, light and radiant heat,
when sent through a tube with a reflecting interior surface, may be
conveyed to great distances with comparatively little loss. In fact,
every experiment on the reflection of light has its analogy in the
reflection of sound. On yonder gallery stands an electric lamp, placed
close to the clock of this lecture-room. An assistant in the gallery
ignites the lamp, and directs its powerful beam upon a mirror placed
here behind the lecture-table. By the act of reflection the divergent
beam is converted into this splendid luminous cone traced out upon
the dust of the room. The point of convergence being marked and the
lamp extinguished, I place my ear at that point. Here every sound-wave
sent forth by the clock and reflected by the mirror is gathered up,
and the ticks are heard as if they came, not from the clock, but from
the mirror. Let us stop the clock, and place a watch _w_, Fig. 7, at
the place occupied a moment ago by the electric light. At this great
distance the ticking of the watch is distinctly heard. The hearing is
much aided by introducing the end _f_ of a glass funnel into the ear,
the funnel here acting the part of an ear-trumpet. We know, moreover,
that in optics the positions of a body and of its image are reversible.
When a candle is placed at this lower focus, you see its image on the
gallery above, and I have only to turn the mirror on its stand to make
the image of the flame fall upon any one of the row of persons who
occupy the front seat in the gallery. Removing the candle, and putting
the watch, _w_, Fig. 8, in its place, the person on whom the light
falls distinctly hears the sound. When the ear is assisted by the glass
funnel, the reflected ticks of the clock in our first experiment are
so powerful as to suggest the idea of something pounding against the
tympanum, while the direct ticks are scarcely if at all, heard.

[Illustration: FIG. 7.]

[Illustration: FIG. 8.]

One of these two parabolic mirrors, _n n′_, Fig. 9, is placed upon
the table, the other, _m m′_, being drawn up to the ceiling of this
theatre; they are five-and-twenty feet apart. When the carbon-points
of the electric light are placed in the focus _a_ of the lower mirror
and ignited, a fine luminous cylinder rises like a pillar to the upper
mirror, which brings the parallel beam to a focus. At that focus
is seen a spot of sunlike brilliancy, due to the reflection of the
light from the surface of a watch, _w_, there suspended. The watch
is ticking, but in my present position I do not hear it. At this
lower focus, _a_, however, we have the energy of every sonorous wave
converged. Placing the ear at _a_, the ticking is as audible as if the
watch were at hand; the sound, as in the former case, appearing to
proceed, not from the watch itself, but from the lower mirror.[15]

[Illustration: FIG. 9.]

Curved roofs and ceilings and bellying sails act as mirrors upon
sound. In our old laboratory, for example, the singing of a kettle
seemed, in certain positions, to come, not from the fire on which it
was placed, but from the ceiling. Inconvenient secrets have been thus
revealed, an instance of which has been cited by Sir John Herschel.[16]
In one of the cathedrals in Sicily the confessional was so placed that
the whispers of the penitents were reflected by the curved roof, and
brought to a focus at a distant part of the edifice. The focus was
discovered by accident, and for some time the person who discovered
it took pleasure in hearing, and in bringing his friends to hear,
utterances intended for the priest alone. One day, it is said, his own
wife occupied the penitential stool, and both he and his friends were
thus made acquainted with secrets which were the reverse of amusing to
one of the party.

When a sufficient interval exists between a direct and a reflected
sound, we hear the latter as an _echo_.

Sound, like light, may be reflected several times in succession, and,
as the reflected light under these circumstances becomes gradually
feebler to the eye, so the successive echoes become gradually feebler
to the ear. In mountain regions this repetition and decay of sound
produce wonderful and pleasing effects. Visitors to Killarney will
remember the fine echo in the Gap of Dunloe. When a trumpet is sounded
in the proper place in the Gap, the sonorous waves reach the ear in
succession after one, two, three, or more reflections from the adjacent
cliffs, and thus die away in the sweetest cadences. There is a deep
_cul-de-sac_, called the Ochsenthal, formed by the great cliffs of the
Engelhörner, near Rosenlaui, in Switzerland, where the echoes warble in
a wonderful manner.

The sound of the Alpine horn, echoed from the rocks of the Wetterhorn
or the Jungfrau, is in the first instance heard roughly. But by
successive reflections the notes are rendered more soft and flute-like,
the gradual diminution of intensity giving the impression that the
source of sound is retreating further and further into the solitudes of
ice and snow. The repetition of echoes is also in part due to the fact
that the reflecting surfaces are at different distances from the hearer.

In large, unfurnished rooms the mixture of direct and reflected sound
sometimes produces very curious effects. Standing, for example, in the
gallery of the Bourse at Paris, you hear the confused vociferation of
the excited multitude below. You see all the motions—of their lips as
well as of their hands and arms. You know they are speaking—often,
indeed, with vehemence—but what they say you know not. The voices mix
with their echoes into a chaos of noise, out of which no intelligible
utterance can emerge. The echoes of a room are materially damped by its
furniture. The presence of an audience may also render intelligible
speech possible where, without an audience, the definition of the
direct voice is destroyed by its echoes. On the 16th of May, 1865,
having to lecture in the Senate House of the University of Cambridge,
I first made some experiments as to the loudness of voice necessary
to fill the room, and was dismayed to find that a friend, placed at a
distant part of the hall, could not follow me because of the echoes.
The assembled audience, however, so quenched the sonorous waves that
the echoes were practically absent, and my voice was plainly heard in
all parts of the Senate House.

Sounds are also said to be reflected from the clouds. Arago reports
that, when the sky is clear, the report of a cannon on an open plain
is short and sharp, while a cloud is sufficient to produce an echo
like the rolling of distant thunder. The subject of aërial echoes will
be subsequently treated at length, when it will be shown that Arago’s
conclusion requires correction.

Sir John Herschel, in his excellent article “Sound,” In the
“Encyclopædia Metropolitana,” has collected with others the following
instances of echoes. An echo in Woodstock Park repeats seventeen
syllables by day and twenty by night; one, on the banks of the Lago del
Lupo, above the fall of Terni, repeats fifteen. The tick of a watch may
be heard from one end of the abbey church of St. Albans to the other.
In Gloucester Cathedral, a gallery of an octagonal form conveys a
whisper seventy-five feet across the nave. In the whispering-gallery of
St. Paul’s, the faintest sound is conveyed from one side to the other
of the dome, but is not heard at any intermediate point. At Carisbrook
Castle, in the Isle of Wight, is a well two hundred and ten feet deep
and twelve wide. The interior is lined by smooth masonry; when a pin
is dropped into the well it is distinctly heard to strike the water.
Shouting or coughing into this well produces a resonant ring of some
duration.[17]


§ 6. _Refraction of Sound_

[Illustration: FIG. 10.]

Another important analogy between sound and light has been established
by M. Sondhauss.[18] When a large lens is placed in front of our
lamp, the lens compels the rays of light that fall upon it to deviate
from their direct and divergent course, and to form a convergent cone
behind it. This refraction of the luminous beam is a consequence of
the retardation suffered by the light in passing through the glass.
Sound may be similarly refracted by causing it to pass through a lens
which retards its motion. Such a lens is formed when we fill a thin
balloon with some gas heavier than air. A collodion balloon, B, Fig.
10, filled with carbonic-acid gas, the envelope being so thin as to
yield readily to the pulses which strike against it, answers the
purpose.[19] A watch, _w_, is hung up close to the lens, beyond which,
and at a distance of four or five feet from the lens, is placed the
ear, assisted by the glass funnel _f f′_. By moving the head about, a
position is soon discovered in which the ticking is particularly loud.
This, in fact, is the focus of the lens. If the ear be moved from this
focus the intensity of the sound falls; if, when the ear is at the
focus, the balloon be removed, the ticks are enfeebled; on replacing
the balloon their force is restored. The lens, in fact, enables us to
hear the ticks distinctly when they are perfectly inaudible to the
unaided ear.

[Illustration: FIG. 11.]

How a sound-wave is thus converged may be comprehended by reference to
Fig. 11. Let _m o n o″_ be a section of the sound-lens, and _a b_ a
portion of a sonorous wave approaching it from a distance. The middle
point, _o_, of the wave first touches the lens, and is first retarded
by it. By the time the ends _a_ and _b_, still moving through air,
reach the balloon, the middle point _o_, pursuing its way through the
heavier gas within, will have only reached _o′_. The wave is therefore
broken at _o_; and the direction of motion being at right angles to the
face of the wave, the two halves will encroach upon each other. This
convergence of the two halves of the wave is augmented on quitting the
lens. For when _o′_ has reached _o″_, the two ends _a_ and _b_ will
have pushed forward to a greater distance, say to _a′_ and _b′_. Soon
afterward the two halves of the wave will cross each other, or in other
words come to a focus, the air at the focus being agitated by the sum
of the motions of the two waves.[20]


§ 7. _Diffraction of Sound: illustrations offered by great Explosions_

When a long sea-roller meets an isolated rock in its passage, it rises
against the rock and embraces it all round. Facts of this nature caused
Newton to reject the undulatory theory of light. He contended that if
light were a product of wave-motion we could have no shadows, because
the waves of light would propagate themselves round opaque bodies as
a wave of water round a rock. It has been proved since his time that
the waves of light do bend round opaque bodies; but with that we have
nothing now to do. A sound-wave certainly bends thus round an obstacle,
though as it diffuses itself in the air at the back of the obstacle it
is enfeebled in power, the obstacle thus producing a partial _shadow_
of the sound. A railway train passing through cuttings and long
embankments exhibits great variations in the intensity of the sound.
The interposition of a hill in the Alps suffices to diminish materially
the sound of a cataract; it is able sensibly to extinguish the tinkle
of the cowbells. Still the sound-shadow is but partial, and the marker
at the rifle-butts never fails to hear the explosion, though he is well
protected from the ball. A striking example of this _diffraction_ of a
sonorous wave was exhibited at Erith after the tremendous explosion of
a powder magazine which occurred there in 1864. The village of Erith
was some miles distant from the magazine, but in nearly all cases the
windows were shattered; and it was noticeable that the windows turned
away from the origin of the explosion suffered almost as much as those
which faced it. Lead sashes were employed in Erith Church, and these,
being in some degree flexible, enabled the windows to yield to pressure
without much fracture of the glass. As the sound-wave reached the
church it separated right and left, and, for a moment, the edifice was
clasped by a girdle of intensely compressed air, every window in the
church, front and back, being bent _inward_. After compression, the air
within the church no doubt dilated, tending to restore the windows to
their first condition. The bending in of the windows, however, produced
but a small condensation of the whole mass of air within the church;
the recoil was therefore feeble in comparison with the pressure, and
insufficient to undo what the latter had accomplished.


§ 8. _Velocity of Sound: relation to Density and Elasticity of Air_

Two conditions determine the velocity of propagation of a sonorous
wave; namely, the elasticity and the density of the medium through
which the wave passes. The elasticity of air is measured by the
pressure which it sustains or can hold in equilibrium. At the sea-level
this pressure is equal to that of a stratum of mercury about thirty
inches high. At the summit of Mont Blanc the barometric column is not
much more than half this height; and, consequently, the elasticity of
the air upon the summit of the mountain is not much more than half what
it is at the sea-level.

If we could augment the elasticity of air, without at the same time
augmenting its density, we should augment the velocity of sound. Or,
if allowing the elasticity to remain constant we could diminish the
density, we should augment the velocity. Now, air in a closed vessel,
where it cannot expand, has its elasticity augmented by heat, while
its density remains unchanged. Through such heated air sound travels
more rapidly than through cold air. Again, air free to expand has its
density lessened by warming, its elasticity remaining the same, and
through such air sound travels more rapidly than through cold air. This
is the case with our atmosphere when heated by the sun.

The velocity of sound in air, _at the freezing temperature_, is 1,090
feet a second.

At all lower temperatures the velocity is less than this, and at all
higher temperatures it is greater. The late M. Wertheim has determined
the velocity of sound in air of different temperatures, and here are
some of his results:

  Temperature of air      Velocity of sound

   0·5° centigrade        1,089 feet
   2·10    ”              1,091  ”
   8·5     ”              1,109  ”
  12·0     ”              1,113  ”
  26·6     ”              1,140  ”

At a temperature of half a degree above the freezing-point of water the
velocity is 1,089 feet a second; at a temperature of 26·6 degrees, it
is 1,140 feet a second, or a difference of 51 feet for 26 degrees; that
is to say, an augmentation of velocity of nearly two feet for every
single degree centigrade.

With the same elasticity the density of hydrogen gas is much less than
that of air, and the consequence is that the velocity of sound in
hydrogen far exceeds its velocity in air. The reverse holds good for
heavy carbonic-acid gas. If density and elasticity vary in the same
proportion, as the law of Boyle and Mariotte proves them to do in
air when the temperature is preserved constant, they neutralize each
other’s effects; hence, if the temperature were the same, the velocity
of sound upon the summits of the highest Alps would be the same as that
at the mouth of the Thames. But, inasmuch as the air above is colder
than that below, the actual velocity on the summits of the mountains
is less than that at the sea-level. To express this result in stricter
language, the velocity is _directly_ proportional to the square root
of the elasticity of the air; it is also _inversely_ proportional to
the square root of the density of the air. Consequently, as in air
of a constant temperature elasticity and density vary in the same
proportion, and act oppositely, the velocity of sound is not affected
by a change of density, if unaccompanied by a change of temperature.

There is no mistake more common than to suppose the velocity of sound
to be augmented by density. The mistake has arisen from a misconception
of the fact that in solids and liquids the velocity is greater than in
gases. But it is the higher elasticity of those bodies, _in relation to
their density_, that causes sound to pass rapidly through them. Other
things remaining the same, an augmentation of density always produces a
diminution of velocity. Were the elasticity of water, which is measured
by its compressibility, only equal to that of air, the velocity of
sound in water, instead of being more than quadruple the velocity in
air, would be only a small fraction of that velocity. Both density and
elasticity, then, must be always borne in mind; the velocity of sound
being determined by neither taken separately, but by the relation of
the one to the other. The effect of small density and high elasticity
is exemplified in an astonishing manner by the luminiferous ether,
which transmits the vibrations of light—not at the rate of so many
feet, but at the rate of nearly two hundred thousand miles a second.

Those who are unacquainted with the details of scientific investigation
have no idea of the amount of labor expended in the determination of
those numbers on which important calculations or inferences depend.
They have no idea of the patience shown by a Berzelius in determining
atomic weights; by a Regnault in determining coefficients of expansion;
or by a Joule in determining the mechanical equivalent of heat. There
is a morality brought to bear upon such matters which, in point of
severity, is probably without a parallel in any other domain of
intellectual action. Thus, as regards the determination of the velocity
of sound in air, hours might be filled with a simple statement of
the efforts made to establish it with precision. The question has
occupied the attention of experimenters in England, France, Germany,
Italy, and Holland. But to the French and Dutch philosophers we owe
the application of the last refinements of experimental skill to the
solution of the problem. They neutralized effectually the influence
of the wind; they took into account barometric pressure, temperature,
and hygrometric condition. Sounds were started at the same moment
from two distant stations, and thus caused to travel from station to
station through the self-same air. The distance between the stations
was determined by exact trigonometrical observations, and means were
devised for measuring with the utmost accuracy the time required by the
sound to pass from the one station to the other. This time, expressed
in seconds, divided into the distance expressed in feet, gave 1,090
feet per second as the velocity of sound through air at the temperature
of 0° centigrade.

The time required by light to travel over all terrestrial distances is
practically zero; and in the experiments just referred to the moment
of explosion was marked by the flash of a gun, the time occupied
by the sound in passing from station to station being the interval
observed between the appearance of the flash and the arrival of the
sound. The velocity of sound in air once established, it is plain that
we can apply it to the determination of distances. By observing, for
example, the interval between the appearance of a flash of lightning
and the arrival of the accompanying thunder-peal, we at once determine
the distance of the place of discharge. It is only when the interval
between the flash and peal is short that danger from lightning is to be
apprehended.


§ 9. _Theoretic Velocity calculated by Newton Laplace’s Correction_

We now come to one of the most delicate points in the whole theory
of sound. The velocity through air has been determined by direct
experiment; but knowing the elasticity and density of the air, it is
possible, without any experiment at all, to calculate the velocity with
which a sound-wave is transmitted through it. Sir Isaac Newton made
this calculation, and found the velocity at the freezing temperature
to be 916 feet a second. This is about one-sixth less than actual
observation had proved the velocity to be, and the most curious
suppositions were made to account for the discrepancy. Newton himself
threw out the conjecture that it was only in passing from particle to
particle of the air that sound required _time_ for its transmission;
that it moved instantaneously _through the particles themselves_.
He then supposed the line along which sound passes to be occupied
by air-particles for one-sixth of its extent, and thus he sought to
make good the missing velocity. The very art and ingenuity of this
assumption were sufficient to throw doubt on it; other theories were
therefore advanced, but the great French mathematician Laplace was the
first to completely solve the enigma. I shall now endeavor to make you
thoroughly acquainted with his solution.

[Illustration: FIG. 12.]

Into this strong cylinder of glass, T U, Fig. 12, which is accurately
bored, and quite smooth within, fits an air-tight piston. By pushing
the piston down, I condense the air beneath it, heat being at the same
time developed. A scrap of amadou attached to the bottom of the piston
is ignited by the heat generated by compression. If a bit of cotton
wool dipped into bisulphide of carbon be attached to the piston, when
the latter is forced down, a flash of light, due to the ignition of
the bisulphide of carbon vapor, is observed within the tube. It is
thus proved that when air is compressed heat is generated. By another
experiment it may be shown that when air is rarefied cold is developed.
This brass box contains a quantity of condensed air. I open the cock,
and permit the air to discharge itself against a suitable thermometer;
the sinking of the instrument immediately declares the chilling of the
air.

All that you have heard regarding the transmission of a sonorous
pulse through air is, I trust, still fresh in your minds. As the
pulse advances it squeezes the particles of air together, and two
results follow from this compression. First, its elasticity is
augmented through the mere augmentation of its density. Secondly, its
elasticity is augmented by the heat of compression. It was the change
of elasticity which resulted from a change of density that Newton took
into account, and he entirely overlooked the augmentation of elasticity
due to the second cause just mentioned. Over and above, then, the
elasticity involved in Newton’s calculation, we have an additional
elasticity due to changes of temperature produced by the sound-wave
itself. When both are taken into account, the calculated and the
observed velocities agree perfectly.

But here, without due caution, we may fall into the gravest error. In
fact, in dealing with Nature, the mind must be on the alert to seize
all her conditions; otherwise we soon learn that our thoughts are not
in accordance with her facts. It is to be particularly noted that the
augmentation of velocity due to the changes of temperature produced by
the sonorous wave itself is totally different from the augmentation
arising from the heating of the general mass of the air. The _average_
temperature of the air is unchanged by the waves of sound. We cannot
have a condensed pulse without having a rarefied one associated with
it. But in the rarefaction, the temperature of the air is as much
lowered as it is raised in the condensation. Supposing, then, the
atmosphere parcelled out into such condensations and rarefactions, with
their respective temperatures, an extraneous sound passing through such
an atmosphere would be as much retarded in the latter as accelerated
in the former, and no variation of the average velocity could result
from such a distribution of temperature.

[Illustration: FIG. 13.]

Whence, then, does the augmentation pointed out by Laplace arise? I
would ask your best attention while I endeavor to make this knotty
point clear to you. If air be compressed it becomes smaller in volume;
if the pressure be diminished, the volume expands. The force which
resists compression, and which produces expansion, is the elastic
force of the air. Thus an external pressure squeezes the air-particles
together; their own elastic force holds them asunder, and the particles
are in equilibrium when these two forces are in equilibrium. Hence it
is that the external pressure is a measure of the elastic force. Let
the middle row of dots, Fig. 13, represent a series of air-particles
in a state of quiescence between the points _a_ and _x_. Then, because
of the elastic force exerted between the particles, if any one of them
be moved from its position of rest, the motion will be transmitted
through the entire series. Supposing the particle _a_ to be driven by
the prong of a tuning-fork, or some other vibrating body, toward _x_,
so as to be caused finally to occupy the position _a′_ in the lowest
row of particles: at the instant the excursion of _a_ commences, its
motion begins to be transmitted to _b_. In the next following moments
_b_ transmits the motion to _c_, _c_ to _d_, _d_ to _e_, and so on. So
that by the time _a_ has reached the position _a′_, the motion will
have been propagated to some point _o′_ of the line of particles more
or less distant from _a′_. The entire series of particles between _a′_
and _o′_ is then in a state of condensation. The distance _a′ o′_,
over which the motion has travelled during the excursion of _a_ to
_a′_, will depend upon the elastic force exerted between the particles.
Fix your attention on any two of the particles, say _a_ and _b_. The
elastic force between them may be figured as a spiral spring, and it
is plain that the more flaccid this spring the more sluggish would be
the communication of the motion from _a_ to _b_; while the stiffer
the spring the more prompt would be the communication of the motion.
What is true of _a_ and _b_ is true for every other pair of particles
between _a_ and _o_. Now the spring between every pair of these
particles _is suddenly stiffened_ by the heat developed along the line
of condensation, and hence the velocity of propagation is augmented
by this heat. Reverting to our old experiment with the row of boys,
it is as if, by the very act of pushing his neighbor, the muscular
rigidity of each boy’s arm was increased, thus enabling him to deliver
his push more promptly than he would have done without this increase
of rigidity. The _condensed_ portion of a sonorous wave is propagated
in the manner here described, and it is plain that the velocity of
propagation is augmented by the heat developed in the condensation.

Let us now turn our thoughts for a moment to the propagation of the
rarefaction. Supposing, as before, the middle row _a x_ to represent
the particles of air in equilibrium under the pressure of the
atmosphere, and suppose the particle _a_ to be suddenly drawn to the
right, so as to occupy the position _a″_ in the highest line of dots:
_a″_ is immediately followed by _b″_, _b″_ by _c″_, _c″_ by _d″_,
_d″_ by _e″_; and thus the rarefaction is propagated backward toward
_x″_, reaching a point _o″_ in the line of particles by the time _a_
has completed its motion to the right. Now, why does _b″_ follow
_a″_ when _a″_ is drawn away from it? Manifestly because the elastic
force exerted between _b″_ and _a″_ is less than that between _b″_
and _c″_. In fact, _b″_ will be driven after _a″_ by a force equal
to the difference of the two elasticities between _a″_ and _b″_ and
between _b″_ and _c″_. The same remark applies to the motion of _c″_
after _b″_, to that of _d″_ after _c″_, in fact, to the motion of each
succeeding particle when it follows its predecessor. The greater the
difference of elasticity on the two sides of any particle the more
promptly will it follow its predecessor. And here observe what the
_cold_ of rarefaction accomplishes. In addition to the diminution of
the elastic force between _a″_ and _b″_ by the withdrawal of _a″_ to a
greater distance, there is a further diminution due to the lowering of
the temperature. _The cold developed augments the difference of elastic
force on which the propagation of the rarefaction depends._ Thus we
see that because the heat developed in the condensation augments the
rapidity of the condensation, and because the cold developed in the
rarefaction augments the rapidity of the rarefaction, the sonorous
wave, which consists of a condensation and a rarefaction, must have its
velocity augmented by the heat _and the cold_ which it develops during
its own progress.

It is worth while fixing your attention here upon the fact that the
distance _a′ o′_, to which the motion has been propagated while _a_ is
moving to the position _a′_, may be vastly greater than that passed
over in the same time by the particle itself. The excursion of _a′_
may not be more than a small fraction of an inch, while the distance
to which the motion is transferred during the time required by _a′_
to perform this small excursion may be many feet, or even many yards.
If this point should not appear altogether plain to you now, it will
appear so by and by.


§ 10. _Ratio of Specific Heats of Air deduced from Velocity of Sound_

Having grasped this, even partially, I will ask you to accompany me to
a remote corner of the domain of physics, with the view, however, of
showing that remoteness does not imply discontinuity. Let a certain
quantity of air at a temperature of 0°, contained in a perfectly
inexpansible vessel, have its temperature raised 1°. Let the same
quantity of air, placed in a vessel which permits the air to expand
when it is heated—the pressure on the air being kept constant during
its expansion—also have its temperature raised 1°. The quantities
of heat employed in the two cases are different. The one quantity
expresses what is called the specific heat of air at constant volume;
the other the specific heat of air at constant pressure.[21] It is an
instance of the manner in which apparently unrelated natural phenomena
are bound together, that from the calculated and observed velocities
of sound in air we can deduce the ratio of these two specific heats.
Squaring Newton’s theoretic velocity and the observed velocity, and
dividing the greater square by the less, we obtain the ratio referred
to. Calling the specific heat at constant volume C^{v}, and that
at constant pressure C^{p}; calling, moreover, Newton’s calculated
velocity V, and the observed velocity V′, Laplace proved that—

  C^{p}   V′^{2}
  ————— = —————
  C^{v}   V^{2}

Inserting the values of V and V′ in this equation, and making the
calculation, we find—

  C^{p}
  ————— = 1·42.
  C^{v}

Thus, without knowing either the specific heat at constant volume or
at constant pressure, Laplace found the ratio of the greater of them
to the less to be 1·42. It is evident from the foregoing formulæ that
the calculated velocity of sound, multiplied by the square root of this
ratio, gives the observed velocity.

But there is one assumption connected with the determination of this
ratio, which must be here brought clearly forth. It is assumed that
the heat developed by compression _remains in the condensed portion of
the wave_, and applies itself there to augment the elasticity; that no
portion of it is lost by radiation. If air were a powerful radiator,
this assumption could not stand. The heat developed in the condensation
could not then remain in the condensation. It would radiate all
round, lodging itself for the most part in the chilled and rarefied
portion of the wave, which would be gifted with a proportionate power
of absorption. Hence the direct tendency of radiation would be to
equalize the temperatures of the different parts of the wave, and
thus to abolish the increase of velocity which called forth Laplace’s
correction.[22]


§ 11. _Mechanical Equivalent of Heat deduced from Velocity of Sound_

The question, then, of the correctness of this ratio involves the other
and apparently incongruous question, whether atmospheric air possesses
any sensible radiative power. If the ratio be correct, the practical
absence of radiative power on the part of air is demonstrated. How then
are we to ascertain whether the ratio is correct or not? By a process
of reasoning which illustrates still further how natural agencies are
intertwined. It was this ratio, looked at by a man of genius, named
Mayer, which helped him to a clearer and a grander conception of the
relation and interaction of the forces of inorganic and organic nature
than any philosopher up to his time had attained. Mayer was the first
to see that the excess 0·42 of the specific heat at constant pressure
over that at constant volume was the quantity of heat consumed in the
work performed by the expanding gas. Assuming the air to be confined
laterally and to expand in a vertical direction, in which direction it
would simply have to lift the weight of the atmosphere, he attempted
to calculate the precise amount of heat consumed in the raising of
this or any other weight. He thus sought to determine the “mechanical
equivalent” of heat. In the combination of his data his mind was clear,
but for the numerical correctness of these data he was obliged to rely
upon the experimenters of his age. Their results, though approximately
correct, were not so correct as the transcendent experimental ability
of Regnault, aided by the last refinements of constructive skill,
afterward made them. Without changing in the slightest degree the
method of his thought or the structure of his calculation, the simple
introduction of the exact numerical data into the formula of Mayer
brings out the true mechanical equivalent of heat.

But how are we able to speak thus confidently of the accuracy of this
equivalent? We are enabled to do so by the labors of an Englishman, who
worked at this subject contemporaneously with Mayer; and who, while
animated by the creative genius of his celebrated German brother,
enjoyed also the opportunity of bringing the inspirations of that
genius to the test of experiment. By the immortal experiments of Mr.
Joule, the mutual convertibility of mechanical work and heat was first
conclusively established. And “Joule’s equivalent,” as it is rightly
called, considering the amount of resolute labor and skill expended
in its determination, is almost identical with that derived from the
formula of Mayer.


§ 12. _Absence of Radiative Power of Air deduced from Velocity of Sound_

Consider now the ground we have trodden, the curious labyrinth of
reasoning and experiment through which we have passed. We started with
the observed and calculated velocities of sound in atmospheric air. We
found Laplace, by a special assumption, deducing from these velocities
the ratio of the specific heat of air at constant pressure to its
specific heat at constant volume. We found Mayer calculating from
this ratio the mechanical equivalent of heat; finally, we found Joule
determining the same equivalent by direct experiments on the friction
of solids and liquids. And what is the result? Mr. Joule’s experiments
prove the result of Mayer to be the true one; they therefore prove the
ratio determined by Laplace to be the true ratio; and, because they do
this, they prove at the same time the practical absence of radiative
power in atmospheric air. It seems a long step from the stirring of
water, or the rubbing together of iron plates in Joule’s experiments,
to the radiation of the atoms of our atmosphere; both questions are,
however, connected by the line of reasoning here followed out.

But the true physical philosopher never rests content with an
inference when an experiment to verify or contravene it is possible.
The foregoing argument is clinched by bringing the radiative power of
atmospheric air to a direct test. When this is done, experiment and
reasoning are found to agree; air being proved to be a body sensibly
devoid of radiative and absorptive power.[23]

But here the experimenter on the transmission of sound through gases
needs a word of warning. In Laplace’s day, and long subsequently, it
was thought that gases of all kinds possessed only an infinitesimal
power of radiation; but that this is not the case is now well
established. It would be rash to assume that, in the case of such
bodies as ammonia, aqueous vapor, sulphurous acid, and olefiant gas,
their enormous radiative powers do not interfere with the application
of the formula of Laplace. It behooves us to inquire whether the
ratio of the two specific heats deduced from the velocity of sound
in these bodies is the true ratio; and whether, if the true ratio
could be found by other methods, its square root, multiplied into the
calculated velocity, would give the observed velocity. From the moment
heat first appears in the condensation and cold in the rarefaction of
a sonorous wave in any of those gases, the radiative power comes into
play to abolish the difference of temperature. The condensed part of
the wave is on this account rendered more flaccid and the rarefied part
less flaccid than it would otherwise be, and with a sufficiently high
radiative power the velocity of sound, instead of coinciding with that
derived from the formula of Laplace, must approximate to that derived
from the more simple formula of Newton.


§ 13. _Velocity of Sound through Gases, Liquids, and Solids_

To complete our knowledge of the transmission of sound through gases,
a table is here added from the excellent researches of Dulong, who
employed in his experiments a method which shall be subsequently
explained:

VELOCITY OF SOUND IN GASES AT THE TEMPERATURE OF 0° C.

                            Velocity

  Air                       1,092  feet
  Oxygen                    1,040   ”
  Hydrogen                  4,164   ”
  Carbonic acid               858   ”
  Carbonic oxide            1,107   ”
  Protoxide of nitrogen       859   ”
  Olefiant gas              1,030   ”

According to theory, the velocities of sound in oxygen and hydrogen are
inversely proportional to the square roots of the densities of the two
gases. We here find this theoretic deduction verified by experiment.
Oxygen being sixteen times heavier than hydrogen, the velocity of
sound in the latter gas ought, according to the above law, to be four
times its velocity in the former; hence, the velocity in oxygen being
1,040, in hydrogen calculation would make it 4,160. Experiment, we see,
makes it 4,164.

The velocity of sound in liquids may be determined theoretically, as
Newton determined its velocity in air; for the density of a liquid is
easily determined, and its elasticity can be measured by subjecting it
to compression. In the case of water, the calculated and the observed
velocities agree so closely as to prove that the changes of temperature
produced by a sound-wave in water have no sensible influence upon the
velocity. In a series of memorable experiments in the Lake of Geneva,
MM. Colladon and Sturm determined the velocity of sound through water,
and made it 4,708 feet a second. By a mode of experiment which you will
subsequently be able to comprehend, the late M. Wertheim determined the
velocity through various liquids, and in the following table I have
collected his results:

TRANSMISSION OF SOUND THROUGH LIQUIDS

  Name of Liquid                  Temperature  Velocity

  River-water (Seine)               15° C.       4,714  feet
      ”          ”                  30           5,013   ”
      ”          ”                  60           5,657   ”
  Sea-water (artificial)            20           4,768   ”
  Solution of common salt           18           5,132   ”
  Solution of sulphate of soda      20           5,194   ”
  Solution of carbonate of soda     22           5,230   ”
  Solution of nitrate of soda       21           5,477   ”
  Solution of chloride of calcium   23           6,493   ”
  Common alcohol                    20           4,218   ”
  Absolute alcohol                  23           3,804   ”
  Spirits of turpentine             24           3,976   ”
  Sulphuric ether                    0           3,801   ”

We learn from this table that sound travels with different velocities
through different liquids; that a salt dissolved in water augments the
velocity, and that the salt which produces the greatest augmentation is
chloride of calcium. The experiments also teach us that in water, as in
air, the velocity augments with the temperature. At a temperature of
15° C., for example, the velocity in Seine water is 4,714 feet, at 30°
it is 5,013 feet, and at 60° 5,657 feet a second.

I have said that from the compressibility of a liquid, determined by
proper measurements, the velocity of sound through the liquid may
be deduced. Conversely, from the velocity of sound in a liquid, the
compressibility of the liquid may be deduced. Wertheim compared a
series of compressibilities deduced from his experiments on sound with
a similar series obtained directly by M. Grassi. The agreement of both,
exhibited in the following table, is a strong confirmation of the
accuracy of the method pursued by Wertheim:

                                   Cubic compressibility

                           ╭——————————————-——^————————————————╮
                            from Wertheim’s     from the direct
                            velocity of sound   experiments of
                                                M. Grassi

  Sea-water                  0·0000467          0·0000436
  Solution of common salt    0·0000349          0·0000321
     ”     carbonate of soda 0·0000337          0·0000297
     ”     nitrate of soda   0·0000301          0·0000295
  Absolute alcohol           0·0000947          0·0000991
  Sulphuric ether            0·0001002          0·0001110

The greater the resistance which a liquid offers to compression, the
more promptly and forcibly will it return to its original volume after
it has been compressed. The less the compressibility, therefore, the
greater is the elasticity, and consequently, other things being equal,
the greater the velocity of sound through the liquid.

We have now to examine the transmission of sound through solids. Here,
as a general rule, the elasticity, as compared with the density, is
greater than in liquids, and consequently the propagation of sound is
more rapid.

In the following table the velocity of sound through various metals, as
determined by Wertheim, is recorded:

VELOCITY OF SOUND THROUGH METALS

  Name of Metal         At 20° C.   At 100° C.  At 200° C.

  Lead                    4,030       3,951      ......
  Gold                    5,717       5,640       5,619
  Silver                  8,553       8,658       8,127
  Copper                 11,666      10,802       9,690
  Platinum                8,815       8,437       8,079
  Iron                   16,822      17,386      15,483
  Iron wire (ordinary)   16,130      16,728      ......
  Cast steel             16,357      16,153      15,709
  Steel wire (English)   15,470      17,201      16,394
  Steel wire             16,023      16,443      ......

As a general rule, the velocity of sound through metals is diminished
by augmented temperature; iron is, however, a striking exception to
this rule, but it is only within certain limits an exception. While,
for example, a rise of temperature from 20° to 100° C. in the case of
copper causes the velocity to fall from 11,666 to 10,802, the same
rise produces in the case of iron an increase of velocity from 16,822
to 17,386. Between 100° and 200°, however, we see that iron falls from
the last figure to 15,483. In iron, therefore, up to a certain point,
the elasticity is augmented by heat; beyond that point it is lowered.
Silver is also an example of the same kind.

The difference of velocity in iron and in air may be illustrated by the
following instructive experiment: Choose one of the longest horizontal
bars employed for fencing in Hyde Park; and let an assistant strike the
bar at one end while the ear of the observer is held close to the bar
at a considerable distance from the point struck. Two sounds will reach
the ear in succession; the first being transmitted through the iron and
the second through the air. This effect was obtained by M. Biot, in his
experiments on the iron water-pipes of Paris.

The transmission of sound through a solid depends on the manner
in which the molecules of the solid are arranged. If the body be
homogeneous and without structure, sound is transmitted through it
equally well in all directions. But this is not the case when the body,
whether inorganic like a crystal or organic like a tree, possesses
a definite structure. This is also true of other things than sound.
Subjecting, for example, a sphere of wood to the action of a magnet,
it is not equally affected in all directions. It is repelled by the
pole of the magnet, but it is most strongly repelled when the force
acts along the fibre. Heat also is conducted with different facilities
in different directions through wood. It is most freely conducted
along the fibre, and it passes more freely across the ligneous layers
than along them. Wood, therefore, possesses _three unequal axes_ of
calorific conduction. These, established by myself, coincide with the
axes of elasticity discovered by Savart. MM. Wertheim and Chevandier
have determined the velocity of sound along these three axes and
obtained the following results:

VELOCITY OF SOUND IN WOOD

  Name of Wood   Along Fibre    Across Rings   Along Rings

  Acacia           15,467          4,840          4,436
  Fir              15,218          4,382          2,572
  Beech            10,965          6,028          4,643
  Oak              12,622          5,036          4,229
  Pine             10,900          4,611          2,605
  Elm              13,516          4,665          3,324
  Sycamore         14,639          4,916          3,728
  Ash              15,314          4,567          4,142
  Alder            15,306          4,491          3,423
  Aspen            16,677          5,297          2,987
  Maple            13,472          5,047          3,401
  Poplar           14,050          4,600          3,444

Separating a cube from the bark-wood of a good-sized tree, where the
rings for a short distance may be regarded as straight: then, if A R,
Fig. 14, be the section of the tree, the velocity of the sound in the
direction _m n_, through such a cube, is greater than in the direction
_a b_.

[Illustration: FIG. 14.]

The foregoing table strikingly illustrates the influence of molecular
structure. The great majority of crystals show differences of the same
kind. Such bodies, for the most part, have their molecules arranged
in different degrees of proximity in different directions, and where
this occurs there are sure to be differences in the transmission and
manifestation of heat, light, electricity, magnetism, and sound.


§ 14. _Hooke’s Anticipation of the Stethoscope_

I will conclude this lecture on the transmission of sound through
gases, liquids, and solids, by a quaint and beautiful extract from
the writings of that admirable thinker, Dr. Robert Hooke. It will be
noticed that the philosophy of the stethoscope is enunciated in the
following passage, and another could hardly be found which illustrates
so well that action of the scientific imagination which, in all great
investigators, is the precursor and associate of experiment:

“There may also be a possibility,” writes Hooke, “of discovering the
internal motions and actions of bodies by the sound they make. Who
knows but that, as in a watch, we may hear the beating of the balance,
and the running of the wheels, and the striking of the hammers, and the
grating of the teeth, and multitudes of other noises; who knows, I say,
but that it may be possible to discover the motions of the internal
parts of bodies, whether animal, vegetable, or mineral, by the sound
they make; that one may discover the works performed in the several
offices and shops of a man’s body, and thereby discover what instrument
or engine is out of order, what works are going on at several times,
and lie still at others, and the like; that in plants and vegetables
one might discover by the noise the pumps for raising the juice, the
valves for stopping it, and the rushing of it out of one passage into
another, and the like? I could proceed further, but methinks I can
hardly forbear to blush when I consider how the most part of men will
look upon this: but, yet again, I have this encouragement, not to think
all these things utterly impossible, though never so much derided
by the generality of men, and never so seemingly mad, foolish, and
fantastic, that as the thinking them impossible cannot much improve my
knowledge, so the believing them possible may, perhaps, be an occasion
of taking notice of such things as another would pass by without
regard as useless. And somewhat more of encouragement I have also from
experience, that I have been able to hear very plainly the beating of a
man’s heart, and it is common to hear the motion of wind to and fro in
the guts, and other small vessels; the stopping of the lungs is easily
discovered by the wheezing, the stopping of the head by the humming and
whistling noises, the slipping to and fro of the joints, in many cases,
by crackling, and the like, as to the working or motion of the parts
one among another; methinks I could receive encouragement from hearing
the hissing noise made by a corrosive menstruum in its operation, the
noise of fire in dissolving, of water in boiling, of the parts of a
bell after that its motion is grown quite invisible as to the eye, for
to me these motions and the other seem only to differ _secundum magis
minus_, and so to their becoming sensible they require either that
their motions be increased, or that the organ be made more nice and
powerful to sensate and distinguish them.”


  NOTE ON THE DIFFRACTION OF SOUND

  The recent explosion of a powder-laden barge in the Regent’s
  Park produced effects similar to those mentioned in § 7.
  The sound-wave bent round houses and broke the windows at
  the back, the coalescence of different portions of the wave
  at special points being marked by intensified local action.
  Close to the place where the explosion occurred the unconsumed
  gunpowder was in the wave, and, as a consequence, the
  dismantled gatekeeper’s lodge was girdled all round by a black
  belt of carbon.


SUMMARY OF CHAPTER I

The sound of an explosion is propagated as a wave or pulse through the
air.

This wave impinging upon the tympanic membrane causes it to shiver, its
tremors are transmitted to the auditory nerve, and along the auditory
nerve to the brain, where it announces itself as sound.

A sonorous wave consists of two parts, in one of which the air is
condensed, and in the other rarefied.

The motion of the sonorous wave must not be confounded with the motion
of the particles which at any moment form the wave. During the passage
of the wave every particle concerned in its transmission makes only a
small excursion to and fro.

The length of this excursion is called the _amplitude_ of the vibration.

Sound cannot pass through a vacuum.

A certain sharpness of shock, or rapidity of vibration, is needed for
the production of sonorous waves in air. It is still more necessary
in hydrogen, because the greater mobility of this light gas tends to
prevent the formation of condensations and rarefactions.

Sound is in all respects reflected like light; it is also refracted
like light; and it may, like light, be condensed by suitable lenses.

Sound is also diffracted, the sonorous wave bending round obstacles;
such obstacles, however, in part shade off the sound.

Echoes are produced by the reflected waves of sound.

In regard to sound and the medium through which it passes, four
distinct things are to be borne in mind—intensity, velocity,
elasticity, and density.

The intensity is proportional to the square of the amplitude as above
defined.

It is also proportional to the square of the maximum velocity of the
vibrating air-particles.

When sound issues from a small body in free air, the intensity
diminishes as the square of the distance from the body increases.

If the wave of sound be confined in a tube with a smooth interior
surface, it may be conveyed to great distances without sensible loss of
intensity.

The velocity of sound in air depends on the elasticity of the air in
relation to its density. The greater the elasticity the swifter is the
propagation; the greater the density the slower is the propagation.

The velocity is directly proportional to the square root of the
elasticity; it is inversely proportional to the square root of the
density.

Hence, if elasticity and density vary in the same proportion, the one
will neutralize the other as regards the velocity of sound.

That they do vary in the same proportion is proved by the law of Boyle
and Mariotte; hence the velocity of sound in air is independent of the
density of the air.

But that this law shall hold good, it is necessary that the dense air
and the rare air should have the same temperature.

The intensity of a sound depends upon the density of the air in which
it is generated, but not on that of the air in which it is heard.

The velocity of sound in air of the temperature 0° C. is 1,090 feet a
second; it augments nearly 2 feet for every degree Centigrade added to
its temperature.

Hence, given the velocity of sound in air, the temperature of the air
may be readily calculated.

The distance of a fired cannon or of a discharge of lightning may be
determined by observing the interval which elapses between the flash
and the sound.

From the foregoing, it is easy to see that if a row of soldiers form a
circle, and discharge their pieces all at the same time, the sound will
be heard as a single discharge by a person occupying the centre of the
circle.

But if the men form a straight row, and if the observer stand at one
end of the row, the simultaneous discharge of the men’s pieces will be
prolonged to a kind of roar.

A discharge of lightning along a lengthy cloud may in this way produce
the prolonged roll of thunder. The roll of thunder, however, must in
part at least be due to echoes from the clouds.

The pupil will find no difficulty in referring many common occurrences
to the fact that sound requires a sensible time to pass through any
considerable length of air. For example, the fall of the axe of a
distant wood-cutter is not simultaneous with the sound of the stroke.
A company of soldiers marching to music along a road cannot march
in time, for the notes do not reach those in front and those behind
simultaneously.

In the condensed portion of a sonorous wave the air is above, in the
rarefied portion of the wave it is below, its average temperature.

This change of temperature, produced by the passage of the sound-wave
itself, virtually augments the elasticity of the air, and makes the
velocity of sound about one-sixth greater than it would be if there
were no change of temperature.

The velocity found by Newton, who did not take this change of
temperature into account, was 916 feet a second.

Laplace proved that by multiplying Newton’s velocity by the square root
of the ratio of the specific heat of air at constant pressure to its
specific heat at constant volume, the actual or observed velocity is
obtained.

Conversely, from a comparison of the calculated and observed
velocities, the ratio of the two specific heats may be inferred.

The mechanical equivalent of heat may be deduced from this ratio; it is
found to be the same as that established by direct experiment.

This coincidence leads to the conclusion that atmospheric air is devoid
of any sensible power to radiate heat. Direct experiments on the
radiative power of air establish the same result.

The velocity of sound in water is more than four times its velocity in
air.

The velocity of sound in iron is seventeen times its velocity in air.

The velocity of sound along the fibre of pine-wood is ten times its
velocity in air.

The cause of this great superiority is that the elasticities of the
liquid, the metal, and the wood, as compared with their respective
densities, are vastly greater than the elasticity of air in relation to
its density.

The velocity of sound is dependent to some extent upon molecular
structure. In wood, for example, it is conveyed with different degrees
of rapidity in different directions.




CHAPTER II

  Physical Distinction between Noise and Music—A Musical
  Tone Produced by Periodic, Noise Produced by Unperiodic,
  Impulses—Production of Musical Sounds by Taps—Production
  of Musical Sounds by Puffs—Definition of Pitch in
  Music—Vibrations of a Tuning-Fork; their Graphic
  Representation on Smoked Glass—Optical Expression of the
  Vibrations of a Tuning-Fork—Description of the Siren—Limits
  of the Ear; Highest and Deepest Tones—Rapidity of Vibration
  Determined by the Siren—Determination of the Lengths
  of Sonorous Waves—Wave-Lengths of the Voice in Man and
  Woman—Transmission of Musical Sounds through Liquids and Solids


In our last chapter we considered the propagation through air of a
sound of momentary duration. We have to-day to consider continuous
sounds, and to make ourselves in the first place acquainted with the
physical distinction between noise and music. As far as sensation goes,
everybody knows the difference between these two things. But we have
now to inquire into the causes of sensation, and to make ourselves
acquainted with the condition of the external air which in one case
resolves itself into music and in another into noise.

We have already learned that what is loudness in our sensations is
outside of us nothing more than width of swing, or _amplitude_, of the
vibrating air-particles. Every other real sonorous impression of which
we are conscious has its correlative without, as a mere form or state
of the atmosphere. Were our organs sharp enough to see the motions
of the air through which an agreeable voice is passing, we might see
stamped upon that air the conditions of motion on which the sweetness
of the voice depends. In ordinary conversation, also, the physical
precedes and arouses the psychical; the spoken language, which is to
give us pleasure or pain, which is to rouse us to anger or soothe us
to peace, existing for a time, between us and the speaker, as a purely
mechanical condition of the intervening air.

Noise affects us as an irregular succession of shocks. We are conscious
while listening to it of a jolting and jarring of the auditory
nerve, while a musical sound flows smoothly and without asperity
or irregularity. How is this smoothness secured? _By rendering the
impulses received by the tympanic membrane perfectly periodic._ A
periodic motion is one that repeats itself. The motion of a common
pendulum, for example, is periodic, but its vibrations are far too
sluggish to excite sonorous waves. To produce a musical tone we
must have a body which vibrates with the unerring regularity of the
pendulum, but which can impart much sharper and quicker shocks to the
air.

Imagine the first of a series of pulses following each other at regular
intervals, impinging upon the tympanic membrane. It is shaken by the
shock; and a body once shaken cannot come instantaneously to rest. The
human ear, indeed, is so constructed that the sonorous motion vanishes
with extreme rapidity, but its disappearance is not instantaneous; and
if the motion imparted to the auditory nerve by each individual pulse
of our series continues until the arrival of its successor, the sound
will not cease at all. The effect of every shock will be renewed before
it vanishes, and the recurrent impulses will link themselves together
to a continuous musical sound. The pulses, on the contrary, which
produce noise, are of irregular strength and recurrence. The action of
noise upon the ear has been well compared to that of a flickering light
upon the eye, both being painful through the sudden and abrupt changes
which they impose upon their respective nerves.

The only condition necessary to the production of a musical sound is
that the pulses should succeed each other in the same interval of time.
No matter what its origin may be, if this condition be fulfilled the
sound becomes musical. If a watch, for example, could be caused to
tick with sufficient rapidity—say one hundred times a second—the ticks
would lose their individuality and blend to a musical tone. And if the
strokes of a pigeon’s wings could be accomplished at the same rate, the
progress of the bird through the air would be accompanied by music.
In the humming-bird the necessary rapidity is attained; and when we
pass on from birds to insects, where the vibrations are more rapid,
we have a musical note as the ordinary accompaniment of the insects’
flight.[24] The puffs of a locomotive at starting follow each other
slowly at first, but they soon increase so rapidly as to be almost
incapable of being counted. If this increase could continue up to fifty
or sixty puffs a second, the approach of the engine would be heralded
by an organ-peal of tremendous power.


§ 2. _Musical Sounds produced by Taps_

Galileo produced a musical sound by passing a knife over the edge of
a piastre. The minute serration of the coin indicated the periodic
character of the motion, which consisted of a succession of taps quick
enough to produce sonorous continuity. Every schoolboy knows how to
produce a note with his slate-pencil. I will not call it musical,
because this term is usually associated with pleasure, and the sound of
the pencil is not pleasant.

[Illustration: FIG. 15.]

The production of a musical sound by taps is usually effected by
causing the teeth of a rotating wheel to strike in quick succession
against a card. This was first illustrated by the celebrated Robert
Hooke,[25] and nearer our own day by the eminent French experimenter
Savart. We will confine ourselves to homelier modes of illustration.
This gyroscope is an instrument consisting mainly of a heavy brass
ring, _d_, Fig. 15, loading the circumference of a disk, through which
and at right angles to its surface, passes a steel axis, delicately
supported at its two ends. By coiling a string round the axis, and
drawing it vigorously out, the ring is caused to spin rapidly; and
along with it rotates a small-toothed wheel, w. On touching this wheel
with the edge of a card _c_, a musical sound of exceeding shrillness is
produced. I place my thumb for a moment against the ring; the rapidity
of its rotation is thereby diminished, and this is instantly announced
by a lowering of the pitch of the note. By checking the motion still
more, the pitch is lowered still further. We are here made acquainted
with the important fact that the pitch of a note depends upon the
rapidity of its pulses.[26] At the end of the experiment you hear the
separate taps of the teeth against the card, their succession not being
quick enough to produce that continuous flow of sound which is the
essence of music. A screw with a milled head attached to a whirling
table, and caused to rotate, produces by its taps against a card a note
almost as clear and pure as that obtained from the toothed wheel of the
gyroscope.

The production of a musical sound by taps may also be pleasantly
illustrated in the following way: In this vise are fixed vertically
two pieces of sheet-lead, with their horizontal edges a quarter of an
inch apart. I lay a bar of brass across them, permitting it to rest
upon the edges, and, tilting the bar a little, set it in oscillation
like a see-saw. After a time, if left to itself, it comes to rest. But
suppose the bar on touching the lead to be always tilted upward by a
force issuing from the lead itself, it is plain that the vibrations
would then be rendered permanent. Now such a force is brought into
play _when the bar is heated_. On its then touching the lead the heat
is communicated, a sudden jutting upward of the lead at the point of
contact being the result. Hence an incessant tilting of the bar from
side to side, so long as it continues sufficiently hot. Substituting
for the brass bar the heated fire-shovel shown in Fig. 16, the same
effect is produced.

[Illustration: FIG. 16.]

In its descent upon the lead the bar taps it gently, the taps being so
slow that you may readily count them. But a mass of metal differently
shaped may be caused to vibrate more briskly, and the taps to succeed
each other more rapidly. When such a heated _rocker_, Fig. 17, is
placed upon a block of lead, the taps hasten to a loud rattle. When,
with the point of a file, the rocker is pressed against the lead,
the vibrations are rendered more rapid, and the taps link themselves
together to a deep musical tone. A second rocker, which oscillates
more quickly than the last, produces music without any other pressure
than that due to its own weight. Pressing it, however, with the file,
the pitch rises, until a note of singular force and purity fills the
room. Relaxing the pressure, the pitch instantly falls; resuming the
pressure, it again rises; and thus by the alternation of the pressure
we obtain great variations of tone. Nor are such rockers essential.
Allowing one face of the clean, square end of a heated poker to rest
upon the block of lead, a rattle is heard; causing another face to rest
upon the block, a clear musical note is obtained. The two faces have
been bevelled differently by a file, so as to secure different rates
of vibration.[27] This curious effect was discovered by Schwartz and
Trevelyan.

[Illustration: FIG. 17.]


§ 3. _Musical Sounds produced by Puffs_

Prof. Robison was the first to produce a musical sound by a quick
succession of _puffs_ of air. His device was the first form of an
instrument which will soon be introduced to you under the name of the
_siren_. Robison describes his experiment in the following words: “A
stop-cock was so constructed that it opened and shut the passage of a
pipe 720 times in a second. The apparatus was fitted to the pipe of a
conduit leading from the bellows to the wind-chest of an organ. The air
was simply allowed to pass gently along this pipe by the opening of the
cock. When this was repeated 720 times in a second, the sound _g in
alt_ was most smoothly uttered, equal in sweetness to a clear female
voice. When the frequency was reduced to 360, the sound was that of a
clear but rather a harsh man’s voice. The cock was now altered in such
a manner that it never shut the hole entirely, but left about one-third
of it open. When this was repeated 720 times in a second, the sound was
uncommonly smooth and sweet. When reduced to 360, the sound was more
mellow than any man’s voice of the same pitch.”

[Illustration: FIG. 18.]

But the difficulty of obtaining the necessary speed renders another
form of the experiment preferable. A disk of Bristol board, B, Fig.
18, twelve inches in diameter, is perforated at equal intervals along
a circle near its circumference. The disk, being strengthened by
a backing of tin, can be attached to a whirling table, and caused
to rotate rapidly. The individual holes then disappear, blending
themselves into a continuous shaded circle. Immediately over this
circle is placed a bent tube, _m_, connected with a pair of acoustic
bellows. The disk is now motionless, the lower end of the tube being
immediately over one of the perforations of the disk. If, therefore,
the bellows be worked, the wind will pass from _m_ through the hole
underneath. But if the disk be turned a little, an unperforated portion
of the disk comes under the tube, the current of air being then
intercepted. As the disk is slowly turned, successive perforations are
brought under the tube, and whenever this occurs a puff of air gets
through. On rendering the rotation rapid, the puffs succeed each other
in very quick succession, producing pulses in the air which blend to a
continuous musical note, audible to you all. Mark how the note varies.
When the whirling table is turned rapidly the sound is shrill; when its
motion is slackened the pitch immediately falls. If instead of a single
glass tube there were two of them, as far apart as two of our orifices,
so that whenever the one tube stood over an orifice, the other should
stand over another, it is plain that if both tubes were blown through,
we should, on turning the disk, get a puff through two holes at the
same time. The intensity of the sound would be thereby augmented, but
the pitch would remain unchanged. The two puffs issuing at the same
instant would act in concert, and produce a greater effect than one
upon the ear. And if instead of two tubes we had ten of them, or better
still, if we had a tube for every orifice in the disk, the puffs from
the entire series would all issue, and would be all cut off at the
same time. These puffs would produce a note of far greater intensity
than that obtained by the alternate escape and interruption of the air
from a single tube. In the arrangement now before you, Fig. 19, there
are nine tubes through which the air is urged—through nine apertures,
therefore, puffs escape at once. On turning the whirling table, and
alternately increasing and relaxing its speed, the sound rises and
falls like the loud wail of a changing wind.

[Illustration: FIG. 19.]


§ 4. _Musical Sounds produced by a Tuning-fork_

Various other means may be employed to throw the air into a state of
periodic motion. A stretched string pulled aside and suddenly liberated
imparts vibrations to the air which succeed each other in perfectly
regular intervals. A tuning-fork does the same. When a bow is drawn
across the prongs of this tuning-fork, Fig. 20, the resin of the bow
enables the hairs to grip the prong, which is thus pulled aside. But
the resistance of the prong soon becomes too strong, and it starts
suddenly back; it is, however, immediately laid hold of again by the
bow, to start back once more as soon as its resistance becomes great
enough. This rhythmic process, continually repeated during the passage
of the bow, finally throws the fork into a state of intense vibration,
and the result is a musical note. A person close at hand could see the
fork vibrating; a deaf person bringing his hand sufficiently near would
feel the shivering of the air. Or causing its vibrating prong to touch
a card, taps against the card link themselves, as in the case of the
gyroscope, to a musical sound, the fork coming rapidly to rest. What we
call silence expresses this absence of motion.

[Illustration: FIG. 20.]

When the tuning-fork is first excited the sound issues from it with
maximum loudness, becoming gradually feebler as the fork continues to
vibrate. A person close to the fork can notice at the same time that
the amplitude, or space through which the prongs oscillate, becomes
gradually less and less. But the most expert ear in this assembly
can detect no change in the pitch of the note. The lowering of the
intensity of a note does not therefore imply the lowering of its pitch.
In fact, though the amplitude changes, the rate of vibration remains
the same. Pitch and intensity must therefore be held distinctly apart;
the latter depends solely upon the amplitude, the former solely upon
the rapidity of vibration.

This tuning-fork may be caused to write the story of its own motion.
Attached to the side of one of its prongs, F, Fig. 21, is a thin
strip of sheet-copper which tapers to a point. When the tuning-fork
is excited it vibrates, and the strip of metal accompanies it in its
vibration. The point of the strip being brought gently down upon a
piece of smoked glass, it moves to and fro over the smoked surface,
leaving a clear line behind. As long as the hand is kept motionless,
the point merely passes to and fro over the same line; but it is plain
that we have only to draw the fork along the glass to produce a sinuous
line, Fig. 21.

[Illustration: FIG. 21.]

When this process is repeated without exciting the fork afresh, the
depth of the indentations diminishes. The sinuous line approximates
more and more to a straight one. This is the visual expression of
decreasing amplitude. When the sinuosities entirely disappear, the
amplitude has become zero, and the sound, which depends upon the
amplitude, ceases altogether.

[Illustration: FIG. 22.]

To M. Lissajous we are indebted for a very beautiful method of giving
optical expression to the vibrations of a tuning-fork. Attached to one
of the prongs of a very large fork is a small metallic mirror, F, Fig.
22, the other prong being loaded with a piece of metal to establish
equilibrium. Permitting a slender beam of intense light to fall upon
the mirror, the beam is thrown back by reflection. In my hands is held
a small looking-glass, which receives the reflected beam, and from
which it is again reflected to the screen, forming a small luminous
disk upon the white surface. The disk is perfectly motionless, but
the moment the fork is set in vibration the reflected beam is tilted
rapidly up and down, the disk describing a band of light three feet
long. The length of the band depends on the amplitude of the vibration,
and you see it gradually shorten as the motion of the fork is expended.
It remains, however, a straight line as long as the glass is held in
a fixed position. But on suddenly turning the glass so as to make
the beam travel from left to right over the screen, you observe the
straight line instantly resolved into a beautiful luminous ripple _m
n_. A luminous impression once made upon the retina lingers there
for the tenth of a second; if then the time required to transfer the
elongated image from side to side of the screen be less than the tenth
of a second, the wavy line of light will occupy for a moment the whole
width of the screen. Instead of permitting the beam from the lamp to
issue through a single aperture, it may be caused to issue through two
apertures, about half an inch asunder, thus projecting two disks of
light, one _above_ the other, upon the screen. When the fork is excited
and the mirror turned, we have a brilliant double sinuous line running
over the dark surface, Fig. 23. turning the diaphragm so as to place
the two disks _beside_ each other, on exciting the fork and moving the
mirror we obtain a beautiful interlacing of the two sinuous lines, Fig.
24.

[Illustration: FIG. 23.]

[Illustration: FIG. 24.]


§ 5. _The Waves of Sound_

How are we to picture to ourselves the condition of the air through
which this musical sound is passing? Imagine one of the prongs of the
vibrating fork swiftly advancing; it compresses the air immediately in
front of it, and when it retreats it leaves a partial vacuum behind,
the process being repeated by every subsequent advance and retreat.
The whole function of the tuning-fork is to carve the air into these
condensations and rarefactions, and they, as they are formed, propagate
themselves in succession through the air. A condensation with its
associated rarefaction constitutes, as already stated, a sonorous
wave. In water the length of a wave is measured from crest to crest;
while, in the case of sound, the _wave-length_ is the distance between
two successive condensations. The condensation of the sound-wave
corresponds to the crest, while the rarefaction of the sound-wave
corresponds to the _sinus_, or depression, of the water-wave. Let the
dark spaces, _a_, _b_, _c_, _d_, Fig. 25, represent the condensations,
and the light ones, _a′_, _b′_, _c′_, _d′_, the rarefactions of the
waves issuing from the fork A B: the wave-length would then be measured
from _a_ to _b_, from _b_ to _c_, or from _c_ to _d_.

[Illustration: FIG. 25.]


§ 6. _Definition of Pitch: Determination of Rates of Vibration_

When two notes from two distinct sources are of the same pitch, their
rates of vibration are the same. If, for example, a string yield the
same note as a tuning-fork, it is because they vibrate with the same
rapidity; and if a fork yield the same note as the pipe of an organ or
the tongue of a concertina, it is because the vibrations of the fork in
the one case are executed in precisely the same time as the vibrations
of the column of air, or of the tongue, in the other. The same holds
good for the human voice. If a string and a voice yield the same note,
it is because the vocal chords of the singer vibrate in the same time
as the string vibrates. Is there any way of determining the actual
number of vibrations corresponding to a musical note? Can we infer from
the pitch of a string, of an organ-pipe, of a tuning-fork, or of the
human voice, the number of waves which it sends forth in a second? This
very beautiful problem is capable of the most complete solution.


§ 7. _The Siren: Analysis of the Instrument_

By the rotation of a perforated pasteboard disk, it has been proved to
you that a musical sound is produced by a quick succession of puffs.
Had we any means of registering the number of revolutions accomplished
by that disk in a minute, we should have in it a means of determining
the number of puffs per minute due to a note of any determinate pitch.
The disk, however, is but a cheap substitute for a far more perfect
apparatus, which requires no whirling table, and which registers its
own rotations with the most perfect accuracy.

I will take the instrument asunder, so that you may see its various
parts. A brass tube, _t_, Fig. 26, leads into a round box, C, closed
at the top by a brass plate _a b_. This plate is perforated with four
series of holes, placed along four concentric circles. The innermost
series contains 8, the next 10, the next 12, and the outermost 16
orifices. When we blow into the tube _t_, the air escapes through the
orifices, and the problem now before us is to convert these continuous
currents into discontinuous puffs. This is accomplished by means of
a brass disk _d e_, also perforated with 8, 10, 12, and 16 holes, at
the same distances from the centre and with the same intervals between
them as those in the top of the box C. Through the centre of the disk
passes a steel axis, the two ends of which are smoothly bevelled off
to points at _p_ and _p′_. My object now is to cause this perforated
disk to rotate over the perforated top _a b_ of the box C. You will
understand how this is done by observing how the instrument is put
together.

[Illustration: FIG. 26.]

[Illustration: FIG. 27.]

In the centre of _a b_, Fig. 26, is a depression _x_ sunk in steel,
smoothly polished and intended to receive the end _p′_ of the axis. I
place the end _p′_ in this depression, and, holding the axis upright,
bring down upon its upper end _p_ a steel cap, finely polished within,
which holds the axis at the top, the pressure both at top and bottom
being so gentle, and the polish of the touching surfaces so perfect,
that the disk can rotate with an exceedingly small amount of friction.
At _c_, Fig. 27, is the cap which fits on to the upper end of the
axis _p p′_. In this figure the disk _d e_ is shown covering the top
of the cylinder C. You may neglect for the present the wheel-work of
the figure. Turning the disk _d e_ slowly round, its perforations
may be caused to coincide or not coincide with those of the cylinder
underneath. As the disk turns, its orifices come alternately over
the perforations of the cylinder and over the spaces between the
perforations. Hence it is plain that if air were urged into C, and
if the disk could be caused to rotate at the same time, we should
accomplish our object, and carve into puffs the streams of air. In
this beautiful instrument the disk is caused to rotate by the very air
currents which it renders intermittent. This is done by the simple
device of causing the perforations to pass _obliquely_ through the top
of the cylinder C, and also obliquely, but oppositely inclined, through
the rotating disk _d e_. The air is thus caused to issue from C, not
vertically, but in side currents, which impinge against the disk and
drive it round. In this way, by its passage through the siren, the air
is molded into sonorous waves.

Another moment will make you acquainted with the recording portion of
the instrument. At the upper part of the steel axis _p p′_, Fig. 27,
is a screw _s_, working into a pair of toothed wheels (seen when the
back of the instrument is turned toward you). As the disk and its
axis turn, these wheels rotate. In front you simply see two graduated
dials, Fig. 28, each furnished with an index like the hand of a clock.
These indexes record the number of revolutions executed by the disk
in any given time. By pushing the button _a_ or _b_ the wheel-work is
thrown into or out of action, thus starting or suspending, in a moment,
the process of recording. Finally, by the pins _m_, _n_, _o_, _p_,
Fig. 27, any series of orifices in the top of the cylinder C can be
opened or closed at pleasure. By pressing _m_, one series is opened; by
pressing _n_, another. By pressing two keys, two series of orifices are
opened; by pressing three keys, three series; and by pressing all the
keys, puffs are caused to issue from the four series simultaneously.
The perfect instrument is now before you, and your knowledge of it is
complete.

[Illustration: FIG. 28.]

This instrument received the name of siren from its inventor, Cagniard
de la Tour. The one now before you is the siren as greatly improved by
Dove. The pasteboard siren, whose performance you have already heard,
was devised by Seebeck, who gave the instrument various interesting
forms, and executed with it many important experiments. Let us now make
the siren sing. By pressing the key _m_, the outer series of apertures
in the cylinder C is opened, and by working the bellows, the air is
caused to impinge against the disk. It begins to rotate, and you hear a
succession of puffs which follow each other so slowly that they may be
counted. But as the motion augments, the puffs succeed each other with
increasing rapidity, and at length you hear a deep musical note. As the
velocity of rotation increases the note rises in pitch; it is now very
clear and full, and as the air is urged more vigorously, it becomes so
shrill as to be painful. Here we have a further illustration of the
dependence of pitch on rapidity of vibration. I touch the side of the
disk and lower its speed; the pitch falls instantly. Continuing the
pressure the tone continues to sink, ending in the discontinuous puffs
with which it began.

Were the blast sufficiently powerful and the siren sufficiently free
from friction, it might be urged to higher and higher notes, until
finally its sound would become inaudible to human ears. This, however,
would not prove the absence of vibratory motion in the air; but would
rather show that our auditory apparatus is incompetent to take up and
translate into sound vibrations whose rapidity exceeds a certain limit.
The ear, as we shall immediately learn, is in this respect similar to
the eye.

By means of this siren we can determine with extreme accuracy the
rapidity of vibration of any sonorous body. It may be a vibrating
string, an organ-pipe, a reed, or the human voice. Operating
delicately, we might even determine from the hum of an insect the
number of times it flaps its wings in a second. I will illustrate the
subject by determining in your presence a tuning-fork’s rapidity of
vibration. From the acoustic bellows I urge the air through the siren,
and, at the same time, draw my bow across the fork. Both now sound
together, the tuning-fork yielding at present the highest note. But the
pitch of the siren gradually rises, and at length you hear the “beats”
so well known to musicians, which indicate that the two notes are not
wide apart in pitch. These beats become slower and slower; now they
entirely vanish, both notes blending as it were to a single stream of
sound.

All this time the clockwork of the siren has remained out of action.
As the second-hand of a watch crosses the number 60, the clockwork is
set going by pushing the button _a_. We will allow the disk to continue
its rotation for a minute, the tuning-fork being excited from time
to time to assure you that the unison is preserved. The second-hand
again approaches 60; as it passes that number the clockwork is stopped
by pushing the button _b_; and then, recorded on the dials, we have
the exact number of revolutions performed by the disk. The number is
1,440. But the series of holes open during the experiment numbers 16;
for every revolution, therefore, we had 16 puffs of air, or 16 waves
of sound. Multiplying 1,440 by 16, we obtain 23,040 as the number of
vibrations executed by the tuning-fork in a minute. Dividing this by
60, we find the number of vibrations executed in a second to be 384.


§ 8. _Determination of Wave-lengths: Time of Vibration_

Having determined the rapidity of vibration, the length of the
corresponding sonorous wave is found with the utmost facility. Imagine
a tuning-fork vibrating in free air. At the end of a second from the
time it commenced its vibrations the foremost wave would have reached
a distance of 1,090 feet in air of the freezing temperature. In the
air of a room which has a temperature of about 15° C., it would reach
a distance of 1,120 in a second. In this distance, therefore, are
embraced 384 sonorous waves. Dividing 1,120 by 384, we find the length
of each wave to be nearly 3 feet. Determining in this way the rates of
vibration of the four tuning-forks now before you, we find them to be
256, 320, 384, and 512; these numbers corresponding to wave-lengths
of 4 feet 4 inches, 3 feet 6 inches, 2 feet 11 inches, and 2 feet 2
inches respectively. The waves generated by a man’s voice in common
conversation are from 8 to 12 feet, those of a woman’s voice are from 2
to 4 feet in length. Hence a woman’s ordinary pitch in the lower sounds
of conversation is more than an octave above a man’s; in the higher
sounds it is two octaves.

And here it is important to note that by the term vibrations is meant
_complete ones_; and by the term sonorous wave is meant a condensation
and its associated rarefaction. By a vibration an excursion _to and
fro_ of the vibrating body is to be understood. Every wave generated
by such a vibration bends the tympanic membrane once in and once
out. These are the definitions of a vibration and of a sonorous wave
employed in England and Germany. In France, however, a vibration
consists of an excursion of the vibrating body _in one direction_,
whether to or fro. The French vibrations, therefore, are only the
halves of ours, and we therefore call them semi-vibrations. In all
cases throughout these chapters, when the word vibration is employed
without qualification, it refers to complete vibrations.

During the time required by each of those sonorous waves to pass
entirely over a particle of air, that particle accomplishes one
complete vibration. It is at one moment pushed forward into the
condensation, while at the next moment it is urged back into the
rarefaction. The time required by the particle to execute a complete
oscillation is, therefore, that required by the sonorous wave _to move
through a distance equal to its own length_. Supposing the length of
the wave to be eight feet, and the velocity of sound in air of our
present temperature to be 1,120 feet a second, the wave in question
will pass over its own length of air in, 1/140th of a second: this is
the time required by every air-particle that it passes to complete an
oscillation.

In air of a definite density and elasticity a certain length of wave
always corresponds to the same pitch. But supposing the density or
elasticity not to be uniform; supposing, for example, the sonorous
waves from one of our tuning-forks to pass from cold to hot air: an
instant augmentation of the wave-length would occur, without any change
of pitch, for we should have no change in the rapidity with which the
waves would reach the ear. Conversely with the same length of wave
the pitch would be higher in hot air than in cold, for the succession
of the waves would be quicker. In an atmosphere of hydrogen, waves of
a certain length would produce a note nearly two octaves higher than
waves of the same length in air; for, in consequence of the greater
rapidity of propagation, the number of impulses received in a given
time in the one case would be nearly four times the number received in
the other.


§ 9. _Definition of an Octave_

Opening the innermost and outermost series of the orifices of our
siren, and sounding both of them, either together or in succession, the
musical ears present at once detect the relationship of the two sounds.
They notice immediately that the sound which issues from the circle of
sixteen orifices is the octave of that which issues from the circle of
eight. But for every wave sent forth by the latter, two waves are sent
forth by the former. In this way we prove that the physical meaning
of the term “octave” is, that it is a note produced by double the
number of vibrations of its fundamental. By multiplying the vibrations
of the octave by two, we obtain _its_ octave, and by a continued
multiplication of this kind we obtain a series of numbers answering to
a series of octaves. Starting, for example, from a fundamental note of
100 vibrations, we should find, by this continual multiplication, that
a note five octaves above it would be produced by 3,200 vibrations.
Thus:

   100     Fundamental note.
     2
  ————
   200     1st octave.
     2
  ————
   400     2d octave.
     2
  ————
   800     3d octave.
     2
  ————
  1600    4th octave.
     2
  ————
  3200     5th octave.

This result is more readily obtained by multiplying the vibrations of
the fundamental note by the fifth power of two. In a subsequent chapter
we shall return to this question of musical intervals. For our present
purpose it is only necessary to define an octave.


§ 10. _Limits of the Ear; and of Musical Sounds_

The ear’s range of hearing is limited in both directions. Savart fixed
the lower limit at eight complete vibrations a second; and to cause
these slowly recurring vibrations to link themselves together he was
obliged to employ shocks of great power. By means of a toothed wheel
and an associated counter, he fixed the upper limit of hearing at
24,000 vibrations a second. Helmholtz has recently fixed the lower
limit at 16 vibrations, and the higher at 38,000 vibrations, a second.
By employing very small tuning-forks, the late M. Depretz showed that
a sound corresponding to 38,000 vibrations a second is audible.[28]
Starting from the note 16, and multiplying continually by 2, or more
compendiously raising 2 to the 11th power, and multiplying this by 16,
we should find that at 11 octaves above the fundamental note the number
of vibrations would be 32,768. Taking, therefore, the limit assigned
by Helmholtz, the entire range of the human ear embraces about eleven
octaves. But all the notes comprised within these limits cannot be
employed in music. The practical range of musical sounds is comprised
between 40 and 4,000 vibrations a second, which amounts, in round
numbers, to seven octaves.[29]

The limits of hearing are different in different persons. While
endeavoring to estimate the pitch of certain sharp sounds, Dr.
Wollaston remarked in a friend a total insensibility to the sound of
a small organ-pipe, which, in respect to acuteness, was far within
the ordinary limits of hearing. The sense of hearing of this person
terminated at a note four octaves above the middle E of the pianoforte.
The squeak of the bat, the sound of a cricket, even the chirrup of the
common house-sparrow, are unheard by some people who for lower sounds
possess a sensitive ear. A difference of a single note is sometimes
sufficient to produce the change from sound to silence. “The suddenness
of the transition,” writes Wollaston, “from perfect hearing to total
want of perception, occasions a degree of surprise which renders an
experiment of this kind with a series of small pipes among several
persons rather amusing. It is curious to observe the change of feeling
manifested by various individuals of the party, in succession, as the
sounds approach and pass the limits of their hearing. Those who enjoy a
temporary triumph are often compelled, in their turn, to acknowledge to
how short a distance their little superiority extends.” “Nothing can be
more surprising,” writes Sir John Herschel, “than to see two persons,
neither of them deaf, the one complaining of the penetrating shrillness
of a sound, while the other maintains there is no sound at all. Thus,
while one person mentioned by Dr. Wollaston could but just hear a
note four octaves above the middle E of the pianoforte, others have
a distinct perception of sounds full two octaves higher. The chirrup
of the sparrow is about the former limit; the cry of the bat about an
octave above it; and that of some insects probably another octave.” In
“The Glaciers of the Alps” I have referred to a case of short auditory
range, noticed by myself in crossing the Wengern Alps in company with
a friend. The grass at each side of the path swarmed with insects,
which to me rent the air with their shrill chirruping. My friend heard
nothing of this, the insect-music lying beyond his limit of audition.


§ 11. _Drum of the Ear. The Eustachian Tube_

Behind the tympanic membrane exists a cavity—the drum of the ear—in
part crossed by a series of bones, and in part occupied by air. This
cavity communicates with the mouth by means of a duct called the
Eustachian tube. This tube is generally closed, the air-space behind
the tympanic membrane being thus shut off from the external air. If,
under these circumstances, the external air becomes denser, it will
press the tympanic membrane inward. If, on the other hand, the air
outside becomes rarer, while the Eustachian tube remains closed, the
membrane will be pressed outward. Pain is felt in both cases, and
partial deafness is experienced. I once crossed the Stelvio Pass by
night in company with a friend who complained of acute pain in the
ears. On swallowing his saliva the pain instantly disappeared. By the
act of swallowing, the Eustachian tube is opened, and thus equilibrium
is established between the external and internal pressure.

It is possible to quench the sense of hearing of low sounds by stopping
the nose and mouth, and trying to expand the chest, as in the act
of inspiration. This effort partially exhausts the space behind the
tympanic membrane, which is then thrown into a state of tension by
the pressure of the outward air. A similar deafness to low sounds is
produced when the nose and mouth are stopped, and a strong effort is
made to expire. In this case air is forced through the Eustachian tube
into the drum of the ear, the tympanic membrane being distended by the
pressure of the internal air. The experiment may be made in a railway
carriage, when the low rumble will vanish or be greatly enfeebled,
while the sharper sounds are heard with undiminished intensity. Dr.
Wollaston was expert in closing the Eustachian tube, and leaving the
space behind the tympanic membrane occupied by either compressed or
rarefied air. He was thus able to cause his deafness to continue
for any required time without effort on his part, always, however,
abolishing it by the act of swallowing. A sudden concussion may produce
deafness by forcing air either into or out of the drum of the ear, and
this _may_ account for a fact noticed by myself in one of my Alpine
rambles. In the summer of 1858, jumping from a cliff on to what was
supposed to be a deep snowdrift, I came into rude collision with a rock
which the snow barely covered. The sound of the wind, the rush of the
glacier-torrents, and all the other noises which a sunny day awakes
upon the mountains, instantly ceased. I could hardly hear the sound of
my guide’s voice. This deafness continued for half an hour; at the end
of which time the blowing of the nose opened, I suppose, the Eustachian
tube, and restored, with the quickness of magic, the innumerable
murmurs which filled the air around me.

Light, like sound, is excited by pulses or waves; and lights of
different colors, like sounds of different pitch, are excited by
different rates of vibration. But in its width of perception the ear
exceedingly transcends the eye; for while the former ranges over eleven
octaves, but little more than a single octave is possible to the
latter. The quickest vibrations which strike the eye, as light, have
only about twice the rapidity of the slowest;[30] whereas the quickest
vibrations which strike the ear, as a musical sound, have more than two
thousand times the rapidity of the slowest.


§ 12. _Helmholtz’s Double Siren_

Prof. Dove, as we have seen, extended the utility of the siren of
Cagniard de la Tour, by providing it with four series of orifices
instead of one. By doubling all its parts, Helmholtz has recently
added vastly to the power of the instrument. The double siren, as it
is called, is now before you, Fig. 29 (next page). It is composed
of two of Dove’s sirens, C and C′, one turned upside down. You will
recognize in the lower siren the instrument with which you are already
acquainted. The disks of the two sirens have a common axis, so that
when one disk rotates the other rotates with it. As in the former case,
the number of revolutions is recorded by clockwork (omitted in the
figure). When air is urged through the tube _t′_ the upper siren alone
sounds; when urged through _t_, the lower one only sounds; when it is
urged simultaneously through _t′_ and _t_, both the sirens sound. With
this instrument, therefore, we are able to introduce much more varied
combinations than with the former one. Helmholtz has also contrived a
means by which not only the disk of the upper siren, but the box C′
above the disk, can be caused to rotate. This is effected by a toothed
wheel and pinion, turned by a handle. Underneath the handle is a dial
with an index, the use of which will be subsequently illustrated.

[Illustration: FIG. 29.]

Let us direct our attention for the present to the upper siren. By
means of an India-rubber tube, the orifice _t′_ is connected with an
acoustic bellows, and air is urged into C′. Its disk turns round, and
we obtain with it all the results already obtained with Dove’s siren.
The pitch of the note is uniform. Turning the handle above, so as to
cause the orifices of the cylinder C′ to _meet_ those of the disk,
the two sets of apertures pass each other more rapidly than when the
cylinder stood still. An instant rise of pitch is the result. By
reversing the motion, the orifices are caused to pass each other more
slowly than when C′ is motionless, and in this case you notice an
instant fall of pitch when the handle is turned. Thus, by imparting
in quick alternation a right-handed and left-handed motion to the
handle, we obtain successive rises and falls of pitch. An extremely
instructive effect of this kind may be observed at any railway station
on the passage of a rapid train. During its approach the sonorous waves
emitted by the whistle are virtually shortened, a greater number of
them being crowded into the ear in a given time. During its retreat
we have a virtual lengthening of the sonorous waves. The consequence
is, that, when approaching, the whistle sounds a higher note, and when
retreating it sounds a lower note, than if the train were still. A fall
of pitch, therefore, is perceived as the train passes the station.[31]
This is the basis of Doppler’s theory of the colored stars. He supposes
that all stars are white, but that some of them are rapidly retreating
from us, thereby lengthening their luminiferous waves and becoming red.
Others are rapidly approaching us, thereby shortening their waves, and
becoming green or blue. The ingenuity of this theory is extreme, but
its correctness is more than doubtful.


§ 13. _Transmission of Musical Sounds by Liquids and Solids_

We have thus far occupied ourselves with the transmission of musical
sounds through air. They are also transmitted by liquids and solids.
When a tuning-fork screwed into a little wooden foot vibrates, nobody,
except the persons closest to it, hears its sound. On dipping the foot
into a glass of water a musical sound is audible: the vibrations having
been transmitted through the water to the air. The tube M N, Fig. 30,
three feet long, is set upright upon a wooden tray A B. The tube ends
in a funnel at the top, and is now filled with water to the brim. The
fork F is thrown into vibration, and on dipping its foot into the
funnel at the top of the tube, a musical sound swells out. I must so
far forestall matters as to remark that in this experiment the tray
is the real sounding body. It has been thrown into vibration by the
fork, but the vibrations have been conveyed to the tray _by the water_.
Through the same medium vibrations are communicated to the auditory
nerve, the terminal filaments of which are immersed in a liquid:
substituting mercury for water, a similar result is obtained.

[Illustration: FIG. 30.]

The siren has received its name from its capacity to sing under water.
A vessel now in front of the table is half filled with water, in which
a siren is wholly immersed. When a cock is turned, the water from the
pipes which supply the house forces itself through the instrument. Its
disk is now rotating, and a sound of rapidly augmenting pitch issues
from the vessel. The pitch rises thus rapidly because the heavy and
powerfully pressed water soon drives the disk up to its maximum speed
of rotation. When the supply is lessened, the motion relaxes and the
pitch falls. Thus, by alternately opening and closing the cock, the
song of the siren is caused to rise and fall in a wild and melancholy
manner. You would not consider such a sound likely to woo mariners to
their doom.

The transmission of musical sounds through solid bodies is also
capable of easy and agreeable illustration. Before you is a wooden
rod, thirty feet long, passing from the table through a window in
the ceiling, into the open air above. The lower end of the rod rests
upon a wooden tray, to which the musical vibrations of a body applied
to the upper end of the rod are to be transferred. An assistant is
above, with a tuning-fork in his hand. He strikes the fork against a
pad; it vibrates, but you hear nothing. He now applies the stem of
the fork to the end of the rod, and instantly the wooden tray upon
the table is rendered musical. The pitch of the sound, moreover, is
exactly that of the tuning-fork; the wood has been passive as regards
pitch, transmitting the precise vibrations imparted to it without any
alteration. With another fork a note of another pitch is obtained. Thus
fifty forks might be employed instead of two, and 300 feet of wood
instead of 30; the rod would transmit the precise vibrations imparted
to it, and no other.

We are now prepared to appreciate an extremely beautiful experiment,
for which we are indebted to Sir Charles Wheatstone. In a room
underneath this, and separated from it by two floors, is a piano.
Through the two floors passes a tin tube 2-1/2 inches in diameter,
and along the axis of this tube passes a rod of deal, the end of
which emerges from the floor in front of the lecture-table. The rod
is clasped by India-rubber bands, which entirely close the tin tube.
The lower end of the rod rests upon the sound-board of the piano, its
upper end being exposed before you. An artist is at this moment engaged
at the instrument, but you hear no sound. When, however, a violin
is placed upon the end of the rod, the instrument becomes instantly
musical, not, however, with the vibrations of its own strings, but
with those of the piano. When the violin is removed, the sound ceases;
putting in its place a guitar, the music revives. For the violin and
guitar we may substitute a plain wooden tray, which is also rendered
musical. Here, finally, is a harp, against the sound-board of which
the end of the deal rod is caused to press; every note of the piano
is reproduced before you. On lifting the harp so as to break the
connection with the piano, the sound vanishes; but the moment the
sound-board is caused to press upon the rod the music is restored. The
sound of the piano so far resembles that of the harp that it is hard
to resist the impression that the music you hear is that of the latter
instrument. An uneducated person might well believe that witchcraft or
“spiritualism” is concerned in the production of this music.

What a curious transference of action is here presented to the mind!
At the command of the musician’s will, the fingers strike the keys;
the hammers strike the strings, by which the rude mechanical shock
is converted into tremors. The vibrations are communicated to the
sound-board of the piano. Upon that board rests the end of the deal
rod, thinned off to a sharp edge to make it fit more easily between the
wires. Through the edge, and afterward along the rod, are poured with
unfailing precision the entangled pulsations produced by the shocks of
those ten agile fingers. To the sound-board of the harp before you the
rod faithfully delivers up the vibrations of which it is the vehicle.
This second sound-board transfers the motion to the air, carving it and
chasing it into forms so transcendently complicated that confusion
alone could be anticipated from the shock and jostle of the sonorous
waves. But the marvellous human ear accepts every feature of the
motion, and all the strife and struggle and confusion melt finally into
music upon the brain.[32]


SUMMARY OF CHAPTER II

A musical sound is produced by sonorous shocks which follow each other
at regular intervals with a sufficient rapidity of succession.

Noise is produced by an irregular succession of sonorous shocks.

A musical sound may be produced by _taps_ which rapidly and regularly
succeed each other. The taps of a card against the cogs of a rotating
wheel are usually employed to illustrate this point.

A musical sound may also be produced by a succession of _puffs_. The
siren is an instrument by which such puffs are generated.

The pitch of a musical note depends solely on the number of vibrations
concerned in its production. The more rapid the vibrations, the higher
the pitch.

By means of the siren the rate of vibration of any sounding body may be
determined. It is only necessary to render the sound of the siren and
that of the body identical in pitch to maintain both sounds in unison
for a certain time, and to ascertain, by means of the counter of the
siren, how many puffs have issued from, the instrument in that time.
This number expresses the number of vibrations executed by the sounding
body.

When a body capable of emitting a musical sound—a tuning-fork, for
example—vibrates, it molds the surrounding air into sonorous waves,
each of which consists of a condensation and a rarefaction.

The length of the sonorous wave is measured from condensation to
condensation, or from rarefaction to rarefaction.

The wave-length is found by dividing the velocity of sound per second
by the number of vibrations executed by the sounding body in a second.

Thus a tuning-fork which vibrates 256 times in a second produces in air
of 15° C., where the velocity is 1,120 feet a second, waves 4 feet 4
inches long. While two other forks, vibrating respectively 320 and 384
times a second, generate waves 3 feet 6 inches, and 2 feet 11 inches
long.

A vibration, as defined in England and Germany, comprises a motion to
_and_ fro. It is a _complete_ vibration. In France, on the contrary, a
vibration comprises a movement to _or_ fro. The French vibrations are
with us semi-vibrations.

The time required by a particle of air over which a sonorous wave
passes to execute a complete vibration is that required by the wave to
move through a distance equal to its own length.

The higher the temperature of the air, the longer is the sonorous
wave corresponding to any particular rate of vibration. Given the
wave-length and the rate of vibration, we can readily deduce the
temperature of the air.

The human ear is limited in its range of hearing musical sounds. If
the vibrations number less than 16 a second, we are conscious only of
the separate shocks. If they exceed 38,000 a second, the consciousness
of sound ceases altogether. The range of the best ear covers about
11 octaves, but an auditory range limited to 6 or 7 octaves is not
uncommon.

The sounds available in music are produced by vibrations comprised
between the limits of 40 and 4,000 a second. They embrace 7 octaves.

The range of the ear far transcends that of the eye, which hardly
exceeds an octave.

By means of the Eustachian tube, which is opened in the act of
swallowing, the pressure of the air on both sides of the tympanic
membrane is equalized.

By either condensing or rarefying the air behind the tympanic membrane,
deafness to sounds of low pitch may be produced.

On the approach of a railway train the pitch of the whistle is higher,
on the retreat of the train the pitch is lower, than it would be if the
train were at rest.

Musical sounds are transmitted by liquids and solids. Such sounds may
be transferred from one room to another; from the ground-floor to the
garret of a house of many stories, for example, the sound being unheard
in the rooms intervening between both, and rendered audible only when
the vibrations are communicated to a suitable sound-board.




CHAPTER III

  Vibration of Strings—How employed in Music—Influence of
  Sound-Boards—Laws of Vibrating Strings—Combination of
  Direct and Reflected Pulses—Stationary and Progressive
  Waves—Nodes and Ventral Segments—Application of Results to the
  Vibrations of Musical Strings—Experiments of Melde—Strings
  set in Vibration by Tuning-Forks—Laws of Vibration thus
  demonstrated—Harmonic Tones of Strings—Definitions of Timbre
  or Quality, or Overtones and Clang—Abolition of Special
  Harmonics—Conditions which affect the Intensity of the
  Harmonic Tones—Optical Examination of the Vibrations of a
  Piano-Wire.


§ 1. _Vibrations of Strings: Use of Sound-Boards_

We have to begin our studies to-day with the vibrations of strings
or wires; to learn how bodies of this form are rendered available
as sources of musical sounds, and to investigate the laws of their
vibrations.

To enable a musical string to vibrate _transversely_, or at right
angles to its length, it must be stretched between two rigid points.
Before you, Fig. 31 (next page), is an instrument employed to stretch
strings, and to render their vibrations audible. From the pin _p_,
to which one end of it is firmly attached, a string passes across
the two bridges B and B′, being afterward carried over the wheel H,
which moves with great freedom. The string is finally stretched by a
weight W, of 28 lbs., attached to its extremity. The bridges B and B′,
which constitute the real ends of the string, are fastened on to the
long wooden box M N. The whole instrument is called a monochord, or
sonometer.

[Illustration: FIG. 31.]

Taking hold of the stretched string B B′ at its middle and plucking it
aside, it springs back to its first position, passes it, returns, and
thus vibrates for a time to and fro across its position of equilibrium.
You hear a sound, but the sonorous waves which at present strike
your ears do not proceed immediately from the string. The amount of
wave-motion generated by so thin a body is too small to be sensible at
any distance. But the string is drawn tightly over the two bridges B
B′; and when it vibrates, its tremors are communicated through these
bridges to the entire mass of the box M N, and to the air within the
box, which thus become the real sounding bodies.

[Illustration: FIG. 32.]

That the vibrations of the string alone are not sufficient to produce
the sound may be thus experimentally demonstrated: A B, Fig. 32 (next
page), is a piece of wood placed across an iron bracket C. From each
end of the piece of wood depends a rope ending in a loop, while
stretching across from loop to loop is an iron bar _m n_. From the
middle of the iron bar hangs a steel wire _s s′_, stretched by a weight
W, of 28 lbs. By this arrangement the wire is detached from all large
surfaces to which it could impart its vibrations. Plucking the wire _s
s′_, it vibrates vigorously, but even those nearest to it do not hear
any sound. The agitation imparted to the air is too inconsiderable to
affect the auditory nerve at any distance. A second wire _t t′_, Fig.
33 (next page), of the same length, thickness, and material as _s s′_,
has one of its ends attached to the wooden tray A B. This wire also
carries a weight W, of 28 lbs. Finally, passing over the bridges B B′
of the sonometer, Fig. 31, is our third wire, in every respect like
the two former, and, like them, stretched by a weight W, of 28 lbs.
When the wire _t t′_, Fig. 33, is caused to vibrate, you hear its sound
distinctly. Though one end only of the wire is connected with the tray
A B, the vibrations transmitted to it are sufficient to convert the
tray into a sounding body. Finally, when the wire of the sonometer
M N, Fig. 31, is plucked, the sound is loud and full, because the
instrument is specially constructed to take up the vibrations of the
wire.

The importance of employing proper sounding apparatus in stringed
instruments is rendered manifest by these experiments. It is not the
strings of a harp, or a lute, or a piano, or a violin, that throw the
air into sonorous vibrations. It is the large surfaces with which the
strings are associated, and the air inclosed by these surfaces. The
goodness of such instruments depends almost wholly upon the quality and
disposition of their sound-boards.[33]

[Illustration: FIG. 33.]

Take the violin as an example. It is, or ought to be, formed of wood
of the most perfect elasticity. Imperfectly elastic wood expends
the motion imparted to it in the friction of its own molecules; the
motion is converted into heat, instead of sound. The strings of the
violin pass from the “tail-piece” of the instrument over the “bridge,”
being thence carried to the “pegs,” the turning of which regulates
the tension of the strings. The bow is drawn across at a point about
one-tenth of the length of the string from the bridge. The two “feet”
of the bridge rest upon the most yielding portion of the “belly” of
the violin, that is, the portion that lies between the two _f_-shaped
orifices. One foot is fixed over a short rod, the “sound post,” which
runs from belly to back through the interior of the violin. This foot
of the bridge is thereby rendered rigid, and it is mainly through
the other foot, which is not thus supported, that the vibrations are
conveyed to the wood of the instrument, and thence to the air within
and without. The sonorous quality of the wood of a violin is mellowed
by age. The very act of playing also has a beneficial influence,
apparently constraining the molecules of the wood, which in the first
instance might be refractory, to conform at last to the requirements of
the vibrating strings.

[Illustration: FIG. 34.]

This is the place to make the promised reference (page 38) to Prof.
Stokes’s explanation of the action of sound-boards. Although the
amplitude of the vibrating board may be very small, still its larger
area renders the abolition of the condensations and rarefactions
difficult. The air cannot move away in front nor slip in behind before
it is sensibly condensed and rarefied. Hence with such vibrating bodies
sound-waves may be generated, and loud tones produced, while the thin
strings that set them in vibration, acting alone, are quite inaudible.

The increase of sound, produced by the stoppage of lateral motion, has
been experimentally illustrated by Prof. Stokes. Let the two black
rectangles in Fig. 34 represent the section of a tuning-fork. After
it has been made to vibrate, place a sheet of paper, or the blade of
a broad knife, with its edge parallel to the axis of the fork, and as
near to the fork as may be without touching. If the obstacle be so
placed that the section of it is A or B, no effect is produced; but
if it be placed at C, so as to prevent the reciprocating to-and-fro
movement of the air, which tends to abolish the condensations and
rarefactions, the sound becomes much stronger.


§ 2. _Laws of Vibrating Strings_

Having thus learned how the vibrations of strings are rendered
available in music, we have next to investigate the laws of such
vibrations. I pluck at its middle point the string B B′, Fig. 31. The
sound heard is the fundamental or lowest note of the string, to produce
which it swings, as a whole, to and fro. By placing a movable bridge
under the middle of the string, and pressing the string against the
bridge, it is divided into two equal parts. Plucking either of those
at its centre, a musical note is obtained, which many of you recognize
as the octave of the fundamental note. In all cases, and with all
instruments, the octave of a note is produced by doubling the number of
its vibrations. It can, moreover, be proved, both by theory and by the
siren, that this half string vibrates with exactly twice the rapidity
of the whole. In the same way it can be proved that one-third of the
string vibrates with three times the rapidity, producing a note a fifth
above the octave, while one-fourth of the string vibrates with four
times the rapidity, producing the double octave of the whole string. In
general terms, _the number of vibrations is inversely proportional to
the length of the string_.

Again, the more tightly a string is stretched the more rapid is its
vibration. When this comparatively slack string is caused to vibrate,
you hear its low fundamental note. By turning a peg, round which one
end of it is coiled, the string is tightened, and the pitch rendered
higher. Taking hold with my left hand of the weight w, attached to the
wire B B′ of our sonometer, and plucking the wire with the fingers of
my right, I alternately press upon the weight and lift it. The quick
variations of tension are expressed by a varying wailing tone. Now,
the number of vibrations executed in the unit of time bears a definite
relation to the stretching force. Applying different weights to the end
of the wire B B′, and determining in each case the number of vibrations
executed in a second, we find the numbers thus obtained to be
_proportional to the square roots of the stretching weights_. A string,
for example, stretched by a weight of one pound, executes a certain
number of vibrations per second; if we wish to double this number, we
must stretch it by a weight of four pounds; if we wish to treble the
number, we must apply a weight of nine pounds, and so on.

The vibrations of a string also depend upon its thickness. Preserving
the stretching weight, the length, and the material of the string
constant, _the number of vibrations varies inversely as the thickness
of the string_. If, therefore, of two strings of the same material,
equally long and equally stretched, the one has twice the diameter
of the other, the thinner string will execute double the number of
vibrations of its fellow in the same time. If one string be three times
as thick as another, the latter will execute three times the number of
vibrations, and so on.

Finally, the vibrations of a string depend upon the density of the
matter of which it is composed. A platinum wire and an iron wire,
for example, of the same length and thickness, stretched by the same
weight, will not vibrate with the same rapidity. For, while the
specific gravity of iron, or in other words its density, is 7·8, that
of platinum is 21·5. All other conditions remaining the same, _the
number of vibrations is inversely proportional to the square root of
the density of the string_. If the density of one string, therefore,
be one-fourth that of another of the same length, thickness, and
tension, it will execute its vibrations twice as rapidly; if its
density be one-ninth that of the other, it will vibrate with three
times the rapidity, and so on. The last two laws, taken together, may
be expressed thus: _The number of vibrations is inversely proportional
to the square root of the weight of the string_.

In the violin and other stringed instruments we avail ourselves of
thickness instead of length to obtain the deeper tones. In the piano
we not only augment the thickness of the wires intended to produce
the bass notes, but we load them by coiling round them an extraneous
substance. They resemble horses heavily jockeyed, and move more slowly
on account of the greater weight imposed upon the force of tension.


§ 3. _Mechanical Illustrations of Vibrations. Progressive and
Stationary Waves. Ventral Segments and Nodes_

These, then, are the four laws which regulate the _transverse_
vibrations of strings. We now turn to certain allied phenomena, which,
though they involve mechanical considerations of a rather complicated
kind, may be completely mastered by an average amount of attention.
And they _must_ be mastered if we would thoroughly comprehend the
philosophy of stringed instruments.

[Illustration: FIG. 35.]

[Illustration: FIG. 36.]

From the ceiling _c_, Fig. 35, of this room hangs an India-rubber
tube twenty-eight feet long. The tube is filled with sand to render
its motions slow and more easily followed by the eye. I take hold of
its free end _a_, stretch the tube a little, and by properly timing
my impulses cause it to swing to and fro as a whole, as shown in
the figure. It has its definite period of vibration dependent on
its length, weight, thickness, and tension, and my impulses must
synchronize with that period.

I now stop the motion, and by a sudden jerk raise a hump upon the tube,
which runs along it as a pulse toward its fixed end; here the hump
reverses itself, and runs back to my hand. At the fixed end of the
tube, in obedience to the law of reflection, the pulse reversed both
its position and the direction of its motion. Supposing _c_, Fig. 36,
to be the fixed end of the tube, and _a_ the end held in the hand:
if the pulse on reaching _c_ have the position shown in (1), after
reflection it will have the position shown in (2). The arrows mark
the direction of progression. The time required for the pulse to pass
from the hand to the fixed end and back is exactly that required to
accomplish one complete vibration of the tube as a whole. It is indeed
the addition of such impulses which causes the tube to continue to
vibrate as a whole.

[Illustration: FIG. 37.]

If, instead of a single jerk, a succession of jerks be imparted,
thereby sending a series of pulses along the tube, every one of them
will be reflected above, and we have now to inquire how the direct and
reflected pulses behave toward each other.

Let the time required by the pulse to pass from my hand to the fixed
end be one second; at the end of half a second it occupies the position
_a b_ (1), Fig. 37, its foremost point having reached the middle of
the tube. At the end of a whole second it would have the position _b
c_ (2), its foremost point having reached the fixed end _c_ of the
tube. At the moment when reflection begins at _c_, let another jerk
be imparted at _a_. The reflected pulse from _c_ moving with the same
velocity as this direct one from _a_, the foremost points of both will
arrive at the centre _b_ (3) at the same moment. What must occur? The
hump _a b_ wishes to move on to _c_, and to do so must move the point
_b_ to the right. The hump _c b_ wishes to move toward _a_, and to do
so must move the point _b_ to the left. The point _b_, urged by equal
forces in two opposite directions at the same time, will not move in
either direction. Under these circumstances, the two halves, _a b_,
_b c_ of the tube will oscillate as if they were independent of each
other (4). Thus by the combination of two _progressive pulses_, the one
direct and the other reflected, we produce two _stationary pulses_ on
the tube _a c_.

The vibrating parts _a b_ and _b c_ are called _ventral segments_; the
point of no vibration _b_ is called a _node_.

The term “pulse” is here used advisedly, instead of the more usual term
_wave_. For a wave embraces two of these pulses. It embraces both the
hump and the depression which follows the hump. The length of a wave,
therefore, is twice that of a ventral segment.

[Illustration: FIG. 38.]

Supposing the jerks to be so timed as to cause each hump to be
one-third of the tube’s length. At the end of one-third of a second
from starting the pulse will be in the position _a b_ (1), Fig. 38. In
two-thirds of a second it will have reached the position _b b′_ (2),
Fig. 38. At this moment let a new pulse be started at _a_; after the
lapse of an entire second from the commencement we shall have two humps
upon the tube, one occupying the position _a b_ (3), the other the
position _b′ c_ (3). It is here manifest that the end of the reflected
pulse from _c_, and the end of the direct one from _a_, will reach
the point _b′_ at the same moment. We shall therefore have the state
of things represented in (4), where _b b′_ wishes to move upward, and
_c b′_ to move downward. The action of both upon the point _b′_ being
in opposite directions, that point will remain fixed. _And from it,
as if it were a fixed point, the pulse b b′ will be reflected, while
the segment b′ c will oscillate as an independent string._ Supposing
that at the moment _b b′_ (4) begins to be reflected at _b′_ we start
another pulse from _a_, it will reach _b_ at the same moment the pulse
reflected from _b′_ reaches it. The pulses will neutralize each other
at _b_, and we shall have there a second node. Thus, by properly timing
our jerks, we divide the rope into three ventral segments, separated
from each other by two nodal points. As long as the agitation continues
the tube will vibrate as in (6).

There is no theoretic limit to the number of nodes and ventral
segments that may be thus produced. By the quickening of the impulses,
the tube is divided into four ventral segments separated by three
nodes; quickening still more we have five ventral segments and four
nodes. With this particular tube the hand may be caused to vibrate
sufficiently quick to produce ten ventral segments, as shown in Fig.
38 (7). When the stretching force is constant, the number of ventral
segments is proportional to the rapidity of the hand’s vibration. To
produce 2, 3, 4, 10 ventral segments requires twice, three times, four
times, ten times the rapidity of vibration necessary to make the tube
swing as a whole. When the vibration is very rapid the ventral segments
appear like a series of shadowy spindles, separated from each other by
dark motionless nodes. The experiment is a beautiful one, and it is
easily performed.

If, instead of moving the hand to-and fro, it be caused to describe a
small circle, the ventral segments become “surfaces of revolution.”
Instead of the hand, moreover, we may employ a hook turned by a
whirling-table. Before you is a cord more rigid than the India-rubber
tube, 25 feet long, with one of its ends attached to a freely-moving
swivel fixed in the ceiling of the room. By turning the whirling-table
to which the other end is attached, this cord may be divided into
as many as 20 ventral segments, separated from each other by their
appropriate nodes. In another arrangement a string of catgut 12 feet
long, with silvered beads strung along it, is stretched horizontally
between a vertical wheel and a free swivel fixed in a rigid stand. On
turning the wheel, and properly regulating both the tension and the
rapidity of rotation, the beaded cord may be caused to rotate as a
whole, and to divide itself successively into 2, 3, 4, or 5 ventral
segments. When we envelop the cord in a luminous beam, every spot of
light on every bead describes a brilliant circle, and a very beautiful
experiment is the result.


§ 4. _Mechanical Illustrations of Damping Various Points of Vibrating
Cord_

The subject of _stationary waves_ was first experimentally treated by
the Messrs. Weber, in their excellent researches on wave-motion. It is
a subject which will well repay your attention by rendering many of the
most difficult phenomena of musical strings perfectly intelligible. It
will make the connection of both classes of vibrations more obvious if
we vary our last experiments. Before you is a piece of India-rubber
tubing, 10 or 12 feet long, stretched from _c_ to _a_, Fig. 39, and
made fast to two pins at _c_ and _a_. The tube is blackened, and behind
it is placed a surface of white paper, to render its motions more
visible. Encircling the tube at its centre _b_ (1) by the thumb and
forefinger of my left hand, and taking the middle of the lower half
_b a_ of the tube in my right, I pluck it aside. Not only does the
lower half swing, but the upper half also is thrown into vibration.
Withdrawing the hands wholly from the tube, its two halves _a b_ and _b
c_ continue to vibrate, being separated from each other by a node _b_
at the centre (2).

[Illustration: FIG. 39.]

I now encircle the tube at a point _b_ (3) one-third of its length
from its lower end _a_, and, taking hold of _a b_ at its centre, pluck
it aside; the length _b c_ above my hand instantly divides into two
vibrating segments. Withdrawing the hands wholly, you see the entire
tube divided into three ventral segments, separated from each other
by two motionless nodes, _b_ and _b′_ (4). I pass on to the point _b_
(5), which marks off one-fourth of the length of the tube, encircle
it, and pluck the shorter segment aside. The longer segment above my
hand divides itself immediately into three vibrating parts. So that, on
withdrawing the hand, the whole tube appears before you divided into
four ventral segments, separated from each other by three nodes _b b′
b″_ (6). In precisely the same way the tube may be divided into five
vibrating segments with four nodes.

This sudden division of the long upper segment of the tube, without any
apparent cause, is very surprising; but if you grant me your attention
for a moment, you will find that these experiments are essentially
similar to those which illustrated the coalescence of direct and
reflected undulations. Reverting for a moment to the latter, you
observed that the to-and-fro motion of the hand through the space of
a single inch was sufficient to make the middle points of the ventral
segments vibrate through a foot or eighteen inches. By being properly
timed the impulses accumulated, until the amplitude of the vibrating
segments exceeded immensely that of the hand which produced them. The
hand, in fact, constituted a nodal point, so small was its comparative
motion. Indeed, it is usual, and correct, to regard the ends of the
tube also as nodal points.

Consider now the case represented in (1), Fig. 39, where the tube was
encircled at its middle, the lower segment _a b_ being thrown into the
vibration corresponding to its length and tension. The circle formed
by the finger and thumb permitted the tube to oscillate at the point
_b_ through the space of an inch; and the vibrations at that point
acted upon the upper half _b c_ exactly as my hand acted when it caused
the tube suspended from the ceiling to swing as a whole, as in Fig.
35. Instead of the timid vibrations of the hand, we have now the timid
vibrations of the lower half of the tube; and these, though narrowed to
an inch at the place clasped by the finger and thumb, soon accumulate,
and finally produce an amplitude, in the upper half, far exceeding
their own. The same reasoning applies to all the other cases of
subdivision. If, instead of encircling a point by the finger and thumb,
and plucking the portion of the tube below it aside, that same point
were taken hold of by the hand and agitated in the period proper to the
lower segment of the tube, precisely the same effect would be produced.
We thus reduce both effects to one and the same cause; namely, the
combination of direct and reflected undulations.

And here let me add that, when the tube was divided by the timid
impulses of the hand, not one of its nodes was, strictly speaking, a
point of no motion; for were the nodes not capable of vibrating through
a very small amplitude, the motion of the various segments of the tube
could not be maintained.


§ 5. _Stationary Water-waves_

What is true of the undulations of an India-rubber tube applies to all
undulations whatsoever. Water-waves, for example, obey the same laws,
and the coalescence of direct and reflected waves exhibits similar
phenomena. This long and narrow vessel with glass sides, Fig. 40, is
a copy of the wave-canal of the brothers Weber. It is filled to the
level A B with colored water. By tilting the end A suddenly, a wave
is generated, which moves on to B, and is there reflected. By sending
forth fresh waves at the proper intervals, the surface is divided into
two stationary undulations. Making the succession of impulses more
rapid we can subdivide the surface into three, four (shown in the
figure), or more stationary undulations, separated from each other by
nodes. The step of a water-carrier is sometimes so timed as to throw
the surface of the water in his vessel into stationary waves, which may
augment in height until the water splashes over the brim. Practice has
taught the water-carrier what to do; he changes his step, alters the
period of his impulses, and thus stops the accumulation of the motion.

[Illustration: FIG. 40.]

In travelling recently in the coupé of a French railway carriage, I had
occasion to place a bottle half filled with water on one of the little
coupé tables. It was interesting to observe it. At times it would be
quite still; at times it would oscillate violently. To the passenger
within the carriage there was no sensible change in the motion of the
train to which the difference could be ascribed. But in the one case
the tremor of the carriage contained no vibrations synchronous with
the oscillating period of the water, while in the other case such
vibrations were present. Out of the confused assemblage of tremors the
water selected the particular constituent which belonged to itself,
and declared its presence when the traveller was utterly unconscious of
its introduction.


§ 6. _Application of Mechanical Illustrations to Musical Strings_

From these comparatively gross, but by no means unbeautiful, mechanical
vibrations, we pass to those of a sounding string. In the experiments
with our monochord, when the wire was to be shortened, a movable bridge
was employed, against which the wire was pressed so as to deprive the
point resting on the bridge of all possibility of motion. This strong
pressure, however, is not necessary. Placing the feather-end of a
goose-quill lightly against the middle of the string, and drawing a
violin-bow over one of its halves, the string yields the octave of the
note yielded by the whole string. The mere _damping_ of the string at
the centre, by the light touch of the feather, is sufficient to cause
the string to divide into two vibrating segments. Nor is it necessary
to hold the feather there throughout the experiment: after having
drawn the bow, the feather may be removed; the string will continue
to vibrate, emitting the same note as before. We have here a case
exactly analogous to that in which the central point of our stretched
India-rubber tube was damped, by encircling it with the finger and
thumb as in Fig. 39 (1). Not only did the half plucked aside vibrate,
but the other half vibrated also. We can, in fact, reproduce, with
the vibrating string, every effect obtained with the tube. This,
however, is a point of such importance as to demand full experimental
illustration.

To prove that when the centre is damped, and the bow drawn across one
of the halves of the string, the other half vibrates, I place across
the middle of the untouched half a little rider of red paper. Damping
the centre and drawing the bow, the string shivers, and the rider is
overthrown, Fig. 41.

[Illustration: FIG. 41.]

[Illustration: FIG. 42.]

When the string is damped at a point which cuts off one-third of its
length, and the bow drawn across the shorter section, not only is this
section thereby thrown into vibration, but the longer section divides
itself into two ventral segments with a node between them. This is
proved by placing small riders of red paper on the ventral segments,
and a rider of blue paper at the node. Passing the bow across the short
segment you observe a fluttering of the red riders, and now they are
completely tossed off, while the blue rider which crosses the node is
undisturbed, Fig. 42.

[Illustration: FIG. 43.]

Damping the string at the end of one-fourth of its length, the bow is
drawn across the shorter section; the remaining three-fourths divide
themselves into three ventral segments, with two nodes between them.
This is proved by the unhorsing of the three riders placed astride
the ventral segments, the two at the nodes keeping their places
undisturbed, Fig. 43.

[Illustration: FIG. 44.]

Finally, damping the string at the end of one-fifth of its length,
and arranging, as before, the red riders on the ventral segments and
the blue ones on the nodes, by a single sweep of the bow the four red
riders are unhorsed, and the three blue ones left undisturbed, Fig.
44. In this way we perform with a sounding string the same series of
experiments that were formerly executed with a stretched India-rubber
tube, the results in both cases being identical.[34]

To make, if possible, this identity still more evident to you, a stout
steel wire 28 feet in length is stretched behind the table from side
to side of the room. I take the central point of this wire between
my finger and thumb, and allow my assistant to pluck one-half of it
aside. It vibrates, and the vibrations transmitted to the other half
are sufficiently powerful to toss into the air a large sheet of paper
placed astride the wire. With this long wire, and with riders not of
one-eighth of a square inch, but of 30, 40, or 50 square inches in
area, we may repeat all the experiments which you have witnessed with
the musical string. The sheets of paper placed across the nodes remain
always in their places, while those placed astride the ventral segments
are tossed simultaneously into the air when the shorter segment of
the wire is set in vibration. In this case, when close to it, you can
actually see the division of the wire.


§ 7. _Melde’s Experiments_

It is now time to introduce to your notice some recent experiments
on vibrating strings, which appeal to the eye with a beauty and a
delicacy far surpassing anything attainable with our monochord. To M.
Melde, of Marburg, we are indebted for this new method of exhibiting
the vibrations of strings. The scale of the experiments will be here
modified so as to suit our circumstances.

[Illustration: FIG. 45.]

First, then, you observe here a large tuning-fork T, Fig. 45, with a
small screw fixed into the top of one of its prongs, by which a silk
string can be firmly attached to the prong. From the fork the string
passes round a distant peg P, by turning which it may be stretched
to any required extent. When the bow is drawn across the fork, an
irregular flutter of the string is the only result. On tightening it,
however, when at the proper tension it expands into a beautiful gauzy
spindle six feet long, more than six inches across at its widest part,
and shining with a kind of pearly lustre. The stretching force at the
present moment is such that the string swings to and fro as a whole,
its vibrations being executed in a vertical plane.

Relaxing the string gradually, when the proper tension has been
reached, it suddenly divides into two ventral segments, separated from
each other by a sharply-defined and apparently motionless node.

While the fork continues vibrating, if the string be relaxed still
further, it divides into three vibrating parts. Slackening it still
more, it divides into four vibrating parts. And thus we might continue
to subdivide the string into ten, or even twenty ventral segments,
separated from each other by the appropriate number of nodes.

[Illustration: FIG. 46.]

When white-silk strings vibrate thus, the nodes appear perfectly fixed,
while the ventral segments form spindles of the most delicate beauty.
Every protuberance of the twisted string, moreover, writes its motion
in a more or less luminous line on the surface of the aërial gauze. The
four nodes of vibration just illustrated are represented in Fig. 46, 1,
2, 3, 4.[35]

When the synchronism between fork and string is perfect, the vibrations
of the string are steady and long-continued. A slight departure
from synchronism, however, introduces unsteadiness, and the ventral
segments, though they may show themselves for a time, quickly disappear.

[Illustration: FIG. 47.]

In the experiments just executed the fork vibrated in the direction of
the length of the string. Every forward stroke of the fork raised a
protuberance, which ran to the fixed end of the string, and was there
reflected; so that when the _longitudinal_ impulses were properly timed
they produced a _transverse_ vibration. Let us consider this further.
One end of this heavy cord is attached to a hook A, Fig. 47, fixed in
the wall. Laying hold of the other end I stretch the cord horizontally,
and then move my hand to and fro in the direction of the cord. It
swings as a whole, and you may notice that always, when the cord is
at the limit of its swing, the hand is in its most forward position.
If it vibrate in a vertical plane, the hand, in order to time the
impulses properly, must be at its forward limit at the moment the cord
reaches the upper boundary, and also at the moment it reaches the lower
boundary of its excursion. A little reflection will make it plain
that, in order to accomplish this, the hand must execute a complete
vibration while the cord executes a semi-vibration; in other words, the
vibrations of the hand must be twice as rapid as those of the cord.

Precisely the same is true of our tuning-fork. When the fork vibrates
in the direction of the string, the number of vibrations which it
executes in a certain time is twice the number executed by the string
itself. And if, while arranged thus, a fork and string vibrate with
sufficient rapidity to produce musical notes, the note of the fork will
be an octave above that of the string.

But if, instead of the hand being moved to and fro in the direction of
this heavy cord, it is moved at right angles to that direction, then
every upward movement of the hand coincides with an upward movement of
the cord; every downward movement of the hand with a downward movement
of the cord. In fact, the vibrations of hand and string, in this case,
synchronize perfectly; and if the hand could emit a musical note, the
cord would, emit a note of the same pitch. The same holds good when a
vibrating fork is substituted for the vibrating hand.

Hence, if the string vibrate as a whole when the vibrations of the
fork are _along_ it, it will divide into two ventral segments when the
vibrations are _across_ it; or, more generally expressed, preserving
the tension constant, whatever be the number of ventral segments
produced by the fork when its vibrations are in the direction of the
string, twice that number will be produced when the vibrations are
transverse to the string. The string A B, for example, Figs. 48 and
49, passing over a pulley B, is stretched by a definite weight (not
shown in the figure). When the tuning-fork vibrates _along_ it, as in
Fig. 48, the string divides into two equal ventral segments. When the
fork is turned so that it shall vibrate at right angles to the string,
the number of ventral segments is four, Fig. 49, or double the former
number. Attaching two strings of the same length to the same fork, the
one parallel and the other perpendicular to the direction of vibration,
and stretching both with equal weights, when the fork is caused to
vibrate, one of them divides itself into twice the number of ventral
segments exhibited by the other.

[Illustration: FIG. 48.]

[Illustration: FIG. 49.]

A number of exquisite effects may be obtained with these vibrating
cords. The path described by any point of any one of them may be
studied, after the manner of Dr. Young, by illuminating that point,
and watching the line of light which it describes. This is well
illustrated by a flat burnished silver wire, twisted so as to form a
spiral surface, from which, at regular intervals, the light flashes
when the wire is illuminated. When the vibration is steady, the
luminous spots describe straight lines of sunlike brilliancy. On
slackening the wire, but not so much as to produce its next higher
subdivision, upon the larger motion of the wire are superposed a
host of minor motions, the combination of all producing scrolls of
marvellous complication and of indescribable splendor.

In reflecting on the best means of rendering these effects visible,
the thought occurred to me of employing a fine platinum wire heated
to redness by an electric current. Such a wire now stretches from a
tuning-fork over a bridge of copper, and then passes round a peg. The
copper bridge on the one hand and the tuning-fork on the other are the
poles of a voltaic battery, from which a current passes through the
wire and causes it to glow. On drawing the bow across the fork, the
wire vibrates as a whole; its two ends are brilliant, while its middle
is dark, being chilled by its rapid passage through the air. Thus you
have a shading off of incandescence from the ends to the centre of the
wire. On relaxing the tension, the wire divides itself into two ventral
segments; on relaxing still further, we obtain three; still further,
and the wire divides into four ventral segments, separated from each
other by three brilliant nodes. Right and left from every node the
incandescence shades away until it disappears. You notice also, when
the wire settles into steady vibration, that the nodes shine out with
greater brilliancy than that possessed by the wire before the vibration
commenced. The reason is this. Electricity passes more freely along a
cold wire than along a hot one. When, therefore, the vibrating segments
are chilled by their swift passage through the air, their conductivity
is improved, more electricity passes through the vibrating than through
the motionless wire, and hence the augmented glow of the nodes. If,
previous to the agitation of the fork, the wire be at a bright-red
heat, when it vibrates its nodes may be raised to the temperature of
fusion.


§ 8. _New Mode of determining the Laws of Vibration_

We may extend the experiments of M. Melde to the establishment of all
the laws of vibrating strings. Here are four tuning-forks, which we may
call _a_, _b_, _c_, _d_, whose rates of vibration are to each other as
the numbers 1, 2, 4, 8. To the largest fork is attached a string, _a_,
stretched by a weight, which causes it to vibrate as a whole. Keeping
the stretching weight the same, I determine the lengths of the same
string, which, when attached to the other three forks, _b_, _c_, _d_,
swing as a whole. The lengths in the four respective cases are as the
numbers 8, 4, 2, 1.

From this follows the first law of vibration, already established (p.
126) by another method; viz., _the length of the string is inversely
proportional to the rapidity of vibration_.[36]

In this case the longest string vibrates as a whole when attached to
the fork _a_. I now transfer the string to _b_, still keeping it
stretched by the same weight. It vibrates when _b_ vibrates; but how?
By dividing into two equal ventral segments. In this way alone can it
accommodate itself to the swifter vibrating period of _b_. Attached
to _c_, the same string separates into four, while when attached
to _d_, it divides into eight ventral segments. The number of the
ventral segments is proportional to the rapidity of vibration. It is
evident that we have here, in a more delicate form, a result which we
have already established in the case of our India-rubber tube set in
motion by the hand. It is also plain that this result might be deduced
theoretically from our first law.

We may extend the experiment. Here are two tuning-forks separated from
each other by the musical interval called a fifth. Attaching a string
to one of the forks, I stretch the string until it divides into two
ventral segments: attached to the other fork, and stretched by the
same weight, it divides instantly into three segments when the fork is
set in vibration. Now, to form the interval of a fifth, the vibrations
of the one fork must be to those of the other in the ratio of 2:3.
The division of the string, therefore, declares the interval. In, the
same way the division of the string in relation to all other musical
intervals may be illustrated.[37]

Again. Here are two tuning-forks, _a_ and _b_, one of which (_a_)
vibrates twice as rapidly as the other. A string of silk is attached to
_a_, and stretched until it synchronizes with the fork, and vibrates as
a whole. Here is a second string of the same length, formed by laying
four strands of the first one side by side. I attach this compound
thread to _b_, and, keeping the tension the same as in the last
experiment, set _b_ in vibration. The compound thread synchronizes with
_b_, and swings as a whole. Hence, as the fork _b_ vibrates with half
the rapidity of _a_, by quadrupling the weight of the string we halved
its rapidity of vibration. In the same simple way it might be proved
that by augmenting the weight of the string nine times we reduce the
number of its vibrations to one-third. We thus demonstrate the law:

_The rapidity of vibration is inversely proportional to the square root
of the weight of the string._

An instructive confirmation of this result is thus obtained: Attached
to this tuning-fork is a silk string six feet long. Two feet of the
string are composed of four strands of the single thread, placed side
by side; the remaining four feet are a single thread. A stretching
force is applied, which causes the string to divide into two ventral
segments. But how does it divide? Not at its centre, as is the case
when the string is of uniform thickness throughout, but at the precise
point where the thick string terminates. This thick segment, two feet
long, is now vibrating at the same rate as the thin segment four feet
long, a result which follows by direct deduction from the two laws
already established.

Here again are two strings of the same length and thickness. One of
them is attached to the fork _a_, the other to the fork _b_, which
vibrates with twice the rapidity of _a_. Stretched by a weight of 20
grains, the string attached to _b_ vibrates as a whole. Substituting
_b_ for _a_, a weight of 80 grains causes the string to vibrate as a
whole. Hence, to double the rapidity of vibration, we must quadruple
the stretching weight. In the same way it might be proved that to
treble the rapidity of vibration we should have to make the stretching
weight ninefold. Hence our third law:

_The rapidity of vibration is proportional to the square root of the
tension._

[Illustration: FIG. 50.]

Let us vary this experiment. This silk cord is carried from the
tuning-fork over the pulley, and stretched by a weight of 80 grains.
The string vibrates as a whole as at A, Fig. 50. By diminishing the
weight the string is relaxed, and finally divides sharply into two
ventral segments, as at B, Fig. 50. What is now the stretching
weight?—20 grains, or one-fourth of the first. With a stretching
weight of almost exactly 9 grains it divides into three segments,
as at C; while with a stretching weight of 5 grains it divides into
four segments, as at D. Thus then, a tension of one-fourth doubles, a
tension of one-ninth trebles, and a tension of one-sixteenth quadruples
the number of ventral segments. In general terms, the number of
segments is inversely proportional to the square root of the tension.
This result may be deduced by reasoning from our first and third laws,
and its realization here confirms their correctness.

Thus, by a series of reasonings and experiments totally different from
those formerly employed, we arrive at the self-same laws. In science,
different lines of reasoning often converge upon the same truth; and
if we only follow them faithfully, we are sure to reach that truth at
last. We may emerge, and often do emerge, from our reasoning with a
contradiction in our hands; but on retracing our steps, we infallibly
find the cause of the contradiction to be due, not to any lack of
constancy in Nature, but of accuracy in man. It is the millions of
experiences of this kind which science furnishes that give us our
present faith in the stability of Nature.


HARMONIC SOUNDS OR OVERTONES


§ 9. _Timbre; Klangfarbe; Clang-tint_

We now approach a portion of our subject which will subsequently
prove to be of the very highest importance. It has been shown by the
most varied experiments that a stretched string can either vibrate
as a whole, or divide itself into a number of equal parts, each of
which vibrates as an independent string. Now it is not possible to
sound the string as a whole without at the same time causing, to a
greater or less extent, its subdivision; that is to say, superposed
upon the vibrations of the whole string we have always, in a greater
or less degree, the vibrations of its aliquot parts. The higher notes
produced by these latter vibrations are called the _harmonics_ of the
string. And so it is with other sounding bodies; we have in all cases
a coexistence of vibrations. Higher tones mingle with the fundamental
one, and it is their intermixture which determines what, for want of
a better term, we call the _quality_ of the sound. The French call it
_timbre_, and the Germans call it _Klangfarbe_.[38] It is this union
of high and low tones that enables us to distinguish one musical
instrument from another. A clarinet and a violin, for example, though
tuned to the same fundamental note, are not confounded; the auxiliary
tones of the one are different from those of the other, and these
latter tones, uniting themselves to the fundamental tones of the two
instruments, destroy the identity of the sounds.

All bodies and instruments, then, employed for producing musical sounds
emit, besides their fundamental tones, others due to higher orders of
vibration. The Germans embrace all such sounds under the general term
_Obertöne_. I think it will be an advantage if we in England adopt the
term _overtones_ as the equivalent of the term employed in Germany. One
has occasion to envy the power of the German language to adapt itself
to requirements of this nature. The term _Klangfarbe_, for example,
employed by Helmholtz is exceedingly expressive, and we need its
equivalent also. Color depends upon rapidity of vibration, blue light
bearing to red the same relation that a high tone does to a low one. A
simple color has but one rate of vibration, and it may be regarded as
the analogue of a simple tone in music. A _tone_, then, may be defined
as the product of a vibration which cannot be decomposed into more
simple ones. A compound color, on the contrary, is produced by the
admixture of two or more simple ones, and an assemblage of tones, such
as we obtain when the fundamental tone and the harmonics of a string
sound together, is called by the Germans a _Klang_. May we not employ
the English word _clang_ to denote the same thing, and thus give the
term a precise scientific meaning akin to its popular one? And may we
not, like Helmholtz, add the word _color_ or _tint_, to denote the
character of the clang, using the term _clang-tint_ as the equivalent
of Klangfarbe?

With your permission I shall henceforth employ these terms; and now it
becomes our duty to look a little more closely than we have hitherto
done into the subdivision of a string into its harmonic segments. Our
monochord with its stretched wire is before you. The scale of the
instrument is divided into 100 equal parts. At the middle point of the
wire stands the number 50; at a point almost exactly one-third of its
length from its end stands the number 33; while at distances equal to
one-fourth and one-fifth of its length from its end stand the numbers
25 and 20 respectively. These numbers are sufficient for our present
purpose. When the wire is plucked at 50 you hear its clang, rather
hollow and dull. When plucked at 33, the clang is different. When
plucked at 25, the clang is different from either of the former. As
we retreat from the centre of the string, the clang-tint becomes more
“brilliant,” the sound more brisk and sharp. What is the reason of
these differences in the sound of the same wire?

The celebrated Thomas Young, once professor in this Institution,
enables us to solve the question. He proved that when any point of a
string is plucked, all the higher tones _which require that point for
a node_ vanish from the clang. Let me illustrate this experimentally.
I pluck the point 50, and permit the string to sound. It may be
proved that the first overtone, which corresponds to a division of
the string into two vibrating parts, is now absent from the clang.
If it were present, the damping of the point 50 would not interfere
with it, for this point would be its node. But on damping the point
50 the fundamental tone is quenched, and no octave of that tone is
heard. Along with the octave its whole progeny of overtones, with rates
of vibration four times, six times, eight times—all even numbers of
times—the rate of the fundamental tone, disappear from the clang. All
these tones require that a node should exist at the centre, where,
according to the principle of Young, it cannot now be formed. Let us
pluck some other point, say 25, and damp 50 as before. The fundamental
tone is now gone, but its octave, clear and full, rings in your ears.
The point 50 in this case not being the one plucked, a node can form
there; it _has_ formed, and the two halves of the string continue
to vibrate after the vibrations of the string as a whole have been
extinguished. Plucking the point 33, the second harmonic or overtone
is absent from the clang. This is proved by damping the point 33. If
the second harmonic were on the string this would not affect it, for 33
is its node. The fundamental is quenched, but no tone corresponding to
a division of the string into three vibrating parts is now heard. The
tone is not heard because it was never there.

All the overtones which depend on this division, those with six times,
nine times, twelve times the rate of vibration of the fundamental one,
are also withdrawn from the clang. Let us now pluck 20, damping 33 as
before. The second harmonic is not extinguished, but continues to sound
clearly and fully after the extinction of the fundamental tone. In
this case the point 33 not being that plucked, a node can form there,
and the string can divide itself into three parts accordingly. In like
manner, if 25 be plucked and then damped, the third harmonic is not
heard; but when a point between 25 and the end of the wire is plucked,
and the point 25 damped, the third harmonic is plainly heard. And thus
we might proceed, the general rule enunciated by Young, and illustrated
by these experiments, being, that when any point of a string is plucked
or struck, or, as Helmholtz adds, agitated with a bow, the harmonic
which requires that point for a node vanishes from the general clang of
the string.


§ 10. _Mingling of Overtones with Fundamental. The Æolian Harp_

You are now in a condition to estimate the influence which these
higher vibrations must have upon the quality of the tone emitted by
the string. The sounds which ring in your ears so plainly after the
fundamental tone is quenched mingled with that note before it was
extinguished. It seems strange that tones of such power could be
so masked by the fundamental one that even the disciplined ear of a
musician is unable to separate the one from the other. But Helmholtz
has shown that this is due to want of practice and attention. The
musician’s faculties were never exercised in this direction. There
are numerous effects which the musician can distinguish, because his
art demands the habit of distinguishing them. But it is no necessity
of his art to resolve the clang of an instrument into its constituent
tones. By attention, however, even the unaided ear can accomplish this,
particularly if the mind be informed beforehand what the ear has to
bend itself to find.

And this reminds me of an occurrence which took place in this room at
the beginning of my acquaintance with Faraday. I wished to show him a
peculiar action of an electro-magnet upon a crystal. Everything was
arranged, when just before the magnet was excited he laid his hand
upon my arm and asked, “What am I to look for?” Amid the assemblage
of impressions connected with an experiment, even this prince of
experimenters felt the advantage of having his attention directed to
the special point to be illustrated. Such help is the more needed
when we attempt to resolve into its constituent parts an effect so
intimately blended as the composite tones of a clang. When we desire
to isolate a particular tone, one way of helping the attention is to
sound that tone feebly on a string of the proper length. Thus prepared,
the ear glides more readily from the single tone to that of the same
pitch in a composite clang, and detaches it more readily from its
companions. In the experiments executed a moment ago, where our aim
in each respective case was to bring out the higher tone of the string
in all its power, we entirely extinguished its fundamental tone. It
may, however, be enfeebled without being destroyed. I pluck this string
at 33, and lay the feather lightly for a moment on the string at 50.
The fundamental tone is thereby so much lowered that its octave can
make itself plainly heard. By again touching the string at 50, the
fundamental tone is lowered still more; so that now its first harmonic
is more powerful than itself. You hear the sound of both, and you
might have heard them in the first instance by a sufficient stretch of
attention.

The harmonics of a string may be augmented or subdued within wide
limits. They may, as we have seen, be masked by the fundamental
tone, and they may also effectually mask it. A stroke with a hard
body is favorable, while a stroke with a soft body is unfavorable
to their development. They depend, moreover, on the promptness with
which the body striking the string retreats after striking. Thus they
are influenced by the weight and elasticity of the hammers in the
pianoforte. They also depend upon the place at which the shock is
imparted. When, for example, a string is struck in the centre, the
harmonics are less powerful than when it is struck near one end.

Helmholtz, who is equally eminent as a mathematician and as an
experimental philosopher, has calculated the theoretic intensity of
the harmonics developed in various ways; that is to say, the actual
_vis viva_ or energy of the vibration, irrespective of its effects
upon the ear. A single example given by him will suffice to illustrate
this subject. Calling the intensity of the fundamental tone, in each
case, 100, that of the second harmonic, when the string was simply
pulled aside at a point one-seventh of its length from its end and then
liberated, was found to be 56·1, or a little better than one-half. When
the string was struck with the hammer of a pianoforte, whose contact
with the string endured for three-sevenths of the period of vibration
of the fundamental tone, the intensity of the same tone was 9. In
this case the second harmonic was nearly quenched. When, however, the
duration of contact was diminished to three-twentieths of the period of
the fundamental, the intensity of the harmonic rose to 357; while, when
the string was sharply struck with a very hard hammer, the intensity
mounted to 505, or to more than quintuple that of the fundamental
tone.[39] Pianoforte manufacturers have found that the most pleasing
tone is excited by the middle strings of their instruments, when the
point against which the hammer strikes is from one-seventh to one-ninth
of the length of the wire from its extremity.

Why should this be the case? Helmholtz has given the answer. Up to
the tones which require these points as nodes the overtones all form
chords with the fundamental; but the sixth and eighth overtones of
the wire do not enter into such chords; they are dissonant tones, and
hence the desirability of doing away with them. This is accomplished by
making the point at which a node is required that on which the hammer
falls. The possibility of the tone forming is thereby shut out, and its
injurious effect is avoided.

The strings of the Æolian harp are divided into harmonic parts by a
current of air passing over them. The instrument is usually placed in
a window between the sash and frame, so as to leave no way open to the
entrance of the air except over the strings. Sir Charles Wheatstone
recommends the stretching of a first violin-string at the bottom of a
door which does not closely fit. When the door is shut, the current
of air entering beneath sets the string in vibration, and when a fire
is in the room, the vibrations are so intense that a great variety
of sounds are simultaneously produced.[40] A gentleman in Basel once
constructed with iron wires a large instrument which he called the
weather-harp or giant-harp, and which, according to its maker, sounded
as the weather changed. Its sounds were also said to be evoked by
changes of terrestrial magnetism. Chladni pointed out the error of
these notions, and reduced the action of the instrument to that of the
wind upon its strings.


§ 11. _Young’s Optical Illustrations_

Finally, with regard to the vibrations of a wire, the experiments
of Dr. Young, who was the first to employ optical methods in such
experiments, must be mentioned. He allowed a sheet of sunlight to
cross a pianoforte-wire, and obtained thus a brilliant dot. Striking
the wire he caused it to vibrate, the dot described a luminous line
like that produced by the whirling of a burning coal in the air, and
the form of this line revealed the character of the vibration. It was
rendered manifest by these experiments that the oscillations of the
wire were not confined to a single plane, but that it described in its
vibrations curves of greater or less complexity. Superposed upon the
vibration of the whole string were partial vibrations, which revealed
themselves as loops and sinuosities. Some of the lines observed by Dr.
Young are given in Fig. 51. Every one of these figures corresponds to a
distinct impression made by the wire upon the surrounding air. The form
of the sonorous wave is affected by these superposed vibrations, and
thus they influence the clang-tint or quality of the sound.

[Illustration: FIG. 51.]


SUMMARY OF CHAPTER III

The amount of motion communicated by a vibrating string to the air is
too small to be perceived as sound, even at a small distance from the
string.

When a broad surface vibrates in air, condensations and rarefactions
are more readily formed than when the vibrating body is of small
dimensions like a string. Hence, when strings are employed as sources
of musical sounds, they are associated with surfaces of larger area
which take up their vibrations, and transfer them to the surrounding
air.

Thus the tone of a harp, a piano, a guitar, or a violin, depends mainly
upon the sound-board of the instrument.

The following four laws regulate the vibrations of strings: The rate
of vibrations is inversely proportional to the length; it is inversely
proportional to the diameter; it is directly proportional to the
square root of the stretching weight or tension; and it is inversely
proportional to the square root of the density of the string.

When strings of different diameters and densities are compared, the law
is, that the rate of vibration is inversely proportional to the square
root of the weight of the string.

When a stretched rope, or an India-rubber tube filled with sand, with
one of its ends attached to a fixed object, receives a jerk at the
other end, the protuberance raised upon the tube runs along it as a
pulse to the fixed end, and, being there reflected, returns to the hand
by which the jerk was imparted.

The time required for the pulse to travel from the hand to the fixed
end of the tube and back is that required by the whole tube to execute
a complete vibration.

When a series of pulses are sent in succession along the tube, the
direct and reflected pulses meet, and by their coalescence divide the
tube into a series of vibrating parts, called _ventral segments_, which
are separated from each other by points of apparent rest called _nodes_.

The number of ventral segments is directly proportional to the rate of
vibration at the free end of the tube.

The hand which produces these vibrations may move through less than an
inch of space; while by the accumulation of its impulses the amplitude
of the ventral segments may amount to several inches, or even to
several feet.

If an India-rubber tube, fixed at both ends, be encircled at its centre
by the finger and thumb, when either of its halves is pulled aside and
liberated, both halves are thrown into a state of vibration.

If the tube be encircled at a point one-third, one-fourth, or one-fifth
of its length from one of its ends, on pulling the shorter segment
aside and liberating it, the longer segment divides itself into two,
three, or four vibrating parts, separated from each other by nodes.

The number of vibrating segments depends upon the rate of vibration at
the point encircled by the finger and thumb.

Here also the amplitude of vibration at the place encircled by the
finger and thumb may not be more than a fraction of an inch, while the
amplitude of the ventral segments may amount to several inches.

A musical string damped by a feather at a point one-half, one-third,
one-fourth, one-fifth, etc., of its length from one of its ends, and
having its shorter segment agitated, divides itself exactly like the
India-rubber tube. Its division may be rendered apparent by placing
little paper riders across it. Those placed at the ventral segments are
thrown off, while those placed at the nodes retain their places.

The notes corresponding to the division of a string into its aliquot
parts are called the _harmonics_ of the string.

When a string vibrates as a whole, it usually divides at the same time
into its aliquot parts. Smaller vibrations are superposed upon the
larger; the tones corresponding to those smaller vibrations, and which
we have agreed to call overtones, mingling at the same time with the
fundamental tone of the string.

The addition of these overtones to the fundamental tone determines the
_timbre_ or _quality_ of the sound, or, as we have agreed to call it,
the _clang-tint_.

It is the addition of such overtones to fundamental tones of the
same pitch which enables us to distinguish the sound of a clarionet
from that of a flute, and the sound of a violin from both. Could the
pure fundamental tones of these instruments be detached, they would
be indistinguishable from each other; but the different admixture
of overtones in the different instruments renders their clang-tints
diverse, and therefore distinguishable.

Instead of the heavy India-rubber tube in the experiment above referred
to, we may employ light silk strings, and, instead of the vibrating
hand, we may employ vibrating tuning-forks, and cause the strings to
swing as a whole, or to divide themselves into any number of ventral
segments. Effects of great beauty are thus obtained, and by experiments
of this character all the laws of vibrating strings may be demonstrated.

When a stretched string is plucked aside or agitated by a bow, all the
overtones which require the agitated point for a node vanish from the
clang of the string.

The point struck by the hammer of the piano is from one-seventh to
one-ninth of the length of the string from its end: by striking this
point, the notes which require it as a node cannot be produced, a
source of dissonance being thus avoided.




CHAPTER IV

  Vibrations of a Rod fixed at Both Ends: its Subdivisions
  and Corresponding Overtones—Vibrations of a Rod fixed
  at One End—The Kaleidophone—The Iron Fiddle and Musical
  Box—Vibrations of a Rod free at Both Ends—The Claque-bois and
  Glass Harmonica—Vibrations of a Tuning-Fork: its Subdivisions
  and Overtones—Vibrations of Square Plates—Chladni’s
  Discoveries—Wheatstone’s Analysis of the Vibrations
  of Plates—Chladni’s Figures—Vibrations of Disks and
  Bells—Experiments of Faraday and Strehlke


§ 1. _Transverse Vibrations of a Rod fixed at Both Ends_

[Illustration: FIG. 52.]

Our last chapter was devoted to the transverse vibrations of strings.
This one I propose devoting to the transverse vibrations of rods,
plates, and bells, commencing with the case of a rod fixed at both
ends. Its modes of vibration are exactly those of a string. It vibrates
as a whole, and can also divide itself into two, three, four, or more
vibrating parts. But, for a reason to be immediately assigned, the laws
which regulate the pitch of the successive notes are entirely different
in the two cases. Thus, when a string divides into two equal parts,
each of its halves vibrates with twice the rapidity of the whole;
while, in the case of the rod, each of its halves vibrates with nearly
three times the rapidity of the whole. With greater strictness, the
ratio of the two rates of vibration is as 9 is to 25, or as the square
of 3 to the square of 5. In Fig. 52, _a a′_, _c c′_, _b b′_, _d d′_,
are sketched the first four modes of vibration of a rod fixed at both
ends: the successive rates of vibration, in the four cases bear to each
other the following relation:

  Number of nodes        0   1   2   3
  Number of vibrations   9  25  49  81

the last row of figures being the squares of the odd numbers 3, 5, 7, 9.

In the case of a string, the vibrations are maintained by a tension
externally applied; in the case of a rod, the vibrations are maintained
by the elasticity of the rod itself. The modes of division are in both
cases the same, but the forces brought into play are different, and
hence also the successive rates of vibration.


§ 2. _Transverse Vibrations of a Rod fixed at One End_

Let us now pass on to the case of a rod fixed at one end and free at
the other. Here also it is the elasticity of the material, and not
any external tension, that sustains the vibrations. Approaching, as
usual, sonorous vibrations through more grossly mechanical ones, I fix
this long rod of iron, _n o_, Fig. 53, in a vise, draw it aside, and
liberate it. To make its vibrations more evident, its shadow is thrown
upon a screen. The rod oscillates as a whole to and fro, between the
points _p p′_. But it is capable of other modes of vibration. Damping
it at the point _a_, by holding it gently there between the finger and
thumb, and striking it sharply between _a_ and _o_, the rod divides
into two vibrating parts, separated by a node as shown in Fig. 54.
You see upon the screen a shadowy spindle between _a_ and the vise
below, and a shadowy fan above _a_, with a black node between both. The
division may be effected without damping _a_, by merely imparting a
sufficiently sharp shock to the rod between _a_ and _o_. In this case,
however, besides oscillating in parts, the rod oscillates as a whole,
the partial oscillations being superposed upon the large one.

[Illustration: FIG. 53.]

[Illustration: FIG. 54.]

[Illustration: FIG. 55.]

You notice, moreover, that the amplitude of the partial oscillations
depends upon the promptness of the stroke. When the stroke is sluggish,
the partial division is but feebly pronounced, the whole oscillation
being most marked. But when the shock is sharp and prompt, the whole
oscillation is feeble, and the partial oscillations are executed
with vigor. If the vibrations of this rod were rapid enough to
produce a musical sound, the oscillation of the rod as a whole would
correspond to its fundamental tone, while the division of the rod into
two vibrating parts would correspond to the first of its overtones.
If, moreover, the rod vibrated as a whole and as a divided rod at
the same time, the fundamental tone and the overtone would be heard
simultaneously. By damping the proper point and imparting the proper
shock, we can still further subdivide the rod, as shown in Fig. 55.


§ 3. _Chladni’s Tonometer: the Iron Fiddle, Musical Box, and the
Kaleidophone_

And now let us shorten our rod, so as to bring its vibrations into
proper relation to our ears. When it is about four inches long,
it emits a low musical sound. When further shortened, the tone is
higher; and, by continuing to shorten the rod, the speed of vibration
is augmented, until finally the sound becomes painfully acute.
These musical vibrations differ only in rapidity from the grosser
oscillations which a moment ago appealed to the eye.

The increase in the rate of vibrations here observed is ruled by a
definite law; the number of vibrations executed at a given time is
inversely proportional to the square of the length of the vibrating
rod. You hear the sound of this strip of brass, two inches long, as
the fiddle-bow is passed over its end. Making the length of the strip
one inch, the sound is the double octave of the last one; the rate of
vibration is augmented four times. Thus, by doubling the length of the
vibrating strip, we reduce its rate of vibration to one-fourth; by
trebling the length, we reduce the rate of vibration to one-ninth; by
quadrupling the length, we reduce the vibrations to one-sixteenth, and
so on. It is plain that, by proceeding in this way, we should finally
reach a length where the vibrations would be sufficiently slow to
be counted. Or, it is plain that, beginning with a long strip whose
vibrations could be counted, we might, by shortening, not only make the
strip sound, but also determine the rates of vibration corresponding to
its different tones. Supposing we start with a strip 36 inches long,
which vibrates once in a second, the strip reduced to 12 inches would,
according to the above law, execute 9 vibrations a second; reduced to
6 inches, it would execute 36, to 3 inches, 144; while, if reduced to
1 inch in length, it would execute 1,296 vibrations in a second. It is
easy to fill the spaces between the lengths here given, and thus to
determine the rate of vibration corresponding to any particular tone.
This method was proposed and carried out by Chladni.

A musical instrument may be formed of short rods. Into this common
wooden tray a number of pieces of stout iron wire of different lengths
are fixed, being ranged in a semicircle. When the fiddle-bow is passed
over the series, we obtain a succession of very pleasing notes. A
competent performer could certainly extract very tolerable music from a
sufficient number of these iron pins. The iron fiddle (_violon de fer_)
is thus formed. The notes of the ordinary musical box are also produced
by the vibrations of tongues of metal fixed at one end. Pins are fixed
in a revolving, cylinder, the free ends of the tongues are lifted by
these pins and then suddenly let go. The tongues vibrate, their length
and strength being so arranged as to produce in each particular case
the proper rapidity of vibration.

Sir Charles Wheatstone has devised a simple and ingenious optical
method for the study of vibrating rods fixed at one end. Attaching
light glass beads, silvered within, to the end of a metal rod, and
allowing the light of a lamp or candle to fall upon the bead, he
obtained a small spot intensely illuminated. When the rod vibrated,
this spot described a brilliant line which showed the character of the
vibration. A knitting-needle, fixed in a vise with a small bead stuck
on to it by marine glue, answers perfectly as an illustration. In
Wheatstone’s more complete instrument, which he calls a kaleidophone,
the vibrating rods are firmly screwed into a massive stand. Extremely
beautiful figures are obtained by this simple contrivance, some of
which may now be projected on a magnified scale upon the screen before
you.

Fixing the rod horizontally in the vise, a condensed beam is permitted
to fall upon the silvered bead, a spot of sunlike brilliancy being
thus obtained. Placing a lens in front of the bead, a bright image of
the spot is thrown upon the screen, the needle is then drawn aside,
and suddenly liberated. The spot describes a ribbon of light, at first
straight, but speedily opening out into an ellipse, passing into a
circle, and then again through a second ellipse back to a straight
line. This is due to the fact that a rod held thus in a vise vibrates
not only in the direction in which it is drawn aside, but also at
right angles to this direction. The curve is due to the combination
of two rectangular vibrations.[41] While the rod is thus swinging as
a whole, it may also divide itself into vibrating parts. By properly
drawing a violin-bow across the needle, this serrated circle, Fig. 56,
is obtained, a number of small undulations being superposed upon the
large one. You moreover hear a musical tone, which you did not hear
when the rod vibrated as a whole only; its oscillations, in fact, were
then too slow to excite such a tone. The vibrations which produce these
sinuosities, and which correspond to the first division of the rod,
are executed with about 6-1/4 times the rapidity of the vibrations of
the rod swinging as a whole. Again I draw the bow; the note rises in
pitch, the serrations run more closely together, forming on the screen
a luminous ripple more minute and, if possible, more beautiful than the
last one, Fig. 57. Here we have the second division of the rod, the
sinuosities of which correspond to 17-13/36 times its rate of vibration
as a whole. Thus every change in the sound of the rod is accompanied by
a change of the figure upon the screen.

[Illustration: FIG. 56.]

[Illustration: FIG. 57.]

The rate of vibration of the rod, as a whole, is to the rate
corresponding to its first division nearly as the square of 2 is to the
square of 5, or as 4:25. From the first division onward the rates of
vibration are approximately proportional to the squares of the series
of odd numbers 3, 5, 7, 9, 11, etc. Supposing the vibrations of the
rod as a whole to number 36, then the vibrations corresponding to this
and to its successive divisions would be expressed approximately by the
following series of number’s:

  36, 225, 625, 1225, 2025, etc.

In Fig. 58, _a_, _b_, _c_, _d_, _e_, are shown the modes of division
corresponding to this series of numbers. You will not fail to observe
that these overtones of a vibrating rod rise far more rapidly in pitch
than the harmonics of a string.

[Illustration: FIG. 58.]

Other forms of vibration may be obtained by smartly striking the rod
with the finger near its fixed end. In fact, an almost infinite variety
of luminous scrolls can be thus produced, the beauty of which may be
inferred from the subjoined figures (see next page) first obtained by
Sir C. Wheatstone. They may be produced by illuminating the bead with
sunlight, or with the light of a lamp or candle. The scrolls, moreover,
may be doubled by employing two candles instead of one. Two spots of
light then appear, each of which describes its own luminous line when
the knitting-needle is set in vibration. In a subsequent lecture we
shall become acquainted with Wheatstone’s application of his method to
the study of rectangular vibrations.

[Illustration: FIG. 59.]


§ 4. _Transverse Vibrations of a Rod free at Both Ends. The Claque-bois
and Glass Harmonica_

[Illustration: FIG. 60.]

From a rod or bar fixed at one end, we will now pass to rods or bars
free at both ends; for such an arrangement has also been employed in
music. By a method afterward to be described, Chladni, the father of
modern acoustics, determined experimentally the modes of vibration
possible to such bars. The simplest mode of division in this case
occurs when the rod is divided by two nodes into three vibrating parts.
This division is easily illustrated by a flexible box ruler, six feet
long. Holding it at about twelve inches from its two ends between the
forefinger and thumb of each hand, and shaking it, or causing its
centre to be struck, it vibrates, the middle segment forming a shadowy
spindle, and the two ends forming fans. The shadow of the ruler on the
screen renders the mode of vibration very evident. In this case the
distance of each node from the end of the ruler is about one-fourth of
the distance between the two nodes. In its second mode of vibration
the rod or ruler is divided into four vibrating parts by three nodes.
In Fig. 60, 1 and 2, these respective modes of division are shown.
Looking at the edge of the ruler 1, the dotted lines cutting _a a′_, _b
b′_, show the manner in which the segments bend up and down when the
first division occurs, while _c c′_, _d d′_, show the mode of vibration
corresponding to the second division. The deepest tone of a rod free at
both ends is higher than the deepest tone of a rod fixed at one end in
the proportion of 4:25. Beginning with the first two nodes, the rates
of vibration of the free bar rise in the following proportion:

  Number of nodes                       2, 3, 4, 5, 6, 7
  Numbers to the squares of which the}  3, 5, 7, 9, 11, 13
  pitch is approximately proportional}

Here, also, we have a similarly rapid rise of pitch to that noticed in
the last two cases.

[Illustration: FIG. 61.]

For musical purposes the first division only of a free rod has been
employed. When bars of wood of different lengths, widths, and depths,
are strung along a cord which passes through the nodes, we have the
_claque-bois_ of the French, an instrument now before you, A B, Fig.
61. Supporting the cord at one end by a hook _k_ and holding it at
the other in the left hand, I run the hammer _h_ along the series of
bars, and produce an agreeable succession of musical tones. Instead
of using the cord, the bars may rest at their nodes on cylinders of
twisted straw; hence the name “straw-fiddle,” sometimes applied to this
instrument. Chladni informs us that it is introduced as a play of bells
(Glockenspiel) into Mozart’s opera of “Die Zauberflöte.” If, instead
of bars of wood, we employ strips of glass, we have the glass harmonica.


§ 5. _Vibrations of a Tuning-fork_

From the vibrations of a bar free at both ends it is easy to pass to
the vibrations of a tuning-fork, as analyzed by Chladni. Supposing _a
a_, Fig. 62, to represent a straight steel bar, with the nodal points
corresponding to its first mode of division marked by the transverse
dots. Let the bar be bent to the form _b b_; the two nodal points still
remain, but they have approached nearer to each other. The tone of the
bent bar is also somewhat lower than that of the straight one. Passing
through various stages of bending, _c c_, _d d_, we at length convert
the bar into a tuning-fork _e e_, with parallel prongs; it still
retains its two nodal points, which, however, are much closer together
than when the bar was straight.

[Illustration: FIG. 62.]

[Illustration: FIG. 63.]

When such a fork sounds its deepest note, its free ends oscillate as in
Fig. 63, where the prongs vibrate between the limits _b_ and _n_, and
_f_ and _m_, and where _p_ and _q_ are the nodes. There is no division
of a tuning-fork corresponding to the division of a straight bar by
three nodes. In its second mode of division, which corresponds to the
first overtone of the fork, we have a node on each prong, and two at
the bottom. The principle of Young, referred to at page 155, extends
also to tuning-forks. To free the fundamental tone from an overtone,
you draw your bow across the fork at the place where the node is
required to form the latter. In the third mode of division there are
two nodes on each prong and one at the bottom; in the fourth division
there are two nodes on each prong and two at the bottom; while in the
fifth division there are three nodes on each prong and one at the
bottom. The first overtone of the fork requires, according to Chladni,
6-1/4 times the number of vibrations of the fundamental tone.

It is easy to elicit the overtones of tuning-forks. Here, for example,
is our old series, vibrating respectively 256, 320, 384, and 512
times in a second. In passing from the fundamental tone to the first
overtone of each you notice that the interval is vastly greater
than that between the fundamental tone and the first overtone of a
stretched string. From the numbers just mentioned we pass at once to
1,600, 2,000, 2,400, and 3,200 vibrations a second. Chladni’s numbers,
however, though approximately correct, are not always rigidly verified
by experiment. A pair of forks, for example, may have their fundamental
tones in perfect unison and their overtones discordant. Two such forks
are now before you. When the fundamental tones of both are sounded, the
unison is perfect; but when the first overtones of both are sounded,
they are not in unison. You hear rapid “beats,” which grate upon the
ear. By loading one of the forks with wax, the two overtones may be
brought into unison; but now the fundamental tones produce loud beats
when sounded together. This could not occur if the first overtone
of each fork was produced by a number of vibrations exactly 6-1/4
times the rate of its fundamental. In a series of forks examined by
Helmholtz, the number of vibrations of the first overtone varied from
5·6 to 6·6 times that of the fundamental.

Starting from the first overtone, and including it, the rates of
vibration of the whole series of overtones are as the squares of the
numbers 3, 5, 7, 9, etc. That is to say, in the time required by the
first overtone to execute 9 vibrations, the second executes 25, the
third 49, the fourth 81, and so on. Thus the overtones of the fork rise
with far greater rapidity than those of a string. They also vanish more
speedily, and hence adulterate to a less extent the fundamental tone by
their admixture.


§ 6. _Chladni’s Figures_

The device of Chladni for rendering these sonorous vibrations visible
has been of immense importance to the science of acoustics. Lichtenberg
had made the experiment of scattering an electrified powder over an
electrified resin-cake, the arrangement of the powder revealing the
electric condition of the surface. This experiment suggested to Chladni
the idea of rendering sonorous vibrations visible by means of sand
strewed upon the surface of the vibrating body. Chladni’s own account
of his discovery is of sufficient interest to justify its introduction
here:

“As an admirer of music, the elements of which I had begun to learn
rather late, that is, in my nineteenth year, I noticed that the science
of acoustics was more neglected than most other portions of physics.
This excited in me the desire to make good the defect, and by new
discovery to render some service to this part of science. In 1785 I
had observed that a plate of glass or metal gave different sounds
when it was struck at different places, but I could nowhere find any
information regarding the corresponding modes of vibration. At this
time there appeared in the journals some notices of an instrument made
in Italy by the Abbé Mazzochi, consisting of bells, to which one or two
violin-bows were applied. This suggested to me the idea of employing a
violin-bow to examine the vibrations of different sonorous bodies. When
I applied the bow to a round plate of glass fixed at its middle it gave
different sounds, which, compared with each other, were (as regards the
number of their vibrations) equal to the squares of 2, 3, 4, 5, etc.;
but the nature of the motions to which these sounds corresponded, and
the means of producing each of them at will, were yet unknown to me.
The experiments on the electric figures formed on a plate of resin,
discovered and published by Lichtenberg, in the memoirs of the Royal
Society of Göttingen, made me presume that the different vibratory
motions of a sonorous plate might also present different appearances,
if a little sand or some other similar substance were spread over the
surface. On employing this means, the first figure that presented
itself to my eyes upon the circular plate already mentioned resembled a
star with ten or twelve rays, and the very acute sound, in the series
alluded to, was that which agreed with the square of the number of
diametrical lines.”


§ 7. _Vibrations of Square Plates: Nodal Lines_

I will now illustrate the experiments of Chladni, commencing with a
square plate of glass held by a suitable clamp at its centre. The plate
might be held with the finger and thumb, if they could only reach far
enough. Scattering fine sand over the plate, the middle point of one of
its edges is damped by touching it with the finger-nail, and a bow is
drawn across the edge of the plate, near one of its corners. The sand
is tossed away from certain parts of the surface, and collects along
two _nodal lines_ which divide the large square into four smaller ones,
as in Fig. 64. This division of the plate corresponds to its deepest
tone.

[Illustration: FIG. 64.]

[Illustration: FIG. 65.]

[Illustration: FIG. 66.]

The signs + and - employed in these figures denote that the two squares
on which they occur are always moving in opposite directions. When the
squares marked + are above the average level of the plate those marked
- are below it; and when those marked - are above the average level
those marked + are below it. The nodal lines mark the boundaries of
these opposing motions. They are the places of transition from the one
motion to the other, and are therefore unaffected by either.

Scattering sand once more over its surface, I damp one of the corners
of the plate, and excite it by drawing the bow across the middle of one
of its sides. The sand dances over the surface, and finally ranges
itself in two sharply-defined ridges along its diagonals, Fig. 65. The
note here produced is a fifth above the last. Again damping two other
points, and drawing the bow across the centre of the opposite side of
the plate, we obtain a far shriller note than in either of the former
cases, and the manner in which the plate vibrates to produce this note
is represented in Fig. 66.

[Illustration: FIG. 67.]

Thus far plates of glass have been employed held by a clamp at the
centre. Plates of metal are still more suitable for such experiments.
Here is a plate of brass, 12 inches square, and supported on a
suitable stand. Damping it with the finger and thumb of my left hand
at two points of its edge, and drawing the bow with my right across
a vibrating portion of the opposite edge, the complicated pattern
represented in Fig. 67 is obtained.

[Illustration: FIG. 68.]

The beautiful series of patterns shown on page 182 were obtained by
Chladni, by damping and exciting square plates in different ways. It is
not only interesting but startling to see the suddenness with which
these sharply-defined figures are formed by the sweep of the bow of a
skilful experimenter.


§ 8. _Wheatstone’s Analysis of the Vibrations of Square Plates_

[Illustration: FIG. 69.]

And now let us look a little more closely into the mechanism of these
vibrations. The manner in which a bar free at both ends divides itself
when it vibrates transversely has been already explained. Rectangular
pieces of glass or of sheet metal—the glass strips of the harmonica,
for example—also obey the laws of free rods and bars. In Fig. 69 is
drawn a rectangle _a_, with the nodes corresponding to its first
division marked upon it, and underneath it is placed a figure showing
the manner in which the rectangle, looked at edgewise, bends up and
down when it is set in vibration.[42] For the sake of plainness the
bending is greatly exaggerated. The figures _b_ and _c_ indicate that
the vibrating parts of the plate alternately rise above and fall below
the average level of the plate. At one moment, for example, the centre
of the plate is above the level and its ends below it, as at _b_; while
at the next moment its centre is below and its two ends above the
average level, as at _c_. The vibrations of the plate consist in the
quick successive assumption of these two positions. Similar remarks
apply to all other modes of division.

Now suppose the rectangle gradually to widen, till it becomes a square.
There then would be no reason why the nodal lines should form parallel
to one pair of sides rather than to the other. Let us now examine
what would be the effect of the coalescence of two such systems of
vibrations.

To keep your conceptions clear, take two squares of glass and draw upon
each of them the nodal lines belonging to a rectangle. Draw the lines
on one plate in white, and on the other in black; this will help you
to keep the plates distinct in your mind as you look at them. Now lay
one square upon the other so that their nodal lines shall coincide, and
then realize with perfect mental clearness both plates in a state of
vibration. Let us assume, in the first instance, that the vibrations
of the two plates are concurrent; that the middle segment and the end
segments of each rise and fall together; and now suppose the vibrations
of one plate transferred to the other. What would be the result?
Evidently vibrations of a double amplitude on the part of the plate
which has received this accession. But suppose the vibrations of the
two plates, instead of being concurrent, to be in exact opposition to
each other—that when the middle segment of the one rises the middle
segment of the other falls—what would be the consequence of adding them
together? Evidently a neutralization of all vibration.

Instead of placing the plates so that their nodal lines coincide, set
these lines at right angles to each other. That is to say, push A over
A′, Fig. 70. In these figures the letter P means positive, indicating,
in the section where it occurs, a motion of the plate upward; while N
means negative, indicating, where it occurs, a motion downward. You
have now before you a kind of check pattern, as shown in the third
square, consisting of a square _s_ in the middle, a smaller square
_b_ at each corner, and four rectangles at the middle portions of the
four sides. Let the plates vibrate, and let the vibrations of their
corresponding sections be concurrent, as indicated by the letters P
and N; and then suppose the vibrations of one of them transferred to
the other. What must result? A moment’s reflection will show you that
the big middle square _s_ will vibrate with augmented energy; the same
is true of the four smaller squares _b_, _b_, _b_, _b_, at the four
corners; but you will at once convince yourselves that the vibrations
in the four rectangles are in opposition, and that where their
amplitudes are equal they will destroy each other. The middle point of
each side of the plate of glass would therefore be a point of rest; the
points where the nodal lines of the two plates cross each other would
also be points of rest. Draw a line through every three of these points
and you will obtain a second square inscribed in the first. The sides
of this square are lines of no motion.

[Illustration: FIG. 70.]

We have thus far been theorizing. Let us now clip a square plate
of glass at a point near the centre of one of its edges, and draw
the bow across the adjacent corner of the plate. When the glass
is homogeneous, a close approximation to this inscribed square is
obtained. The reason is that when the plate is agitated in this manner
the two sets of vibrations which we have been considering actually
coexist in the plate, and produce the figure due to their combination.

Again, place the squares of glass one upon the other exactly as in
the last case; but now, instead of supposing them to concur in their
vibrations, let their corresponding sections oppose each other: that
is, let A cover A′, Fig. 71. Then it is manifest that on superposing
the vibrations the middle point of our middle square must be a point of
rest; for here the vibrations are equal and opposite. The intersections
of the nodal lines are also points of rest, and so also is every corner
of the plate itself, for here the added vibrations are also equal and
opposite. We have thus fixed four points of rest on each diagonal of
the square. Draw the diagonals, and they will represent the nodal lines
consequent on the superposition of the two vibrations.

[Illustration: FIG. 71.]

These two systems actually coexist in the same plate when the centre is
clamped and one of the corners touched, while the fiddle-bow is drawn
across the middle of one of the sides. In this case the sand which
marks the lines of rest arranges itself along the diagonals. This, in
its simplest possible form, is Sir C. Wheatstone’s analysis of these
superposed vibrations.


§ 9. _Vibrations of Circular Plates_

Passing from square plates to round ones, we also obtain various
beautiful effects. This disk of brass is supported horizontally upon
an upright stand: it is blackened, and fine white sand is scattered
lightly over it. The disk is capable of dividing itself in various
ways, and of emitting notes of various pitch. I sound the lowest
fundamental note of the disk by touching its edge at a certain point
and drawing the bow across the edge at a point 45° distant from the
damped one. You hear the note and you see the sand. It quits the four
quadrants of the disk, and ranges itself along two of the diameters,
Fig. 72, A (next page). When a disk divides itself thus into four
vibrating segments, it sounds its deepest note. I stop the vibration,
clear the disk, and once more scatter sand over it. Damping its edge,
and drawing the bow across it at a point 30° distant from the damped
one, the sand immediately arranges itself in a star. We have here six
vibrating segments, separated from each other by their appropriate
nodal lines, Fig. 72, B. Again I damp a point, and agitate another
nearer to the damped one than in the last instance; the disk divides
itself into eight vibrating segments with lines of sand between them,
Fig. 72, C. In this way the disk may be subdivided into ten, twelve,
fourteen, sixteen sectors, the number of sectors being always an _even_
one. As the division becomes more minute the vibrations become more
rapid, and the pitch consequently more high. The note emitted by the
sixteen segments into which the disk is now divided is so acute as to
be almost painful to the ear. Here you have Chladni’s first discovery.
You can understand his emotion on witnessing this wonderful effect,
“which no mortal had previously seen.” By rendering the centre of the
disk free, and damping appropriate points of the surface, nodal circles
and other curved lines may be obtained.

[Illustration: FIG. 72.]

The rate of vibration of a disk is directly proportional to its
thickness, and inversely proportional to the square of its diameter. Of
these three disks two have the same diameter, but one is twice as thick
as the other; two of them are of the same thickness, but one has half
the diameter of the other. According to the law just enunciated, the
rules of vibration of the disks are as the numbers 1, 2, 4. When they
are sounded in succession, the musical ears present can testify that
they really stand to each other in the relation of a note, its octave,
and its double octave.


§ 10. _Strehlke and Faraday’s Experiments: Deportment of Light Powders_

The actual movement of the sand toward the nodal lines may be studied
by clogging the sand with a semi-fluid substance. When gum is
employed to retard the motion of the particles, the curves which
they individually describe are very clearly drawn upon the plates. M.
Strehlke has sketched these appearances, and from him the patterns A,
B, C, Fig. 73, are borrowed.

[Illustration: FIG. 73.]

[Illustration: FIG. 74.]

[Illustration: FIG. 75.]

[Illustration: FIG. 76.]

An effect of vibrating plates which long perplexed experimenters is
here to be noticed. When with the sand strewed over a plate a little
fine dust is mingled, say the fine seed of lycopodium, this light
substance, instead of collecting along the nodal lines, forms little
heaps at the places of most violent motion. It is heaped at the four
corners of the plate, Fig. 74, at the four sides of the plate, Fig. 75,
and lodged between the nodal lines of the plate, Fig. 76. These three
figures represent the three states of vibration illustrated in Figs.
64, 65, and 66. The dust chooses in all cases the place of greatest
agitation. Various explanations of this effect had been given, but it
was reserved for Faraday to assign its extremely simple cause. The
light powder is entangled by the little whirlwinds of air produced by
the vibrations of the plate: it cannot escape from the little cyclones,
though the heavier sand particles are readily driven through them.
When, therefore, the motion ceases, the light powder settles down at
the places where the vibration was a maximum. In vacuo no such effect
is observed: here all powders, light and heavy, move to the nodal lines.


§ 11. _Vibration of Bells: Means of rendering them visible_

[Illustration: FIG. 77.]

The vibrating segments and nodes of a bell are similar to those of
a disk. When a bell sounds its deepest note, the coalescence of its
pulses causes it to divide into four vibrating segments, separated from
each other by four nodal lines, which run up from the sound-bow to the
crown of the bell. The place where the hammer strikes is always the
middle of a vibrating segment; the point diametrically opposite is also
the middle of such a segment. Ninety degrees from these points, we have
also vibrating segments, while at 45 degrees right and left of them we
come upon the nodal lines. Supposing the strong, dark circle in Fig. 77
to represent the circumference of the bell in a state of quiescence,
then when the hammer falls on any one of the segments _a_, _c_, _b_, or
_d_, the sound-bow passes periodically through the changes indicated
by the dotted lines. At one moment it is an oval, with _a b_ for its
longest diameter; at the next moment it is an oval, with _c d_ for its
longest diameter. The changes from one oval to the other constitute,
in fact, the vibrations of the bell. The four points _n, n, n, n_,
where the two ovals intersect each other, are the nodes. As in the case
of a disk, the number of vibrations executed by a bell in a given time
varies directly as the thickness, and inversely as the square of the
bell’s diameter.

Like a disk, also, a bell can divide itself into any even number of
vibrating segments, but not into an odd number. By damping proper
points in succession the bell can be caused to divide into 6, 8, 10,
and 12 vibrating parts. Beginning with the fundamental note, the number
of vibrations corresponding to the respective divisions of a bell, as
of a disk, is as follows:

  Number of divisions                           4, 6, 8, 10, 12
  Numbers the squares of which express the  }
    rates of vibration                      }   2, 3, 4, 5, 6

Thus, if the vibrations of the fundamental tone be 40, that of the next
higher tone will be 90, the next 160, the next 250, the next 360, and
so on. If the bell be thin, the tendency to subdivision is so great
that it is almost impossible to bring out the pure fundamental tone
without the admixture of the higher ones.

I will now repeat before you a homely, but an instructive experiment.
This common jug, when a fiddle-bow is drawn across its edge, divides
into four vibrating segments exactly like a bell. The jug is provided
with a handle; and you are to notice the influence of this handle upon
the tone. When the fiddle-bow is drawn across the edge at a point
diametrically opposite to the handle, a certain note is heard. When
it is drawn at a point 90° from the handle, the same note is heard.
In both these cases the handle occupies the middle of a vibrating
segment, loading that segment by its weight. But I now draw the bow at
an angular distance of 45° from the handle; the note is sensibly higher
than before. The handle in this experiment occupies a node; it no
longer loads a vibrating segment, and hence the elastic force, having
to cope with less weight, produces a more rapid vibration. Chladni
executed with a teacup the experiment here made with a jug. Now bells
often exhibit round their sound-bows an absence of uniform thickness
tantamount to the want of symmetry in the case of our jug; and we shall
learn subsequently that the intermittent sound of many bells, noticed
more particularly when their tones are dying out, is produced by the
combination of two distinct rates of vibration, which have this absence
of uniformity for their origin.

There are no points of absolute rest in a vibrating bell, for the
nodes of the higher tones are not those of the fundamental one. But
it is easy to show that the various parts of the sound-bow, when the
fundamental tone is predominant, vibrate with very different degrees of
intensity. Suspending a little ball of sealing-wax _a_, Fig. 78 (next
page), by a string, and allowing it to rest gently against the interior
surface of an inverted bell, it is tossed to and fro when the bell is
thrown into vibration. But the rattling of the sealing-wax ball is far
more violent when it rests against the vibrating segments than when it
rests against the nodes. Permitting the ivory bob of a short pendulum
to rest in succession against a vibrating segment and against a node of
the “Great Bell” of Westminster, I found that in the former position
it was driven away five inches, in the latter only two inches and
three-quarters, when the hammer fell upon the bell.

[Illustration: FIG. 78.]

Could the “Great Bell” be turned upside down and filled with water,
on striking it the vibrations would express themselves in beautiful
ripples upon the liquid surface. Similar ripples may be obtained with
smaller bells, or even with finger and claret glasses, but they would
be too minute for our present purpose. Filling a large hemispherical
glass with water, and passing the fiddle-bow across its edge, large
crispations immediately cover its surface. When the bow is vigorously
drawn, the water rises in spray from the four vibrating segments.
Projecting, by means of a lens, a magnified image of the illuminated
water-surface upon the screen, pass the bow gently across the edge of
the glass, or rub the finger gently along the edge. You hear this low
sound, and at the same time observe the ripples breaking, as it were,
in visible music over the four sectors of the figure.

You know the experiment of Leidenfrost which proves that, if water
be poured into a red-hot silver basin, it rolls about upon its own
vapor. The same effect is produced if we drop a volatile liquid, like
ether, on the surface of warm water. And, if a bell-glass be filled
with ether or with alcohol, a sharp sweep of the bow over the edge of
the glass detaches the liquid spherules, which, when they fall back,
do not mix with the liquid, but are driven over the surface on wheels
of vapor to the nodal lines. The warming of the liquid, as might be
expected, improves the effect. M. Melde, to whom we are indebted for
this beautiful experiment, has given the drawings, Figs. 79 and 80,
representing what occurs when the surface is divided into four and into
six vibrating parts. With a thin wineglass and strong brandy the effect
may also be obtained.[43]

[Illustration: FIG. 79.]

[Illustration: FIG. 80.]

The glass and the liquid within it vibrate here together, and
everything that interferes with the perfect continuity of the entire
mass disturbs the sonorous effect. A crack in the glass passing from
the edge downward extinguishes its sounding power. A rupture in the
continuity of the liquid has the same effect. When a glass containing
a solution of carbonate of soda is struck with a bit of wood, you hear
a clear musical sound. But when a little tartaric acid is added to the
liquid, it foams, and a dry, unmusical collision takes the place of the
musical sound. As the foam disappears, the sonorous power returns, and
now that the liquid is once more clear, you hear the musical ring as
before.

[Illustration: FIG. 81.]

The ripples of the tide leave their impressions upon the sand over
which they pass. The ripples produced by sonorous vibrations have been
proved by Faraday competent to do the same. Attaching a plate of glass
to a long flexible board, and pouring a thin layer of water over the
surface of the glass, on causing the board to vibrate its tremors chase
the water into a beautiful mosaic of ripples. A thin stratum of sand
strewed upon the plate is acted upon by the water, and carved into
patterns, of which Fig. 81 is a reduced specimen.


SUMMARY OF CHAPTER IV

A rod fixed at both ends and caused to vibrate transversely divides
itself in the same manner as a string vibrating transversely.

But the succession of its overtones is not the same as those of
a string, for while the series of tones emitted by the string is
expressed by the natural numbers 1, 2, 3, 4, 5, etc., the series of
tones emitted by the rod is expressed by the squares of the odd numbers
3, 5, 7, 9, etc.

A rod fixed at one end can also vibrate as a whole, or can divide
itself into vibrating segments separated from each other by nodes.

In this case the rate of vibration of the fundamental tone is to that
of the first overtone as 4:25, or as the square of 2 to the square
of 5. From the first division onward the rates of vibration are
proportional to the squares of the odd numbers 3, 5, 7, 9, etc.

With rods of different lengths the rate of vibration is inversely
proportional to the square of the length of the rod.

Attaching a glass bead silvered within to the free end of the rod, and
illuminating the bead, the spot of light reflected from it describes
curves of various forms when the rod vibrates. The kaleidophone of
Wheatstone is thus constructed.

The iron fiddle and the musical box are instruments whose tones are
produced by rods, or tongues, fixed at one end and free at the other.

A rod free at both ends can also be rendered a source of sonorous
vibrations. In its simplest mode of division it has two nodes, the
subsequent overtones correspond to divisions by 3, 4, 5, etc., nodes.
Beginning with its first mode of division, the tones of such a rod are
represented by the squares of the odd numbers 3, 5, 7, 9, etc.

The claque-bois, straw-fiddle, and glass harmonica are instruments
whose tones are those of rods or bars free at both ends, and supported
at their nodes.

When a straight bar, free at both ends, is gradually bent at its
centre, the two nodes corresponding to its fundamental tone gradually
approach each other. It finally assumes the shape of a timing-fork
which, when it sounds its fundamental note, is divided by two nodes
near the base of its two prongs into three vibrating parts.

There is no division of a tuning-fork by three nodes.

In its second mode of division, which corresponds to the first overtone
of the fork, there is a node on each prong and two others at the bottom
of the fork.

The fundamental tone of the fork is to its first overtone approximately
as the square of 2 is to the square of 5. The vibrations of the first
overtone are, therefore, about 6-1/4 times as rapid as those of the
fundamental. From the first overtone onward the successive rates of
vibration are as the squares of the odd numbers 3, 5, 7, 9, etc.

We are indebted to Chladni for the experimental investigation of all
these points. He was enabled to conduct his inquiries by means of the
discovery that, when sand is scattered over a vibrating surface, it is
driven from the vibrating portions of the surface, and collects along
the nodal lines.

Chladni embraced in his investigations plates of various forms. A
square plate, for example, clamped at the centre, and caused to emit
its fundamental tone, divides itself into four smaller squares by lines
parallel to its sides.

The same plate can divide itself into four triangular vibrating parts,
the nodal lines coinciding with the diagonals. The note produced in
this case is a fifth above the fundamental note of the plate.

The plate may be further subdivided, sand-figures of extreme beauty
being produced; the notes rise in pitch as the subdivision of the plate
becomes more minute.

These figures may be deduced from the coalescence of different systems
of vibration.

When a circular plate clamped at its centre sounds its fundamental
tone, it divides into four vibrating parts, separated by four radial
nodal lines.

The next note of the plate corresponds to a division into six vibrating
sectors, the next note to a division into eight sectors; such a plate
can divide into any even number of vibrating sectors, the sand-figures
assuming beautiful stellar forms.

The rates of vibration corresponding to the divisions of a disk are
represented by the squares of the numbers 2, 3, 4, 5, 6, etc. In other
words, the rates of vibration are proportional to the squares of the
numbers representing the sectors into which the disk is divided.

When a bell sounds its deepest note it is divided into four vibrating
parts separated from each other by nodal lines, which run upward from
the sound-bow and cross each other at the crown.

It is capable of the same subdivisions as a disk; the succession of its
tones being also the same.

The rate of vibration of a disk or bell is directly proportional to the
thickness and inversely proportional to the square of the diameter.




CHAPTER V


  Longitudinal Vibrations of a Wire—Relative Velocities of
  Sound in Brass and Iron—Longitudinal Vibrations of Rods
  fixed at One End—Of Rods free at Both Ends—Divisions and
  Overtones of Rods vibrating longitudinally—Examination
  of Vibrating Bars by Polarized Light—Determination of
  Velocity of Sound in Solids—Resonance—Vibrations of
  Stopped Pipes: their Divisions and Overtones—Relation
  of the Tones of Stopped Pipes to those of Open
  Pipes—Condition of Column of Air within a Sounding
  Organ-Pipe—Reeds and Reed-Pipes—The Voice—Overtones of
  the Vocal Chords—The Vowel Sounds—Kundt’s Experiments—New
  Methods of determining the Velocity of Sound


§ 1. _Longitudinal Vibrations of Wires and Rods: Conversion of
Longitudinal into Transverse Vibrations_

We have thus far occupied ourselves exclusively with transversal
vibrations; that is to say, vibrations executed at right angles to
the lengths of the strings, rods, plates, and bells subjected to
examination. A string is also capable of vibrating in the direction of
its length, but here the power which enables it to vibrate is not a
tension applied externally, but the elastic force of its own molecules.
Now this molecular elasticity is much greater than any that we can
ordinarily develop by stretching the string, and the consequence is
that the sounds produced by the _longitudinal vibrations_ of a string
are, as a general rule, much more acute than those produced by its
transverse vibrations. These longitudinal vibrations may be excited by
the oblique passage of a fiddle-bow; but they are more easily produced
by passing briskly along the string a bit of cloth or leather on which
powdered resin has been strewed. The resined fingers answer the same
purpose.

When the wire of our monochord is plucked aside, you hear the sound
produced by its transverse vibrations. When resined leather is rubbed
along the wire, a note much more piercing than the last is heard. This
is due to the longitudinal vibrations of the wire. Behind the table
is stretched a stout iron wire 23 feet long. One end of it is firmly
attached to an immovable wooden tray, the other end is coiled round a
pin fixed firmly into one of our benches. With a key this pin can be
turned, and the wire stretched so as to facilitate the passage of the
rubber. Clasping the wire with the resined leather, and passing the
hand to and fro along it, a rich, loud musical sound is heard. Halving
the wire at its centre, and rubbing one of its halves, the note heard
is the octave of the last: the vibrations are twice as rapid. When the
wire is clipped at one-third of its length and the shorter segment
rubbed, the note is a fifth above the octave. Taking one-fourth of its
length and rubbing as before, the note yielded is the double octave
of that of the whole wire, being produced by four times the number of
vibrations. Thus, in longitudinal as well as in transverse vibrations,
the number of vibrations executed in a given time _is inversely
proportional to the length of the wire_.

And notice the surprising power of these sounds when the wire is rubbed
vigorously. With a shorter length, the note is so acute, and at the
same time so powerful, as to be hardly bearable. It is not the wire
itself which produces this intense sound; it is the wooden tray at
its end to which its vibrations are communicated. And, the vibrations
of the wire being longitudinal, those of the tray, which is at right
angles to the wire, must be transversal. We have here, indeed, an
instructive example of the conversion of longitudinal into transverse
vibrations.


§ 2. _Longitudinal Pulses in Iron and Brass: their Relative Velocities
determined_

Causing the wire to vibrate again longitudinally through its entire
length, my assistant shall at the same time turn the key at the end,
thus changing the tension. You notice no variation of the note. The
longitudinal vibrations of the wire, unlike the transverse ones, are
independent of the tension. Beside the iron wire is stretched a second,
of brass, of the same length and thickness. I rub them both. Their
tones are not the same; that of the iron wire is considerably the
higher of the two. Why? Simply because the velocity of the sound-pulse
is greater in iron than in brass. The pulses in this case pass to and
fro from end to end of the wire. At one moment the wire pushes the
tray at its end; at the next moment it pulls the tray, this pushing
and pulling being due to the passage of the pulse to and fro along the
whole wire. The time required for a pulse to run from one end to the
other _and back_ is that of a complete vibration. In that time the wire
imparts one push and one pull to the wooden tray at its end; the wooden
tray imparts one complete vibration to the air, and the air bends once
in and once out the tympanic membrane. It is manifest that the rapidity
of vibration, or, in other words, the pitch of the note, depends upon
the velocity with which the sound-pulse is transmitted through the
wire.

And now the solution of a beautiful problem falls of itself into our
hands. By shortening the brass wire we cause it to emit a note of
the same pitch as that emitted by the other. You hear both notes now
sounding in unison, and the reason is that the sound-pulse travels
through these 23 feet of iron wire, and through these 15 feet 6 inches
of brass wire, in the same time. These lengths are in the ratio of
11:17, and these two numbers express the relative velocities of sound
in brass and iron. In fact, the former velocity is 11,000 feet, and the
latter 17,000 feet a second. The same method is of course applicable to
many other metals.

[Illustration: FIG. 82.]

When a wire or string, vibrating longitudinally, emits its lowest
note, there is no node whatever upon it; the pulse, as just stated,
runs to and fro along the whole length. But, like a string vibrating
transversely, it can also subdivide itself into ventral segments
separated by nodes. By damping the centre of the wire we make that
point a node. The pulses here run from both ends, meet in the centre,
recoil from each other, and return to the ends, where they are
reflected as before. The note produced is the octave of the fundamental
note. The next higher note corresponds to the division of the wire
into three vibrating segments, separated from each other by two nodes.
The first of these three modes of vibration is shown in Fig. 82, _a_
and _b_; the second at _c_ and _d_; the third at _e_ and _f_; the
nodes being marked by dotted transverse lines, and the arrows in each
case pointing out the direction in which the pulse moves. The rates of
vibration follow the order of the numbers 1, 2, 3, 4, 5, etc., just as
in the case of a wire vibrating transversely.

A _rod_ or _bar_ of wood or metal, with its two ends fixed, and
vibrating longitudinally, divides itself in the same manner as the
wire. The succession of tones is also the same in both cases.


§ 3. _Longitudinal Vibrations of Rods fixed at One End: Musical
Instruments formed on this Principle_

[Illustration: FIG. 83.]

Rods and bars _with one end fixed_ are also capable of vibrating
longitudinally. A smooth wooden or metal rod, for example, with one
of its ends fixed in a vise, yields a musical note, when the resined
fingers are passed along it. When such a note yields its lowest note,
it simply elongates and shortens in quick alternation; there is, then,
no node upon the rod. The pitch of the note is inversely proportional
to the length of the rod. This follows necessarily from the fact that
the time of a complete vibration is the time required for the sonorous
pulse to run twice to and fro over the rod. The first overtone of a
rod, fixed at one end, corresponds to its division by a node at a
point one-third of its length from its free end. Its second overtone
corresponds to a division by two nodes, the highest of which is at
a point one-fifth of the length of the rod from its free end, the
remainder of the rod being divided into two equal parts by the second
node. In Fig. 83, _a_ and _b_, _c_ and _d_, _e_ and _f_, are shown the
conditions of the rod answering to its first three modes of vibration:
the nodes, as before, are marked by dotted lines, the arrows in the
respective cases marking the direction of the pulses.

[Illustration: FIG. 84.]

The order of the tones of a rod fixed at one end and vibrating
longitudinally is that of the odd numbers 1, 3, 5, 7, etc. It is easy
to see that this must be the case. For the time of vibration of _c_ or
_d_ is that of the segment above the dotted line: and the length of
this segment being only one-third that of the whole rod, its vibrations
must be three times as rapid. The time of vibration in _e_ or _f_ is
also that of its highest segment, and as this segment is one-fifth
of the length of the whole rod, its vibrations must be five times as
rapid. Thus the order of the tones must be that of the odd numbers.

Before you, Fig. 84, is a musical instrument, the sounds of which
are due to the longitudinal vibrations of a number of deal rods of
different lengths. Passing the resined fingers over the rods in
succession, a series of notes of varying pitch is obtained. An expert
performer might render the tones of this instrument very pleasant to
you.


§ 4. _Vibrations of Rods free at Both Ends_

[Illustration: FIG. 85.]

Rods _with both ends free_ are also capable of vibrating
longitudinally, and producing musical tones. The investigation of this
subject will lead us to exceedingly important results. Clasping a long
glass tube exactly at its centre, and passing a wetted cloth over one
of its halves, a clear musical sound is the result. A solid glass rod
of the same length would yield the same note. In this case the centre
of the tube is a node, and the two halves elongate and shorten in quick
alternation. M. König, of Paris, has provided us with an instrument
which will illustrate this action. A rod of brass, _a b_, Fig. 85, is
held at its centre by the clamp _s_, while an ivory ball, suspended by
two strings from the points, _m_ and _n_, of a wooden frame, is caused
to rest against the end, _b_, of the brass rod. Drawing gently a bit of
resined leather over the rod near _a_, it is thrown into longitudinal
vibration. The centre, _s_, is a node; but the uneasiness of the
ivory ball shows you that the end, _b_, is in a state of tremor. I
apply the rubber still more briskly. The ball, _b_, rattles, and now
the vibration is so intense that the ball is rejected with violence
whenever it comes into contact with the end of the rod.


§ 5. _Fracture of Glass Tube by Sonorous Vibrations_

When the wetted cloth is passed over the surface of a glass tube
the film of liquid left behind by the cloth is seen forming narrow
tremulous rings all along the rod. Now this shivering of the liquid is
due to the shivering of the glass underneath it, and it is possible
so to augment the intensity of the vibration that the glass shall
actually go to pieces. Savart was the first to show this. Twice in this
place I have repeated this experiment, sacrificing in each case a fine
glass tube 6 feet long and 2 inches in diameter. Seizing the tube at
its centre C, Fig. 86, I swept my hand vigorously to and fro along C
D, until finally the half most distant from my hand was shivered into
annular fragments. On examining these it was found that, narrow as they
were, many of them were marked by circular cracks indicating a still
more minute subdivision.

In this case also the rapidity of vibration is inversely proportional
to the length of the rod. A rod of half the length vibrates
longitudinally with double the rapidity, a rod of one-third the length
with treble the rapidity, and so on. The time of a complete vibration
being that required by the pulse to travel to and fro over the rod, and
that time being directly proportional to the length of the rod, the
rapidity of vibration must, of necessity, be in the inverse proportion.

This division of a rod by a single node at its centre corresponds to
the deepest tone produced by its longitudinal vibration. But, as in
all other cases hitherto examined, such rods can subdivide themselves
further. Holding the long glass rod _a e_, Fig. 87, at a point _b_,
midway between its centre and one of its ends, and rubbing its short
section, _a b_, with a wet cloth, the point _b_ becomes a node, a
second node, _d_, being formed at the same distance from the opposite
end of the rod. Thus we have the rod divided into three vibrating
parts, consisting of one whole ventral segment, _b d_, and two half
ones, _a b_ and _d e_. The sound corresponding to this division of the
rod is the octave of its fundamental note.

[Illustration: FIG. 86.]

[Illustration: FIG. 87.]

You have now a means of checking me. For, if the second mode of
division just described produces the octave of the fundamental note,
and if a rod of half the length produces the same octave, then the
whole rod held at a point one-fourth of its length from one of its ends
ought to emit the same note as the half rod held in the middle. When
both notes are sounded together they are heard to be identical in pitch.

[Illustration: FIG. 88.]

Fig. 88, _a_ and _b_, _c_ and _d_, _e_ and _f_, shows the three first
divisions of a rod free at both ends and vibrating longitudinally. The
nodes, as before, are marked by transverse dots, the direction of the
pulses being shown by arrows. The order of the tones is that of the
numbers, 1, 3, 4, etc.


§ 6. _Action of Sonorous Vibrations on Polarized Light_

When a tube or rod vibrating longitudinally yields its fundamental
tone, its two ends are in a state of free vibration, the glass
there suffering neither strain nor pressure. The case at the centre
is exactly the reverse; here there is no vibration, but a quick
alternation of tension and compression. When the sonorous pulses meet
at the centre they squeeze the glass; when they recoil they strain it.
_Thus while at the ends we have the maximum of vibration, but no change
of density, at the centre we have the maximum changes of density, but
no vibration._

We have now cleared the way for the introduction of a very beautiful
experiment made many years ago by Biot, but never, to my knowledge,
repeated on the scale here presented to you. The beam from our electric
lamp, L, Fig. 89, being sent through a prism, B, of bi-refracting spar,
a beam of polarized light is obtained. This beam impinges on a second
prism of spar, _n_, but, though both prisms are perfectly transparent,
the light which has passed through the first cannot get through the
second. By introducing a piece of glass between the two prisms, and
subjecting the glass to either strain or pressure, the light is enabled
to pass through the entire system.

[Illustration: FIG. 89.]

I now introduce between the prisms B and _n_ a rectangle, _s s′_, of
plate glass, 6 feet long, 2 inches wide, and one-third of an inch
thick, which is to be thrown into longitudinal vibration. The beam
from L passes through the glass at a point near its centre, which is
held in a vise, _c_, so that when a wet cloth is passed over one of
the halves, _c s′_, of the strip, the centre will be a node. During
its longitudinal vibration the glass near the centre is, as already
explained, alternately strained and compressed; and this successive
strain and pressure so changes the condition of the light as to
enable it to pass through the second prism. The screen is now dark;
but on passing the wetted cloth briskly over the glass a brilliant
disk of light, three feet in diameter, flashes out upon the screen.
The vibration quickly subsides, and the luminous disk as quickly
disappears, to be, however, revived at will by the passage of the
wetted cloth along the glass.

The light of this disk appears to be continuous, but it is really
intermittent, for it is only when the glass is under strain or pressure
that the light can get through. In passing from strain to pressure,
and from pressure to strain, the glass is for a moment in its natural
state, which, if it continued, would leave the screen dark. But the
impressions of brightness, due to the strains and pressures, remain
far longer upon the retina than is necessary to abolish the intervals
of darkness; hence the screen appears illuminated by a continuous
light. When the glass rectangle is shifted so as to cause the beam
of polarized light to pass through it close to its end, _s_, the
longitudinal vibrations of the glass have no effect whatever upon the
polarized beam.

Thus, by means of this subtile investigator, we demonstrate that, while
the centre of the glass, where the vibration is _nil_, is subjected to
quick alternations of strain and pressure, the ends of the rectangle,
where the vibration is a maximum, suffer neither.[44]


§ 7. _Vibrations of Rods of Wood: Determination of Relative Velocities
in Different Woods_

Rods of wood and metal also yield musical tones when they vibrate
longitudinally. Here, however, the rubber employed is not a wet cloth,
but a piece of leather covered with powdered resin. The resined fingers
also elicit the music of the rods. The modes of vibration here are
those already indicated, the pitch, however, varying with the velocity
of the sonorous pulse in the respective substances. When two rods of
the same length, the one of deal, the other of Spanish mahogany, are
sounded together, the pitch of the one is much lower than that of
the other. Why? Simply because the sonorous pulses pass more slowly
through the mahogany than through the deal. Can we find the relative
velocity of sound through both? With the greatest ease. We have only to
carefully shorten the mahogany rod till it yields the same note as the
deal one. The notes, rendered approximate by the first trials, are now
identical. Through this rod of mahogany 4 feet long, and through this
rod of deal 6 feet long, the sound-pulse passes in the same time, and
these numbers express the relative velocities of sound through the two
substances.

Modes of investigation, which could only be hinted at in our earlier
lectures, are now falling naturally into our hands. When in the first
lecture the velocity of sound in air was spoken of, many possible
methods of determining this velocity must have occurred to your minds,
because here we have miles of space to operate upon. Its velocity
through wood or metal, where such distances are out of the question,
is determined in the simple manner just indicated. From the notes
which they emit when suitably prepared, we may infer with certainty
the _relative_ velocities of sound through different solid substances;
and determining the ratio of the velocity in any one of them to
its velocity in air, we are able to draw up a table of absolute
velocities. But how is air to be introduced into the series? We shall
soon be able to answer this question, approaching it, however, through
a number of phenomena with which, at first sight, it appears to have no
connection.


RESONANCE


§ 8. _Experiments with Resonant Jars. Analysis and Explanation_

The series of tuning-forks now before you have had their rates of
vibration determined by the siren. One, you will remember, vibrates
256 times in a second, the length of its sonorous wave being 4 feet 4
inches. It is detached from its case, so that when struck against a pad
you hardly hear it. When held over this glass jar, A B, Fig. 90, 18
inches deep, you still fail to hear the sound of the fork. Preserving
the fork in its position, I pour water with the least possible noise
into the jar. The column of air underneath the fork shortens, the
sound augments in intensity, and when the water has reached a certain
level it bursts forth with extraordinary power. A greater quantity of
water causes the sound to sink, and become finally inaudible, as at
first. By pouring the water carefully out, a point is reached where the
reinforcement of the sound again occurs. Experimenting thus, we learn
that there is one particular length of the column of air which, when
the fork is placed above it, produces a maximum augmentation of the
sound. This reinforcement of the sound is named _resonance_.

Operating in the same way with all the forks in succession, a column of
air is found for each, which yields a maximum resonance. These columns
become shorter as the rapidity of vibration increases. In Fig. 91 the
series of jars is represented, the number of vibrations to which each
resounds being placed above it.

[Illustration: FIG. 90.]

[Illustration: FIG. 91.]

What is the physical meaning of this very wonderful effect? To solve
this question we must revive our knowledge of the relation of the
motion of the fork itself to the motion of the sonorous wave produced
by the fork. Supposing a prong of this fork, which executes 256
vibrations in a second, to vibrate between the points _a_ and _b_, Fig.
92, in its motion from _a_ to _b_ the fork generates half a sonorous
wave, and as the length of the whole wave emitted by this fork is 4
feet 4 inches, at the moment the prong reaches _b_ the foremost point
of the sonorous wave will be at C, 2 feet 2 inches distant from the
fork. The motion of the wave, then, is vastly greater than that of
the fork. In fact, the distance _a b_ is, in this case, not more than
one-twentieth of an inch, while the wave has passed over a distance
of 26 inches. With forks of lower pitch the difference would be still
greater.

[Illustration: FIG. 92.]

Our next question is, what is the length of the column of air which
resounds to this fork? By measurement with a two-foot rule it is found
to be 13 inches. But the length of the wave emitted by the fork is 52
inches; hence _the length of the column of air which resounds to the
fork is equal to one-fourth of the length of the sound-wave produced by
the fork_. This rule is general, and might be illustrated by any other
of the forks instead of this one.

[Illustration: FIG. 93.]

Let the prong, vibrating between the limits _a_ and _b_, be placed over
its resonant jar, A B, Fig. 93. In the time required by the prong to
move from _a_ to _b_, the condensation it produces runs down to the
bottom of the jar, is there reflected, and, as the distance to the
bottom and back is 26 inches, the reflected wave will reach the fork at
the moment when it is on the point of returning from _b_ to _a_. The
rarefaction of the wave is produced by the retreat of the prong from
_b_ to _a_. This rarefaction will also run to the bottom of the jar and
back, overtaking the prong just as it reaches the limit, _a_, of its
excursion. It is plain from this analysis that the vibrations of the
fork are perfectly synchronous with the vibrations of the aërial column
A B; and in virtue of this synchronism the motion accumulates in the
jar, spreads abroad in the room, and produces this vast augmentation of
the sound.

When we substitute for the air in one of these jars a gas of different
elasticity, we find the length of the resounding column to be
different. The velocity of sound through coal-gas is to its velocity
in air about as 8:5. Hence, to synchronize with our fork, a jar filled
with coal-gas must be deeper than one filled with air. I turn this
jar, 18 inches long, upside down, and hold close to its open mouth our
agitated tuning-fork. It is scarcely audible. The jar, with air in it,
is 5 inches too deep for this fork. Let coal-gas now enter the jar. As
it ascends the note at a certain point swells out, proving that for the
more elastic gas a depth of 18 inches is not too great. In fact, it
is not great enough; for if too much gas be allowed to enter the jar
the resonance is weakened. By suddenly turning the jar upright, still
holding the fork close to its mouth, the gas escapes, and at the point
of proper admixture of gas and air the note swells out again.[45]


§ 9. _Reinforcement of Bell by Resonance_

This fine, sonorous bell, Fig. 94, is thrown into intense vibration
by the passage of a resined bow across its edge. You hear its sound,
pure, but not very forcible. When, however, the open mouth of this
large tube, which is closed at one end, is brought close to one of the
vibrating segments of the bell, the tone swells into a musical roar.
As the tube is alternately withdrawn and advanced, the sound sinks and
swells in this extraordinary manner.

[Illustration: FIG. 94.]

The second tube, open at both ends, is capable of being lengthened
and shortened by a telescopic slider. When brought near the vibrating
bell, the resonance is feeble. On lengthening the tube by drawing out
the slider at a certain point, the tone swells out as before. If the
tube be made longer, the resonance is again enfeebled. Note the fact,
which shall be explained presently, that the open tube which gives the
maximum resonance is exactly twice the length of the closed one. For
these fine experiments we are indebted to Savart.


§ 10. _Expenditure of Motion in Resonance_

With the India-rubber tube employed in our third chapter it was found
necessary to time the impulses properly, so as to produce the various
ventral segments. I could then feel that the muscular work performed,
when the impulses were properly timed, was greater than when they were
irregular. The same truth may be illustrated by a claret-glass half
filled with water. Endeavor to move your hand to and fro, in accordance
with the oscillating period of the water: when you have thoroughly
established synchronism, the work thrown upon the hand apparently
augments the weight of the water. So likewise with our tuning-fork;
when its impulses are timed to the vibrations of the column of air
contained in this jar, its work is greater than when they are not so
timed. As a consequence of this the tuning-fork comes sooner to rest
when it is placed over the jar than when it is permitted to vibrate
either in free air, or over a jar of a depth unsuited to its periods of
vibration.[46]

Reflecting on what we have now learned, you would have little
difficulty in solving the following beautiful problem: You are provided
with a tuning-fork and a siren, and are required by means of these two
instruments to determine the velocity of sound in air. To solve this
problem you lack, if anything, the mere power of manipulation which
practice imparts. You would first determine, by means of the siren, the
number of vibrations executed by the tuning-fork in a second; you would
then determine the length of the column of air which resounds to the
fork. This length multiplied by 4 would give you, approximately, the
wave-length of the fork, and the wave-length multiplied by the number
of vibrations in a second would give you the velocity in a second.
Without quitting your private room, therefore, you could solve this
important problem. We will go on, if you please, in this fashion,
making our footing sure as we advance.


§ 11. _Resonators of Helmholtz_

[Illustration: FIG. 94_a_.]

Helmholtz has availed himself of the principle of resonance in
analyzing composite sounds. He employs little hollow spheres, called
_resonators_, one of which is shown in Fig. 94_a_. The small projection
_b_, which has an orifice, is placed in the ear, while the sound-waves
enter the hollow sphere through the wide aperture at _a_. Reinforced
by the resonance of such a cavity, and rendered thereby more powerful
than its companions, a particular note of a composite clang may be in a
measure isolated and studied alone.


ORGAN-PIPES


§ 12. _Principles of Resonance applied to Organ-Pipes_

Thus disciplined we are prepared to consider the subject of
organ-pipes, which is one of great importance. Before me on the table
are two resonant jars, and in my right hand and my left are held two
tuning-forks. I agitate both, and hold them over this jar. One of them
only is heard. Held over the other jar, the other fork alone is heard.
Each jar selects that fork whose periods of vibration synchronize with
its own. And instead of two forks suppose several of them to be held
over the jar; from the confused assemblage of pulses thus generated,
the jar would select and reinforce that one which corresponds to its
own period of vibration.

When I blow across the open mouth of the jar, or, better still, for
the jar is too wide for this experiment, when I blow across the
open end of a glass tube, _t u_, Fig. 95, of the same length as the
jar, a fluttering of the air is thereby produced, an assemblage of
pulses at the open mouth of the tube being generated. And what is the
consequence? The tube selects that pulse of the flutter which is in
synchronism with itself, and raises it to a musical sound. The sound,
you perceive, is precisely that obtained when the proper tuning-fork is
placed over the tube. The column of air within the tube has, in this
case, virtually created its own tuning-fork; for by the reaction of its
pulses upon the sheet of air issuing from the lips it has compelled
that sheet to vibrate in synchronism with itself, and made it thus act
the part of the tuning-fork.

[Illustration: FIG. 95.]

Selecting for each of the other tuning-forks a resonant tube, in every
case, on blowing across the open end of the tube, a tone is produced
identical in pitch with that obtained through resonance.

When different tubes are compared, the rate of vibration is found to
be inversely proportional to the length of the tube. These three tubes
are 24, 12, and 6 inches long, respectively. I blow gently across the
24-inch tube, and bring out its fundamental note; similarly treated,
the 12-inch tube yields the octave of the note of the 24-inch. In like
manner the 6-inch tube yields the octave of the 12-inch. It is plain
that this must be the case; for, the rate of vibration depending on
the distance which the pulse has to travel to complete a vibration,
if in one case this distance be twice what it is in another, the rate
of vibration must be twice as slow. In general terms, the rate of
vibration is inversely proportional to the length of the tube through
which the pulse passes.


§ 13. _Vibrations of Stopped Pipes: Modes of Division: Overtones_

But that the current of air should be thus able to accommodate itself
to the requirements of the tube, it must enjoy a certain amount of
_flexibility_. A little reflection will show you that the power of
the reflected pulse over the current must depend to some extent on
the force of the current. A stronger current, like a more powerfully
stretched string, requires a great force to deflect it, and when
deflected vibrates more quickly. Accordingly, to obtain the fundamental
note of this 24-inch tube, we must blow very gently across its open
end; a rich, full, and forcible musical tone is then produced. With
a little stronger blast the sound approaches a mere rustle; blowing
stronger still, a tone is obtained of much higher pitch than the
fundamental one. This is the first overtone of the tube, to produce
which the column of air within it has divided itself into two vibrating
parts, with a node between them. With a still stronger blast a still
higher note is obtained. The tube is now divided into three vibrating
parts, separated from each other by two nodes. Once more I blow
with sudden strength; a higher note than any before obtained is the
consequence.

In Fig. 96 are represented the divisions of the column of air
corresponding to the first three notes of a tube stopped at one end.
At _a_ and _b_, which correspond to the fundamental note, the column
is undivided; the bottom of the tube is the only node, and the pulse
simply moves up and down from top to bottom, as denoted by the arrows.
In _c_ and _d_, which correspond to the first overtone of the tube, we
have one nodal surface shown by dots at _x_, against which the pulses
abut, and from which they are reflected as from a fixed surface. This
nodal surface is situated at one-third of the length of the tube from
its open end. In _e_ and _f_, which correspond to the second overtone,
we have two nodal surfaces, the upper one, _x′_, of which is at
one-fifth of the length of the tube from its open end, the remaining
four-fifths being divided into two equal parts by the second nodal
surface. The arrows, as before, mark the directions of the pulses.

[Illustration: FIG. 96.]

We have now to inquire into the relation of these successive notes to
each other. The space from node to node has been called all through
“a ventral segment”; hence the space between the middle of a ventral
segment and a node is a semi-ventral segment. You will readily bear in
mind the law that _the number of vibrations is directly proportional
to the number of semi-ventral segments_ into which the column of air
within the tube is divided. Thus, when the fundamental note is sounded,
we have but a single semi-ventral segment, as at _a_ and _b_. The
bottom here is a node, and the open end of the tube, where the air
is agitated, is the middle of a ventral segment. The mode of division
represented in _c_ and _d_ yields three semi-ventral segments; in _e_
and _f_ we have five. The vibrations, therefore, corresponding to this
series of notes, augment in the proportion of the series of odd numbers
1:3:5. Could we obtain still higher notes, their relative rates of
vibration would continue to be represented by the odd numbers 7, 9, 11,
13, etc.

It is evident that this _must_ be the law of succession. For the time
of vibration in _c_ or _d_ is that of a stopped tube of the length
_x y_; but this length is one-third of the length of the whole tube,
consequently its vibrations must be three times as rapid. The time
of vibration in _e_ or _f_ is that of a stopped tube of the length
_x′ y′_, and inasmuch as this length is one-fifth that of the whole
tube, its vibrations must be five times as rapid. We thus obtain the
succession 1, 3, 5; if we pushed matters further we should obtain the
continuation of the series of odd numbers.

[Illustration: FIG. 97.]

And here it is once more in your power to subject my statements to an
experimental test. Here are two tubes, one of which is three times
the length of the other. I sound the fundamental note of the longest
tube, and then the next note above the fundamental. The vibrations of
these two notes are stated to be in the ratio of 1:3. This latter note,
therefore, ought to be of precisely the same pitch as the fundamental
note of the shorter of the two tubes. When both tubes are sounded their
notes are identical.

It is only necessary to place a series of such tubes of different
lengths thus together to form that ancient instrument, Pan’s pipes, P
P′, Fig. 97 (page 223), with which we are so well acquainted.

The successive divisions, and the relation of the overtones of a rod
fixed at one end (described in page 205), are plainly identical with
those of a column of air in a tube stopped at one end, which we have
been here considering.


§ 14. _Vibrations of Open Pipes: Modes of Division: Overtones_

From tubes closed at one end, and which, for the sake of brevity,
may be called stopped tubes, we now pass to tubes open at both ends,
which we shall call open tubes. Comparing, in the first instance, a
stopped tube with an open one of the same length, we find the note of
the latter an octave higher than that of the former. This result is
general. To make an open tube yield the same note as a closed one, it
must be twice the length of the latter. And, since the length of a
closed tube sounding its fundamental note is one-fourth of the length
of its sonorous wave, the length of an open tube is one-half that of
the sonorous wave that it produces.

It is not easy to obtain a sustained musical note by blowing across the
end of an open glass tube; but a mere puff of breath across the end
reveals the pitch to the disciplined ear. In each case it is that of a
closed tube half the length of the open one.

There are various ways of agitating the air at the ends of pipes and
tubes, so as to throw the air-columns within them into vibration. In
organ-pipes this is done by blowing a thin sheet of air against a sharp
edge. You will have no difficulty in understanding the construction
of an open organ-pipe, from this model, Fig. 98, one side of which has
been removed so that you may see its inner parts. Through the tube _t_
the air passes from the wind-chest into the chamber, C, which is closed
at the top, save a narrow slit, _e d_, through which the compressed air
of the chamber issues. This thin air-current breaks against the sharp
edge, _a b_, and there produces a fluttering noise, and the proper
pulse of this flutter is converted by the resonance of the pipe above
into a musical sound. The open space between the edge, _a b_, and the
slit below it is called the _embouchure_. Fig. 99 represents a stopped
pipe of the same length as that shown in Fig. 98, and hence producing a
note an octave lower.

[Illustration: FIG. 98.]

[Illustration: FIG. 99.]

Instead of a fluttering sheet of air, a tuning-fork whose rate of
vibration synchronizes with that of the organ-pipe may be placed at
the embouchure, as at A B, Fig. 100. The pipe will resound. Here,
for example, are four open pipes of different lengths, and four
tuning-forks of different rates of vibration. Striking the most slowly
vibrating fork, and bringing it near the embouchure of the longest
pipe, the pipe _speaks_ powerfully. When blown into, the same pipe
yields a tone identical with that of the tuning-fork. Going through all
the pipes in succession, we find in each case that the note obtained
by blowing into the pipe is exactly that produced when the proper
tuning-fork is placed at the embouchure. Conceive now the four forks
placed together near the same embouchure; we should have pulses of four
different periods there excited; but out of the four the pipe would
select only one. And if four hundred vibrating forks could be placed
there instead of four, the pipe would still make the proper selection.
This it also does when for the pulses of tuning-forks we substitute
the assemblage of pulses created by the current of air when it strikes
against the sharp upper edge of the embouchure.

[Illustration: FIG. 100.]

The heavy vibrating mass of the tuning-fork is practically uninfluenced
by the motion of the air within the pipe. But this is not the case when
air itself is the vibrating body. Here, as before explained, the pipe
creates, as it were, its own tuning-fork, by compelling the fluttering
stream at its embouchure to vibrate in periods answering to its own.

[Illustration: FIG. 101.]

The condition of the air within an open organ-pipe, when its
fundamental note is sounded, is that of a rod free at both ends, held
at its centre, and caused to vibrate longitudinally. The two ends
are places of vibration, the centre is a node. Is there any way of
_feeling_ the vibrating air-column so as to determine its nodes and
its places of vibration? The late excellent William Hopkins has taught
us the following mode of solving this problem: Over a little hoop is
stretched a thin membrane, forming a little tambourine. The front of
this organ-pipe, P P′, Fig. 101, is of glass, through which you can see
the position of any body within it. By means of a string, the little
tambourine, _m_, can be raised or lowered at pleasure through the
entire length of the pipe. When held above the upper end of the pipe,
you hear the loud buzzing of the membrane. When lowered into the pipe,
it continues to buzz for a time; the sound becoming gradually feebler,
and finally ceasing totally. It is now in the middle of the pipe, where
it cannot vibrate, because the air around it is at rest. On lowering it
still further, the buzzing sound instantly recommences, and continues
down to the bottom of the pipe. Thus, as the membrane is raised and
lowered in quick succession, during every descent and ascent, we have
two periods of sound separated from each other by one of silence. If
sand were strewed upon the membrane, it would dance above and below,
but it would be quiescent at the centre. We thus prove experimentally
that, when an organ-pipe sounds its fundamental note, it divides itself
into two semi-ventral segments separated by a node.

[Illustration: FIG. 102.]

What is the condition of the air at this node? Again, that of the
middle of a rod, free at both ends, and yielding the fundamental note
of its longitudinal vibration. The pulses reflected from both ends
of the rod, or of the column of air, meet in the middle, and produce
compression; they then retreat and produce rarefaction. Thus, while
there is no vibration in the centre of an organ-pipe, the air there
undergoes the greatest changes of density. At the two ends of the pipe,
on the other hand, the air-particles merely swing up and down without
sensible compression or rarefaction.

If the sounding pipe were pierced at the centre, and the orifice
stopped by a membrane, the air, when condensed, would press the
membrane outward, and, when rarefied, the external air would press
the membrane inward. The membrane would therefore vibrate in unison
with the column of air. The organ-pipe, Fig. 102, is so arranged that
a small jet of gas, _b_, can be lighted opposite the centre of the
pipe, and there acted upon by the vibrations of a membrane. Two other
gas-jets, _a_ and _c_, are placed nearly midway between the centre and
the two ends of the pipe. The three burners, _a_, _b_, _c_, are fed in
the following manner: through the tube, _t_, the gas enters the hollow
chamber, _e d_, from which issue three little bent tubes, shown in the
figure, each communicating with a capsule closed underneath by the
membrane. This is in direct contact with the air of the organ-pipe.
From the three capsules issue the three little burners, with their
flames, _a_, _b_, _c_.

Blowing into the pipe so as to sound its fundamental note, the three
flames are agitated, but the central one is most so. Lowering the
flames so as to render them very small, and blowing again, the central
flame, _b_, is extinguished, while the others remain lighted. The
experiment may be performed half a dozen times in succession; the
sounding of the fundamental note always quenches the middle flame.

By blowing more sharply into the pipe, it is caused to yield its first
overtone. The middle node no longer exists. The centre of the pipe is
now a place of maximum vibration, while two nodes are formed midway
between the centre and the two ends. But if this be the case, and if
the flame opposite the node be always blown out, then, when the first
overtone of this pipe is sounded, the two flames _a_ and _c_ ought
to be extinguished, while the central flame remains lighted. This is
the case. When the first harmonic is sounded the two nodal flames are
infallibly extinguished, while the flame _b_ in the middle of the
ventral segment is not sensibly disturbed.

There is no theoretic limit to the subdivision of an organ-pipe,
either stopped or open. In stopped pipes we begin with 1 semi-ventral
segment, and pass on to 3, 5, 7, etc., semi-ventral segments, the
number of vibrations of the successive notes augmenting in the same
ratio. In open pipes we begin with 2 semi-ventral segments, and pass on
to 4, 6, 8, 10, etc., the number of vibrations of the successive notes
augmenting in the same ratio; that is to say, in the ratio 1:2:3:4:5,
etc. When, therefore, we pass from the fundamental tone to the first
overtone in an open pipe, we obtain the octave of the fundamental. When
we make the same passage in a stopped pipe, we obtain a note a fifth
above the octave. No intermediate modes of vibration are in either
case possible. If the fundamental tone of a stopped pipe be produced
by 100 vibrations a second, the first overtone will be produced by 300
vibrations, the second by 500, and so on. Such a pipe, for example,
cannot execute 200 or 400 vibrations in a second. In like manner the
open pipe, whose fundamental note is produced by 100 vibrations a
second, cannot vibrate 150 times in a second, but passes, at a jump, to
200, 300, 400, and so on.

[Illustration: FIG. 103.]

In open pipes, as in stopped ones, the number of vibrations executed in
the unit of time is inversely proportional to the length of the pipe.
This follows from the fact, already dwelt upon so often, that the time
of a vibration is determined by the distance which the sonorous pulse
has to travel to complete a vibration.

In Fig. 103, _a_ and _b_ (at the bottom) represent the division of an
open pipe corresponding to its fundamental tone; _c_ and _d_ represent
the division corresponding to its first, _e_ and _f_ the division
corresponding to its second overtone, the dots marking the nodes. The
distance _m n_ is one-half, _o p_ is one-fourth, and _s t_ is one-sixth
of the whole length of the pipe. But the pitch of _a_ is that of a
stopped pipe equal in length to _m n_; the pitch of _c_ is that of a
stopped pipe of the length _o p_; while the pitch of _e_ is that of a
stopped pipe of the length _s t_. Hence, as these lengths are in the
ratio of 1/2:1/4:1/6, or as 1:1/2:1/3, the rates of vibration must be
as the reciprocals of these, or as 3:2:1. From the mere inspection,
therefore, of the respective modes of vibration, we can draw the
inference that the succession of tones of an open pipe must correspond
to the series of natural numbers.

The pipe _a_, Fig. 103, has been purposely drawn twice the length of
_a_, Fig. 93 (p. 215). It is perfectly manifest that to complete a
vibration the pulse has to pass over the same distance in both pipes,
and hence that the pitch of the two pipes must be the same. The open
pipe, _a n_, consists virtually of two stopped ones, with the central
nodal surface at _m_ as their common base. This shows the relation of a
stopped pipe to an open one to be that which experiment establishes.


§ 15. _Velocity of Sound in Gases, Liquids, and Solids determined by
Musical Vibrations_

We have already learned that the _relative_ velocities of sound in
different solid bodies may be determined from the notes which they
emit when thrown into longitudinal vibration. It was remarked at the
time that to draw up a table of _absolute_ velocities we only required
the accurate comparison of the velocity in any one of those solids
with the velocity in air. We are now in a condition to supply this
comparison. For we have learned that the vibrations of the air in an
organ-pipe open at both ends are executed precisely as those of a rod
free at both ends. Any difference of rapidity, therefore, between the
vibrations of a rod and of an open organ-pipe of the same length must
be due solely to the different velocities with which the sonorous
pulses are propagated through them. Take therefore an organ-pipe of a
certain length, emitting a note of a certain pitch, and find the length
of a rod of pine which yields the same note. This length would be ten
times that of the organ-pipe, which would prove the velocity of sound
in pine to be ten times its velocity in air. But the absolute velocity
in air is 1,090 feet a second; hence the absolute velocity in pine is
10,900 feet a second, which is that given in our first chapter (p. 74).
To the celebrated Chladni we are indebted for this beautiful mode of
determining the velocity of sound in solid bodies.

We had also in our first lecture a table of the velocities of sound in
other gases than air. I am persuaded that you could tell me, after due
reflection, how this table was constructed. It would only be necessary
to find a series of organ-pipes which, when filled with the different
gases, yield the same note; the lengths of these pipes would give the
relative velocities of sound through the gases. Thus we should find the
length of a pipe filled with hydrogen to be four times that of a pipe
filled with oxygen, yielding the same note, and this would prove the
velocity of sound in the former to be four times its velocity in the
latter.

But we had also a table of velocities through various liquids. How was
it constructed? By forcing the liquids through properly constructed
organ-pipes, and comparing their musical tones. Thus, in water it
requires a pipe a little better than four feet long to produce the
note of an air-pipe one foot long; and this proves the velocity of
sound in water to be somewhat more than four times its velocity in
air. My object here is to give you a clear notion of the way in
which scientific knowledge enables us to cope with these apparently
insurmountable problems. It is not necessary to go into the niceties
of these measurements. You will, however, readily comprehend that all
the experiments with gases might be made with the same organ-pipe, the
velocity of sound in each respective gas being immediately deduced from
the pitch of its note. With a pipe of constant length the pitch, or, in
other words, the number of vibrations, would be directly proportional
to the velocity. Thus, comparing oxygen with hydrogen, we should find
the note of the latter to be the double octave of that of the former,
whence we should infer the velocity of sound in hydrogen to be four
times its velocity in oxygen. The same remark applies to experiments
with liquids. Here also the same pipe may be employed throughout, the
velocities being inferred from the notes produced by the respective
liquids.

In fact, the length of an open pipe being, as already explained,
one-half the length of its sonorous wave, it is only necessary to
determine, by means of the siren, the number of vibrations executed
by the pipe in a second, and to multiply this number by twice the
length of the pipe, in order to obtain the velocity of sound in the
gas or liquid within the pipe. Thus, an open pipe 26 inches long and
filled with air executes 256 vibrations in a second. The length of its
sonorous wave is twice 26 inches, or 4-1/3 feet: multiplying 256 by
4-1/3 we obtain 1,120 feet per second as the velocity of sound through
air of this temperature. Were the tube filled with carbonic-acid gas,
its vibrations would be slower: were it filled with hydrogen, its
vibrations would be quicker; and applying the same principle, we should
find the velocity of sound in both these gases.

So likewise the length of a solid rod free at both ends, and sounding
its fundamental note, is half that of the sonorous wave in the
substance of the solid. Hence we have only to determine the rate of
vibration of such a rod, and multiply it by twice the length of the
rod, to obtain the velocity of sound in the substance of the rod. You
can hardly fail to be impressed by the power which physical science has
given us over these problems; nor will you refuse your admiration to
that famous old investigator, Chladni, who taught us how to master them
experimentally.


REEDS AND REED-PIPES

The construction of the siren and our experiments with that instrument
are, no doubt, fresh in your recollection. Its musical sounds are
produced by the cutting up into puffs of a series of air-currents.
The same purpose is effected by a vibrating reed, as employed in the
accordion, the concertina, and the harmonica. In these instruments it
is not the vibrations of the reed itself which, imparted to the air,
and transmitted through it to our organs of hearing, produce the music;
the function of the reed is _constructive_, not _generative_; it molds
into a series of discontinuous puffs that which without it would be a
continuous current of air.

[Illustration: FIG. 104.]

Reeds, if associated with organ-pipes, sometimes command, and are
sometimes commanded by, the vibrations of the column of air. When they
are stiff they rule the column; when they are flexible the column rules
them. In the former case, to derive any advantage from the air-column,
its length ought to be so regulated that either its fundamental tone
or one of its overtones shall correspond to the rate of vibration of
the reed. The metal reed commonly employed in organ-pipes is shown in
Fig. 104, A and B, both in perspective and in section. It consists
of a long and flexible strip of metal, V V, placed in a rectangular
orifice, through which the current of air enters the pipe. The manner
in which the reed and pipe are associated is shown in Fig. 105. The
front, _b c_, of the space containing the flexible tongue is of
glass, so that you may see the tongue within it. A conical pipe, A
B, surmounts the reed.[47] The wire _w i_, shown pressing against the
reed, is employed to lengthen or shorten it, and thus to vary within
certain limits its rate of vibration. At one time in the practice of
music the reed closed the aperture by simply falling against its sides;
every stroke of the reed produced a tap, and these linked themselves
together to an unpleasant, screaming sound, which materially injured
that of the associated organ-pipe. This was mitigated, but not removed,
by permitting the reed to strike against soft leather; but the reed
now employed is the _free reed_, which vibrates to and fro between
the sides of the aperture, almost, but not quite, filling it. In this
way the unpleasantness referred to is avoided. When reed and pipe
synchronize perfectly, the sound is most pure and forcible; a certain
latitude, however, is possible on both sides of perfect synchronism.
But if the discordance be pushed too far, the pipe ceases to be of any
use. We then obtain the sound due to the vibrations of the reed alone.

[Illustration: FIG. 105.]

Flexible wooden reeds, which can accommodate themselves to the
requirements of the pipes above them, are also employed in organ-pipes.
Perhaps the simplest illustration of the action of the reed commanded
by its aërial column is furnished by a common wheaten straw. At about
an inch from a knot, at _r_, I bury my penknife in this straw, _s r′_,
Fig. 106, to a depth of about one-fourth of the straw’s diameter, and,
turning the blade flat, pass it upward toward the knot, thus raising
a strip of the straw nearly an inch in length. This strip, _r r′_,
is to be our reed, and the straw itself is to be our pipe. It is now
eight inches long. When blown into, it emits this decidedly musical
sound. When cut so as to make its length six inches, the pitch is
higher; with a length of four inches, the pitch is higher still; and
with a length of two inches, the sound is very shrill indeed. In these
experiments the reed was compelled to accommodate itself throughout to
the requirements of the vibrating column of air.

[Illustration: FIG. 106.]

The clarinet is a reed-pipe. It has a single broad tongue, with which
a long, cylindrical tube is associated. The reed-end of the instrument
is grasped by the lips, and by their pressure the slit between the reed
and its frame is narrowed to the required extent. The overtones of a
clarinet are different from those of a flute. A flute is an open pipe,
a clarinet a stopped one, the end at which the reed is placed answering
to the closed end of the pipe. The tones of a flute follow the order
of the natural numbers 1, 2, 3, 4, etc., or of the even numbers 2,
4, 6, 8, etc.; while the tones of a clarinet follow the order of the
odd numbers 1, 3, 5, 7, etc. The intermediate notes are supplied by
opening the lateral orifices of the instrument. Sir C. Wheatstone
was the first to make known this difference between the flute and
clarinet, and his results agree with the more thorough investigations
of Helmholtz. In the hautboy and bassoon we have two reeds inclined
to each other at a sharp angle, with a slit between them, through
which the air is urged. The pipe of the hautboy is _conical_, and its
overtones are those of an open pipe—different, therefore, from those
of a clarinet. The pulpy end of a straw of green corn may be split by
squeezing it, so as to form a double reed of this kind, and such a
straw yields a musical tone. In the horn, trumpet, and serpent, the
performer’s lips play the part of the reed. These instruments are
formed of long, conical tubes, and their overtones are those of an
open organ-pipe. The music of the older instruments of this class was
limited to their overtones, the particular tone elicited depending on
the force of the blast and the tension of the lips. It is now usual to
fill the gaps between the successive overtones by means of keys, which
enable the performer to vary the length of the vibrating column of air.


§ 16. _The Voice_

The most perfect of reed instruments is the organ of voice. The vocal
organ in man is placed at the top of the trachea or wind-pipe, the
head of which is adjusted for the attachment of certain elastic bands
which almost close the aperture. When the air is forced from the lungs
through the slit which separates these _vocal chords_, they are thrown
into vibration; by varying their tension, the rate of vibration is
varied, and the sound changed in pitch. The vibrations of the vocal
chords are practically unaffected by the resonance of the mouth, though
we shall afterward learn that this resonance, by reinforcing one or the
other of the tones of the vocal chords, influences in a striking manner
the quality of the voice. The sweetness and smoothness of the voice
depend on the perfect closure of the slit of the glottis at regular
intervals during the vibration.

[Illustration: FIG. 107.]

The vocal chords may be illuminated and viewed in a mirror, placed
suitably at the back of the mouth. Varied experiments of this kind
have been executed by Sig. Garcia.[48] I once sought to project
the larynx of M. Czermak upon a screen in this room, but with only
partial success. The organ may, however, be viewed directly in the
laryngoscope; its motions, in singing, speaking, and coughing, being
strikingly visible. It is represented at rest in Fig. 107. The
roughness of the voice in colds is due, according to Helmholtz, to
mucous flocculi, which get into the slit of the glottis, and which are
seen by means of the laryngoscope. The squeaking falsetto voice, with
which some persons are afflicted, Helmholtz thinks, may be produced
by the drawing aside of the mucous layer which ordinarily lies under
and loads the vocal chords. Their edges thus become sharper and their
weight less; while, their elasticity remaining the same, they are
shaken into more rapid tremors. The promptness and accuracy with which
the vocal chords can change their tension, their form, and the width of
the slit between them, to which must be added the elective resonance of
the cavity of the mouth, render the voice the most perfect of musical
instruments.

[Illustration: FIG. 108.]

The celebrated comparative anatomist, John Müller, imitated the
action of the vocal chords by means of bands of India-rubber. He
closed the open end of a glass tube by two strips of this substance,
leaving a slit between them. On urging air through the slit, the bands
were thrown into vibration, and a musical tone produced. Helmholtz
recommends the form shown in Fig. 108, where the tube, instead of
ending in a section at right angles to its axis, terminates in two
oblique sections, over which the bands of India-rubber are drawn.
The easiest mode of obtaining sounds from reeds of this character is
to roll round the end of a glass tube a strip of thin India-rubber,
leaving about an inch of the substance projecting beyond the end of
the tube. Taking two opposite portions of the projecting rubber in the
fingers, and stretching it, a slit is formed, the blowing through which
produces a musical sound, which varies in pitch, as the sides of the
slit vary in tension.


§ 17. _Vowel Sounds_

The formation of the vowel sounds of the human voice excited long ago
philosophic inquiry. We can distinguish one vowel sound from another,
while assigning to both the same pitch and intensity. What, then, is
the quality which renders the distinction possible? In the year 1779
this was made a prize question by the Academy of St. Petersburg, and
Kratzenstein gained the prize for the successful manner in which he
imitated the vowel sounds by mechanical arrangements. At the same time
Von Kempelen, of Vienna, made similar and more elaborate experiments.
The question was subsequently taken up by Mr. Willis, who succeeded
beyond all his predecessors in the experimental treatment of the
subject. The true theory of vowel sounds was first stated by Sir C.
Wheatstone, and quite recently they have been made the subject of
exhaustive inquiry by Helmholtz. You will find little difficulty in
comprehending their origin.

Mounted on the acoustic bellows, without any pipe associated with it,
when air is urged through its orifice, a free reed speaks in this
forcible manner. When upon the frame of the reed a pyramidal pipe
is fixed, you notice a change in the sound; and by pushing my flat
hand over the open end of the pipe, the similarity between the sound
produced and that of the human voice is unmistakable. Holding the palm
of the hand over the end of the pipe so as to close it altogether, and
then raising the hand twice in quick succession, the word “mamma” is
heard as plainly as if it were uttered by an infant. For this pyramidal
tube I now substitute a shorter one, and with it make the same
experiment. The “mamma” now heard is exactly such as would be uttered
by a child with a stopped nose. Thus, by associating with a vibrating
reed a suitable pipe, we can impart to the sound the qualities of the
human voice.

In the organ of voice, the reed is formed by the vocal chords, and
associated with this reed is the resonant cavity of the mouth,
which can so alter its shape as to resound, at will, either to the
fundamental tone of the vocal chords or to any of their overtones. With
the aid of the mouth, therefore, we can _mix together_ the fundamental
tone and the overtones of the voice in different proportions. Different
vowel sounds are due to different admixtures of this kind. Striking
one of this series of tuning-forks, and placing it before my mouth, I
adjust the size of that cavity until it resounds forcibly to the fork.
Then, without altering in the least the shape or size of my mouth, I
urge air through the glottis. The vowel sound “U” (_oo_ in hoop) is
produced, and no other. I strike another fork, and, placing it in front
of the mouth, adjust the cavity to resonance. Then removing the fork
and urging air through the glottis, the vowel sound “O,” and it only,
is heard. I strike a third fork, adjust my mouth to it, and then urge
air through the larynx; the vowel sound _ah!_ and no other, is heard.
In all these cases the vocal chords have been in the same constant
condition. They have generated throughout the same fundamental tone and
the same overtones, the changes of sound which you have heard being due
solely to the fact that different tones in the different cases were
reinforced by the resonance of the mouth. Donders first proved that the
mouth resounded differently for the different vowels.

In the formation of the different vowel sounds the resonant cavity of
the mouth undergoes, according to Helmholtz, the following changes:

For the production of the sound “U” (_oo_ in hoop), the lips must
be pushed forward, so as to make the cavity of the mouth as deep as
possible, and the orifice of the mouth, by the contraction of the lips,
as small as possible. This arrangement corresponds to the deepest
resonance of which the mouth is capable. The fundamental tone itself of
the vocal chords is here reinforced, while the higher tones retreat.

The vowel “O” requires a somewhat wider opening of the mouth. The
overtones which lie in the neighborhood of the middle _b_ of the
soprano come out strongly in the case of this vowel.

When “Ah” is sounded, the mouth assumes the shape of a funnel, widening
outward. It is thus tuned to a note an octave higher than in the case
of the vowel “O.” Hence, in sounding “Ah,” those overtones are most
strengthened which lie near the higher _b_ of the soprano. As the mouth
is in this case wide open, all the other overtones are also heard,
though feebly.

In sounding “A” and “E,” the hinder part of the mouth is deepened,
while the front of the tongue rises against the gums and forms a tube;
this yields a higher resonance-tone, rising gradually from “A” to “E,”
while the hinder hollow space yields a lower resonance-tone, which is
deepest when “E” is sounded.

These examples sufficiently illustrate the subject of vowel sounds. We
may blend in various ways the elementary tints of the solar spectrum,
producing innumerable composite colors by their admixture. Out of
violet and red we produce purple, and out of yellow and blue we produce
white. Thus also may elementary sounds be blended so as to produce all
possible varieties of clang-tint. After having resolved the human voice
into its constituent tones, Helmholtz was able to imitate these tones
by tuning-forks, and, by combining them appropriately together, to
produce the sounds of all the vowels.


§ 18. _Kundt’s Experiments: New Modes of determining Velocity of Sound_

Unwilling to interrupt the continuity of our reasonings and experiments
on the sound of organ-pipes, and their relations to the vibrations of
solid rods, I have reserved for the conclusion of this discourse some
reflections and experiments which, in strictness, belong to an earlier
portion of the chapter. You have already heard the tones, and made
yourselves acquainted with the various modes of division of a glass
tube, free at both ends, when thrown into longitudinal vibration. When
it sounds its fundamental tone, you know that the two halves of such
a tube lengthen and shorten in quick alternation. If the tube were
stopped at its ends, the closed extremities would throw the air within
the tube into a state of vibration; and if the velocity of sound in air
were equal to its velocity in glass, the air of the tube would vibrate
in synchronism with the tube itself. But the velocity of sound in air
is far less than its velocity in glass, and hence, if the column of air
is to synchronize with the vibrations of the tube, it can only do so
by dividing itself into vibrating segments of a suitable length. In an
investigation of great interest, recently published in “Poggendorff’s
Annalen,” M. Kundt, of Berlin, has taught us how these segments may
be rendered visible. Into this six-foot tube is introduced the light
powder of lycopodium, being shaken all over the interior surface. A
small quantity of the powder clings to that surface. Stopping the ends
of the tube, holding its centre by a fixed clamp, and sweeping a wet
cloth briskly over one of its halves, instantly the powder, which a
moment ago clung to its interior surface, falls to the bottom of the
tube in the forms shown in Fig. 109, the arrangement of the lycopodium
marking the manner in which the column of air has been divided. Every
node here is encircled by a ring of dust, while from node to node the
dust arranges itself in transverse streaks along the ventral segments.

[Illustration: FIG. 109.]

You will have little difficulty in seeing that we perform here, with
air, substantially the same experiment as that of M. Melde with a
vibrating string. When the string was too long to vibrate as a whole,
it met the requirements of the tuning-fork to which it was attached
by dividing into ventral segments. Now, in all cases, the length from
a node to its next neighbor is half that of the sonorous wave: how
many such half-waves then have we in our tube in the present instance?
Sixteen (the figure shows only four of them). But the length of our
glass tube vibrating thus longitudinally is also half that of the
sonorous wave _in glass_. Hence, in the case before us, with the same
rate of vibration, the length of the semi-wave in glass is sixteen
times the length of the semi-wave in air. In other words, the velocity
of sound in glass is sixteen times its velocity in air. Thus, by a
single sweep of the wet rubber, we solve a most important problem.
But, as M. Kundt has shown, we need not confine ourselves to air.
Introducing any other gas into the tube, a single stroke of our wet
cloth enables us to determine the relative velocity of sound in that
gas and in glass. When hydrogen is introduced, the number of ventral
segments is less than in air; when carbonic acid is introduced, the
number is greater.

From the known velocity of sound in air, coupled with the length of
one of these dust segments, we can immediately deduce the number of
vibrations executed in a second by the tube itself. Clasping a glass
tube at its centre and drawing my wetted cloth over one of its halves,
I elicit this shrill note. The length of every dust segment, now
within the tube, is 3 inches. Hence the length of the aërial sonorous
wave corresponding to this note is 6 inches. But the velocity of
sound in air of our present temperature is 1,120 feet per second; a
distance which would embrace 2,240 of our sonorous waves. This number,
therefore, expresses the number of vibrations per second executed by
the glass tube now before us.

Instead of damping the centre of the tube, and making it a nodal point,
we may employ any other of its subdivisions. Laying hold of it, for
example, at a point midway between its centre and one of its ends, and
rubbing it properly, it divides into three vibrating parts, separated
by two nodes. We know that in this division the note elicited is the
octave of that heard when a single node is formed at the middle of the
tube; for the vibrations are twice as rapid. If therefore we divide the
tube, having air within it, by two nodes instead of one, the number of
ventral segments revealed by the lycopodium dust will be thirty-two
instead of sixteen. The same remark applies, of course, to all other
gases.

Filling a series of four tubes with air, carbonic acid, coal-gas,
and hydrogen, and then rubbing each so as to produce two nodes, M.
Kundt found the number of dust segments formed within the tube in the
respective cases to be as follows:


  Air                 32 dust segments
  Carbonic acid       40      ”
  Coal-gas            20      ”
  Hydrogen             9      ”

Calling the velocity in air unity, the following fractions express the
ratio of this velocity to those in the other gases:

                      32
  Carbonic acid       —— = 0·8
                      40
                      32
  Coal-gas            —— = 1·6
                      20
                      32
  Hydrogen            —— = 3·56
                       9

[Illustration: FIG. 110.]

Referring to a table introduced in our first chapter, we learn that
Dulong by a totally different mode of experiment found the velocity in
carbonic acid to be 0.86, and in hydrogen 3·8 times the velocity in
air. The results of Dulong were deduced from the sounds of organ-pipes
filled with the various gases; but here, by a process of the utmost
simplicity, we arrive at a close approximation to his results. Dusting
the interior surfaces of our tubes, filling them with the proper
gases, and sealing their ends, they may be preserved for an indefinite
time. By properly shaking one of them at any moment, its inner surface
becomes thinly coated with the dust; and afterward a single stroke
of the wet cloth produces the division from which the velocity of
sound in the gas may be immediately inferred. Savart found that a
spiral nodal line is formed round a tube or rod when it vibrates
longitudinally, and Seebeck showed that this line was produced, not
by longitudinal, but by secondary transverse vibrations. Now this
spiral nodal line tends to complicate the division of the dust in our
present experiments. It is, therefore, desirable to operate in a manner
which shall altogether avoid the formation of this line; M. Kundt has
accomplished this, by exciting the longitudinal vibrations in one tube,
and producing the division into ventral segments in another, into which
the first fits like a piston. Before you is a tube of glass, Fig. 110,
seven feet long, and two inches internal diameter. One end of this
tube is filled by the movable stopper _b_. The other end, K K, is also
stopped by a cork, through the centre of which passes the narrower tube
A _a_, which is firmly clasped at its middle by the cork, K K. The end
of the interior tube, A _a_, is also closed with a projecting stopper,
_a_, almost sufficient to fill the larger tube, but still fitting into
it so loosely that the friction of _a_ against the interior surface is
too slight to interfere practically with its vibrations. The interior
surface between _a_ and _b_ being lightly coated with the lycopodium
dust, a wet cloth is passed briskly over A K; instantly the dust
between _a_ and _b_ divides into a number of ventral segments. When
the length of the column of air, _a b_, is equal to that of the glass
tube, A _a_, the number of ventral segments is sixteen. When, as in
the figure, _a b_ is only one-half the length of A _a_, the number of
ventral segments is eight.

But here you must perceive that the method of experiment is capable
of great extension. Instead of the glass tube, A _a_, we may employ a
rod of any other solid substance—of wood or metal, for example, and
thus determine the relative velocity of sound in the solid and in air.
In the place of the glass tube, for example, a rod of brass of equal
length may be employed. Rubbing its external half by a resined cloth,
it divides the column _a b_ into the number of ventral segments proper
to the metal’s rate of vibrations. In this way M. Kundt operated with
brass, steel, glass, and copper, and his results prove the method to
be capable of great accuracy. Calling, as before, the velocity of
sound in air unity, the following numbers expressive of the ratio of
the velocity of sound in brass to its velocity in air were obtained in
three different series of experiments:

  1st experiment           10·87
  2d experiment            10·87
  3d experiment            10·86

The coincidence is here extraordinary. To illustrate the possible
accuracy of the method, the length of the individual dust segments was
measured. In a series of twenty-seven experiments, this length was
found to vary between 43 and 44 millimètres (each millimètre 1/25th of
an inch), never rising so high as the latter and never falling so low
as the former. The length of the metal rod, compared with that of one
of the segments capable of this accurate measurement, gives us at once
the velocity of sound in the metal, as compared with its velocity in
air.

Three distinct experiments, performed in the same manner on steel,
gave the following velocities, the velocity through air, as before,
being regarded as unity:

  1st experiment           15·34
  2d experiment            15·33
  3d experiment            15·34

Here the coincidence is quite as perfect as in the case of brass.

In glass, by this new mode of experiment, the velocity was found to be

  15·25.[49]

Finally, in copper the velocity was found to be

  11·96.

[Illustration: FIG. 111.]

These results agree extremely well with those obtained by other
methods. Wertheim, for example, found the velocity of sound in steel
wire to be 15·108; M. Kundt finds it to be 15·34: Wertheim also found
the velocity in copper to be 11·17; M. Kundt finds it to be 11·96. The
differences are not greater than might be produced by differences in
the materials employed by the two experimenters.

The length of the aërial column may or may not be an exact multiple
of the wave-length, corresponding to the rod’s rate of vibration. If
not, the dust segments usually take the form shown in Fig. 111. But
if, by means of the stopper, _b_, the column of air be made an exact
multiple of the wave-length, then the dust quits the vibrating segments
altogether, and forms, as in Fig. 112, little isolated heaps at the
nodes.


§ 19. _Explanation of a Difficulty_

And here a difficulty presents itself. The stopped end _b_ of the tube
Fig. 110 is, of course, a place of no vibration, where in all cases a
nodal dust-heap is formed; but, whenever the column of air was an exact
multiple of the wave-length, M. Kundt always found a dust-heap close to
the end _a_ of the vibrating rod also. Thus the point from which all
the vibration emanated seemed itself to be a place of no vibration.

[Illustration: FIG. 112.]

This difficulty was pointed out by M. Kundt, but he did not attempt its
solution. We are now in a condition to explain it. In Lecture III. it
was remarked that in strictness a node is not a place of no vibration;
that it is a place of _minimum_ vibration; and that, by the addition of
the minute pulses which the node permits, vibrations of vast amplitude
may be produced. The ends of M. Kundt’s tube are such points of minimum
motion, the lengths of the vibrating segments being such that, by the
coalescence of direct and reflected pulses, the air at a distance of
half a ventral segment from the end of the tube vibrates much more
vigorously than that at the end of the tube itself. This addition of
impulses is more perfect when the aërial column is an exact multiple
of the wave-length, and hence it is that, in this case, the vibrations
become sufficiently intense to sweep the dust altogether away from
the vibrating segments. The same point is illustrated by M. Melde’s
tuning-forks, which, though they are the sources of all the motion, are
themselves nodes.

An experiment of Helmholtz’s is here capable of instructive
application. Upon the string of the sonometer described in our third
lecture I place the iron stem of this tuning-fork, which executes 512
complete vibrations in a second. At present you hear no augmentation of
the sound of the fork; the string remains quiescent. But on moving the
fork along the string, at the number 33, a loud, swelling note issues
from the string. At this particular tension the length 33 exactly
synchronizes with the vibrations of the fork. By the intermediation
of the string, therefore, the fork is enabled to transfer its motion
to the sonometer, and through it to the air. The sound continues as
long as the fork vibrates, but the least movement to the right or to
the left from this point causes a sudden fall of the sound. Tightening
the string, the note disappears; for it requires a greater length of
this more highly tensioned string to respond to the fork. But, on
moving the fork further away, at the number 36 the note again bursts
forth. Tightening still more, 40 is found to be the point of maximum
power. When the string is slackened, it must, of course, be shortened
in order to make it respond to the fork. Moving the fork now toward
the end of the string, at the number 25 the note is found as before.
Again, shifting the fork to 35, nothing is heard; but, by the cautious
turning of the key, the point of synchronism, if I may use the term,
is moved further from the end of the string. It finally reaches the
fork, and at that moment a clear, full note issues from the sonometer.
In all cases, before the exact point is attained, and immediately in
its vicinity, we hear “beats,” which, as we shall afterward understand,
are due to the coalescence of the sound of the fork with that of the
string, when they are nearly, but not quite, in unison with each other.

In these experiments, though the fork was the source of all the motion,
_the point on which it rested was a nodal point_. It constituted
the comparatively fixed extremity of the wire, whose vibrations
synchronized with those of the fork. The case is exactly analogous to
that of the hand holding the India-rubber tube, and to the tuning-fork
in the experiments of M. Melde. It is also an effect precisely the same
in kind as that observed by M. Kundt, where the part of the column of
air in contact with the end of his vibrating rod proved to be a node
instead of the middle of a ventral segment.


ADDENDUM REGARDING RESONANCE

The resonance of caves and of rocky inclosures is well known. Bunsen
notices the thunder-like sound produced when one of the steam jets
of Iceland breaks out near the mouth of a cavern. Most travellers in
Switzerland have noticed the deafening sound produced by the fall of
the Reuss at the Devil’s Bridge. The sound heard when a hollow shell
is placed close to the ear is a case of resonance. Children think
they hear in it the sound of the sea. The noise is really due to the
reinforcement of the feeble sounds with which even the stillest air is
pervaded, and also in part to the noise produced by the pressure of the
shell against the ear itself. By using tubes of different lengths, the
variation of the resonance with the length of the tube may be studied.
The channel of the ear itself is also a resonant cavity. When a poker
is held by two strings, and when the fingers of the hands holding the
poker are thrust into the ears on striking the poker against a piece
of wood, a sound is heard as deep and sonorous as that of a cathedral
bell. When open, the channel of the ear resounds to notes whose periods
of vibration are about 3,000 per second. This has been shown by
Helmholtz, and Madame Seiler has found that dogs which howl to music
are particularly sensitive to the same notes. We may expect from Mr.
Francis Galton interesting results in connection with this subject.


SUMMARY OF CHAPTER V

When a stretched wire is suitably rubbed, in the direction of its
length, it is thrown into longitudinal vibrations: the wire can either
vibrate as a whole or divide itself into vibrating segments separated
from each other by nodes.

The tones of such a wire follow the order of the numbers 1, 2, 3, 4,
etc.

The _transverse_ vibrations of a rod fixed at both ends do not follow
the same order as the transverse vibrations of a stretched wire; for
here the forces brought into play, as explained in Lecture IV., are
different. But the longitudinal vibrations of a stretched wire do
follow the same order as the longitudinal vibrations of a rod fixed at
both ends, for here the forces brought into play are the same, being in
both cases the elasticity of the material.

A rod fixed at one end vibrates longitudinally as a whole, or it
divides into two, three, four, etc., vibrating parts, separated from
each other by nodes. The order of the tones of such a rod is that of
the odd numbers 1, 3, 5, 7, etc.

A rod free at both ends can also vibrate longitudinally. Its lowest
note corresponds to a division of the rod into two vibrating parts by
a node at its centre. The overtones of such a rod correspond to its
division into three, four, five, etc., vibrating parts, separated from
each other by two, three, four, etc., nodes. The order of the tones of
such a rod is that of the numbers 1, 2, 3, 4, 5, etc.

We may also express the order by saying that while the tones of a rod
fixed at both ends follow the order of the odd numbers 1, 3, 5, 7,
etc., the tones of a rod free at both ends follow the order of the even
numbers 2, 4, 6, 8, etc.

At the points of maximum vibration the rod suffers no change of
density; at the nodes, on the contrary, the changes of density reach
a maximum. This may be proved by the action of the rod upon polarized
light.

Columns of air of definite length resound to tuning-forks of definite
rates of vibration.

The length of a tube filled with air, and closed at one end, which
resounds to a fork is one-fourth of the length of the sonorous wave
produced by the fork.

This resonance is due to the synchronism which exists between the
vibrating period of the fork and that of the column of air.

By blowing across the mouth of a tube closed at one end, we produce a
flutter of the air, and some pulse of this flutter may be raised by the
resonance of the tube to a musical sound.

The sound is the same as that obtained when a tuning-fork, whose rate
of vibration is that of the tube, is placed over the mouth of the tube.

When a tube closed at one end—a stopped organ-pipe, for example—sounds
its lowest note, the column of air within it is undivided by a node.
The overtones of such a column correspond to its division into parts,
like those of a rod fixed at one end and vibrating longitudinally. The
order of its tones is that of the odd numbers 1, 3, 5, 7, etc. That
this must be the order follows from the manner in which the column is
divided.

In organ-pipes the air is agitated by causing it to issue from a narrow
slit, and to strike upon a cutting edge. Some pulse of the flutter thus
produced is raised by the resonance of the pipe to a musical sound.

When, instead of the aërial flutter, a tuning-fork of the proper rate
of vibration is placed at the embouchure of an organ-pipe, the pipe
_speaks_ in response to the fork. In practice, the organ-pipe virtually
creates its own tuning-fork, by compelling the sheet of air at its
embouchure to vibrate in periods synchronous with its own.

An open organ-pipe yields a note an octave higher than that of a closed
pipe of the same length. This relation is a necessary consequence of
the respective modes of vibration.

When, for example, a stopped organ-pipe sounds its deepest note, the
column of air, as already explained, is undivided. When an open pipe
sounds its deepest note, the column is divided by a node at its centre.
The open pipe in this case virtually consists of two stopped pipes with
a common base. Hence it is plain that the fundamental note of an open
pipe must be the same as that of a stopped pipe of half its length.

The length of a stopped pipe is one-fourth that of the sonorous wave
which it produces, while the length of an open pipe is one-half that of
its sonorous wave.

The order of the tones of an open pipe is that of the even numbers 2,
4, 6, 8, etc., or of the natural numbers 1, 2, 3, 4, etc.

In both stopped and open pipes the number of vibrations executed in a
given time is inversely proportional to the length of the pipe.

The places of maximum vibration in organ-pipes are places of minimum
changes of density; while at the places of minimum vibration the
changes of density reach a maximum.

The velocities of sound in gases, liquids, and solids may be inferred
from the tones which equal lengths of them produce; or they may be
inferred from the lengths of these substances which yield equal tones.

Reeds, or vibrating tongues, are often associated with vibrating
columns of air. They consist of flexible laminæ, which vibrate to
and fro in a rectangular orifice, thus rendering intermittent the
air-current passing through the orifice.

The action of the reed is the same as that of the siren.

The flexible wooden reeds sometimes associated with organ-pipes are
compelled to vibrate in unison with the column of air in the pipe;
other reeds are too stiff to be thus controlled by the vibrating air.
In this latter case the column of air is taken of such a length that
its vibrations synchronize with those of the reed.

By associating suitable pipes with reeds we impart to their tones the
qualities of the human voice.

The vocal organ in man is a reed instrument, the vibrating reed in this
case being elastic bands placed at the top of the trachea, and capable
of various degrees of tension.

The rate of vibration of these vocal chords is practically uninfluenced
by the resonance of the mouth; but the mouth, by changing its shape,
can be caused to resound to the fundamental tone, or to any of the
overtones of the vocal chords.

By the strengthening of particular tones through the resonance of the
mouth, the clang-tint of the voice is altered.

The different vowel-sounds are produced by different admixtures of the
fundamental tone and the overtones of the vocal chords.

When the solid substance of a tube stopped at one, or at both ends, is
caused to vibrate longitudinally, the air within it is also thrown into
vibration.

By covering the interior surface of the tube with a light powder,
the manner in which the aërial column divides itself may be rendered
apparent. From the division of the column the velocity of sound in
the substance of the tube, compared with its velocity in air, may be
inferred.

Other gases may be employed instead of air, and the velocity of sound
in these gases, compared with its velocity in the substance of the
tube, may be determined.

The end of a rod vibrating longitudinally may be caused to agitate a
column of air contained in a tube, compelling the air to divide itself
into ventral segments. These segments may be rendered visible by light
powders, and from them the velocity of sound in the substance of the
vibrating rod, compared with its velocity in air, may be inferred.

In this way the relative velocities of sound in all solid substances
capable of being formed into rods, and of vibrating longitudinally, may
be determined.




CHAPTER VI

  Singing Flames—Influence of the Tube surrounding
  the Flame—Influence of Size of Flame—Harmonic Notes
  of Flames—Effect of Unisonant Notes on Singing
  Flames—-Action of Sound on Naked Flames—Experiments with
  Fish-Tail and Bat’s-Wing Burners—Experiments on Tall
  Flames—Extraordinary Delicacy of Flames as Acoustic
  Reagents—The Vowel-Flame—Action of Conversational
  Tones upon Flames—Action of Musical Sounds on
  Smoke-Jets—Constitution of Water-Jets—Plateau’s Theory
  of the Resolution of a Liquid Vein into Drops—Action of
  Musical Sounds on Water-Jets—A Liquid Vein may compete in
  Point of Delicacy with the Ear


§ 1. _Rhythm of Friction: Musical Flow of a Liquid through a Small
Aperture_

Friction is always rhythmic. When a resined bow is passed across a
string, the tension of the string secures the perfect rhythm of the
friction. When the wetted finger is moved round the edge of a glass,
the breaking up of the friction into rhythmic pulses expresses itself
in music. Savart’s beautiful experiments on the flow of liquids through
small orifices bear immediately upon this question. We have here the
means of verifying his results. The tube A B, Fig. 113, is filled with
water, its extremity, B, being closed by a plate of brass, which is
pierced by a circular orifice of a diameter equal to the thickness of
the plate. Removing a little peg which stops the orifice, the water
issues from it, and as it sinks in the tube a musical note of great
sweetness issues from the liquid column. This note is due to the
intermittent flow of the liquid through the orifice, by which the whole
column above it is thrown into vibration. The tendency to this effect
shows itself when tea is poured from a teapot, in the circular ripples
that cover the falling liquid. The same intermittence is observed in
the black, dense smoke which rolls in rhythmic rings from the funnel
of a steamer. The unpleasant noise of unoiled machinery is also a
declaration of the fact that the friction is not uniform, but is due to
the alternate “bite” and release of the rubbing surfaces.

[Illustration: FIG. 113.]

Where gases are concerned, friction is of the same intermittent
character. A rifle-bullet sings in its passage through the air; while
to the rubbing of the wind against the boles and branches of the trees
are to be ascribed the “waterfall tones” of an agitated pine-wood. Pass
a steadily-burning candle rapidly through the air; an indented band of
light, declaring intermittence, is often the consequence, while the
almost musical sound which accompanies the appearance of this band
is the audible expression of the rhythm. On the other hand, if you
blow gently against a candle-flame, the fluttering noise announces
a rhythmic action. We have already learned what can be done when a
pipe is associated with such a flutter; we have learned that the pipe
selects a special pulse from the flutter, and raises it by resonance
to a musical sound. In a similar manner the noise of a flame may be
turned to account. The blow-pipe flame of our laboratory, for example,
when inclosed within an appropriate tube, has its flutter raised to a
_roar_. The special pulse first selected soon reacts upon the flame
so as to abolish in a great degree the other pulses, compelling the
flame to vibrate in periods answering to the selected one. And this
reaction can become so powerful—the timed shock of the reflected pulses
may accumulate to such an extent—as to beat the flame, even when very
large, into extinction.


§ 2. _Musical Flames_

Nor is it necessary to produce this flutter by any extraneous means.
When a gas-flame is simply inclosed within a tube, the passage of the
air over it is usually sufficient to produce the necessary rhythmic
action, so as to cause the flame to burst spontaneously into song. This
flame-music may be rendered exceedingly intense. Over a flame issuing
from a ring burner with twenty-eight orifices, I place a tin tube 5
feet long and 2-1/2 inches in diameter. The flame flutters at first,
but it soon chastens its impulses into perfect periodicity, and a deep
and clear musical tone is the result. By lowering the gas the note now
sounded is caused to cease, but, after a momentary interval of silence,
another note, which is the octave of the last, is yielded by the flame.
The first note was the fundamental note of the surrounding tube;
this second note is its first harmonic. Here, as in the case of open
organ-pipes, we have the aërial column dividing itself into vibrating
segments, separated from each other by nodes.

[Illustration: FIG. 114.]

A still more striking effect is obtained with this larger tube, _a
b_, Fig. 114, 15 feet long and 4 inches wide, which was made for a
totally different purpose. It is supported by a steady stand, _s s′_,
and into it is lifted the tall burner, shown enlarged at B. You hear
the incipient flutter: you now hear the more powerful sound. As the
flame is lifted higher the action becomes more violent, until finally
a storm of music issues from the tube. And now all has suddenly
ceased; the reaction of its own pulses upon the flame has beaten it
into extinction. I relight the flame and make it very small. When
raised within the tube, the flame again sings, but it is one of the
harmonics of the tube that you now hear. On turning the gas fully on,
the note ceases—all is silent for a moment; but the storm is brewing,
and soon it bursts forth, as at first in a kind of hurricane of sound.
By lowering the flame the fundamental note is abolished, and now you
hear the first harmonic of the tube. Making the flame still smaller,
the first harmonic disappears, and the second is heard. Your ears
being disciplined to the apprehension of these sounds, I turn the gas
once more fully on. Mingling with the deepest note you notice the
harmonics, as if struggling to be heard amid the general uproar of
the flame. With a large Bunsen’s rose burner, the sound of this tube
becomes powerful enough to shake the floor and seats, and the large
audience that occupies the seats of this room, while the extinction of
the flame, by the reaction of the sonorous pulses, announces itself by
an explosion almost as loud as a pistol-shot. It must occur to you that
a chimney is a tube of this kind upon a large scale, and that the roar
of a flame in a chimney is simply a rough attempt at music.

[Illustration: FIG. 115.]

Let us now pass on to shorter tubes and smaller flames. Placing tubes
of different lengths over eight small flames, each of them starts into
song, and you notice that as the tubes lengthen the tones deepen. The
lengths of these tubes are so chosen that they yield in succession
the eight notes of the gamut. Round some of them you observe a
paper slider, _s_, Fig. 115, by which the tube can be lengthened or
shortened. If while the flame is sounding the slider be raised, the
pitch instantly falls; if lowered, the pitch rises. These experiments
prove the flame to be governed by the tube. By the reaction of
the pulses, reflected back upon the flame, its flutter is rendered
perfectly periodic, the length of that period being determined, as in
the case of organ-pipes, by the length of the tube.

[Illustration: FIG. 116.]

The fixed stars, especially those near the horizon, shine with an
unsteady light, sometimes changing color as they twinkle. I have often
watched at night, upon the plateaux of the Alps, the alternate flash of
ruby and emerald in the lower and larger stars. If you place a piece of
looking-glass so that you can see in it the image of such a star, on
tilting the glass quickly to and fro, the line of light obtained will
not be continuous, but will form a string of colored beads of extreme
beauty. The same effect is obtained when an opera-glass is pointed to
the star and shaken. This experiment shows that in the act of twinkling
the light of the star is quenched at intervals; the dark spaces
between the bright beads corresponding to the periods of extinction.
Now, our singing flame is a _twinkling_ flame. When it begins to sing
you observe a certain quivering motion which may be analyzed with a
looking-glass, or an opera-glass, as in the case of the star.[50] I can
now see the image of this flame in a small looking-glass. On tilting
the glass, so as to cause the image to form a circle of light, the
luminous band is not seen to be continuous, as it would be if the flame
were perfectly steady; it is resolved into a beautiful chain of flames,
Fig. 116.


§ 3. _Experimental Analysis of Musical Flame_

[Illustration: FIG. 117.]

With a larger, brighter, and less rapidly-vibrating flame, you may all
see this intermittent action. Over this gas-flame, _f_, Fig. 117, is
placed a glass tube, A B, 6 feet long and 2 inches in diameter. The
back of the tube is blackened, so as to prevent the light of the flame
from falling directly upon the screen, which it is now desirable to
have as dark as possible. In front of the tube is placed a concave
mirror, M, which forms upon the screen an enlarged image of the
flame. I turn the mirror with my hand and cause the image to pass
over the screen. Were the flame silent and steady, we should obtain a
_continuous_ band of light; but it quivers, and emits at the same time
a deep and powerful note. On twirling the mirror, therefore, we obtain,
instead of a continuous band, a luminous chain of images. By fast
turning, these images are drawn more widely apart; by slow turning,
they are caused to close up, the chain of flames passing through the
most beautiful variations. Clasping the lower end, B, of the tube
with my hand, I so impede the air as to stop the flame’s vibration; a
continuous band is the consequence. Observe the suddenness with which
this band breaks up into a rippling line of images the moment my hand
is removed and the current of air is permitted to pass over the flame.


§ 4. _Rate of Vibration of Flame: Toepler’s Experiment_

When a _small_ vibrating coal-gas flame is carefully examined by the
rotating mirror, the beaded line is a series of yellow-tipped flames,
each resting upon a base of the richest blue. In some cases I have
been unable to observe any union of one flame with another; the spaces
between the flames being absolutely dark to the eye. But if dark, the
flame must have been totally extinguished at intervals, a residue of
heat, however, remaining sufficient to reignite the gas. This is at
least possible, for gas may be ignited by non-luminous air.[51] By
means of the siren, we can readily determine the number of times this
flame extinguishes and relights itself in a second. As the note of the
instrument approaches that of the flame, unison is preceded by these
well-known beats, which become gradually less rapid, and now the two
notes melt into perfect unison. Maintaining the siren at this pitch for
a minute, at the end of that time I find recorded upon our dials 1,700
revolutions. But the disk being perforated by 16 holes, it follows
that every revolution corresponds to 16 pulses. Multiplying 1,700 by
16, we find the number of pulses in a minute to be 27,200. This number
of times did our little flame extinguish and rekindle itself during
the continuance of the experiment; that is to say, it was put out and
relighted 453 times in a second.

A flash of light, though instantaneous, makes an impression upon the
retina which endures for the tenth of a second or more. A flying
rifle-bullet, illuminated by a single flash of lightning, would seem
to stand still in the air for the tenth of a second. A black disk with
radial white strips, when rapidly rotated, causes the white and black
to blend to an impure gray; while a spark of electricity, or a flash
of lightning, reduces the disk to apparent stillness, the white radial
strips being for a time plainly seen. Now, the singing flame is a
flashing flame, and M. Toepler has shown that by causing a striped disk
to rotate at the proper speed in the presence of such a flame it is
brought to apparent stillness, the white stripes being rendered plainly
visible. The experiment is both easy and interesting.


§ 5. _Harmonic Sounds of Flame_

A singing flame yields so freely to the pulses falling upon it that
it is almost wholly governed by the surrounding tube; _almost_, but
not altogether. The pitch of the note depends in some measure upon
the size of the flame. This is readily proved, by causing two flames
to emit the same note, and then slightly altering the size of either
of them. The unison is instantly disturbed by beats. By altering the
size of a flame we can also, as already illustrated, draw forth the
harmonic overtones of the tube which surrounds it. This experiment is
best performed with hydrogen, its combustion being much more vigorous
than that of ordinary gas. When a glass tube 7 feet long is placed over
a large hydrogen-flame, the fundamental note of the tube is obtained,
corresponding to a division of the column of air within it by a single
node at the centre. Placing a second tube, 3 feet 6 inches long, over
the same flame, no musical sound whatever is obtained; the large flame,
in fact, is not able to accommodate itself to the vibrating period
of the shorter tube. But, on lessening the flame, it soon bursts
into vigorous song, its note being the octave of that yielded by the
longer tube. I now remove the shorter tube, and once more cover the
flame with the longer one. It no longer sounds its fundamental notes,
but the precise note of the shorter tube. To accommodate itself to
the vibrating period of the diminished flame, the longer column of
air divides itself like an open organ-pipe when it yields its first
harmonic. By varying the size of the flame, it is possible, with
the tube now before you, to obtain a series of notes whose rates of
vibration are in the ratio of the numbers 1:2:3:4:5; that is to say,
the fundamental tone and its first four harmonics.

These sounding flames, though probably never before raised to the
intensity in which they have been exhibited here to-day, are of old
standing. In 1777, the sounds of a hydrogen-flame were heard by Dr.
Higgins. In 1802, they were investigated to some extent by Chladni, who
also refers to an incorrect account of them given by De Luc. Chladni
showed that the tones are those of the open tube which surrounds the
flame, and he succeeded in obtaining the first two harmonics. In 1802,
G. De la Rive experimented on this subject. Placing a little water in
the bulb of a thermometer, and heating it, he showed that musical tones
of force and sweetness could be produced by the periodic condensation
of the vapor in the stem of the thermometer. He accordingly referred
the sounds of flames to the alternate expansion and condensation of
the aqueous vapor produced by the combustion. We can readily imitate
his experiments. Holding, with its stem oblique, a thermometer-bulb
containing water in the flame of a spirit-lamp the sounds are heard
soon after the water begins to boil. In 1818, however, Faraday showed
that the tones are produced when the tube surrounding the flame is
placed in air of a temperature higher than 100° C., condensation being
then impossible. He also showed that the tones could be obtained from
flames of carbonic oxide, where aqueous vapor is entirely out of the
question.


§ 6. _Action of Extraneous Sounds on Flame: Experiments of Schaffgotsch
and Tyndall_

After these experiments, the first novel acoustic observation on flames
was made in Berlin by the late Count Schaffgotsch, who showed that when
an ordinary gas-flame was surmounted by a short tube, a strong falsetto
voice pitched to the note of the tube, or to its higher octave, caused
the flame to quiver. In some cases when the note of the tube was high,
the flame could even be extinguished by the voice.

In the spring of 1857, this experiment came to my notice. No directions
were given in the short account of the observation published in
“Poggendorff’s Annalen”; but, in endeavoring to ascertain the
conditions of success, a number of singular effects forced themselves
upon my attention. Meanwhile, Count Schaffgotsch also followed up the
subject. To a great extent we travelled over the same ground, neither
of us knowing how the other was engaged; but, so far as the experiments
then executed are common to us both, to Count Schaffgotsch belongs the
priority.

Let me here repeat his first observation. Within this tube, 11 inches
long, burns a small gas-flame, bright and silent. The note of the
tube has been ascertained, and now, standing at some distance from
the flame, I sound that note; the flame _quivers_. To obtain the
_extinction_ of the flame it is necessary to employ a burner with a
very narrow aperture, from which the gas issues under considerable
pressure. On gently sounding the note of the tube surrounding such a
flame, it quivers; but on throwing more power into the voice the flame
is extinguished.

The cause of the quivering of the flame will be best revealed by an
experiment with the siren. As the note of the siren approaches that of
the flame you hear beats, and at the same time you observe a dancing of
the flame synchronous with the beats. The jumps succeed each other more
slowly as unison is approached. They cease when the unison is perfect,
and they begin again as soon as the siren is urged beyond unison,
becoming more rapid as the discordance is increased. The cause of the
quiver observed by M. Schaffgotsch was revealed to me. The flame jumped
because the note of the tube surrounding it was nearly, but not quite,
in unison with the voice of the experimenter.

That the jumping of the flame proceeds in exact accordance with the
beats is well shown by a tuning-fork, which yields the same note as
the flame. Loading such a fork with a bit of wax, so as to throw it
slightly out of unison, and bringing it, when agitated, near the tube
in which the flame is singing, the beats and the leaps of the flame
occur at the same intervals. When the fork is placed over a resonant
jar, all of you can hear the beats, and see at the same time the
dancing of the flame. By changing the load upon the tuning-fork, or by
slightly altering the size of the flame, the rate at which the beats
succeed each other may be altered; but in all cases the jumps address
the eye at the moments when the beats address the ear.

While executing these experiments I noticed that, on raising my voice
to the proper pitch, a flame which had been burning silently in its
tube began to sing. The same observation had, without my knowledge,
been made a short time previously by Count Schaffgotsch. A tube, 12
inches long, is placed over this flame, which occupies a position about
an inch and a half from the lower end of the tube. When the proper note
is sounded the flame trembles, but it does not sing. When the tube
is lowered until the flame is three inches from its end, the song is
spontaneous. Between these two positions there is a third, at which, if
the flame be placed, it will burn silently; but if it be excited by
the voice it will sing, and continue to sing.

[Illustration: FIG. 118.]

Even when the back is turned toward the flame the sonorous pulses run
round the body, reach the tube, and call forth the song. A pitch-pipe,
or any other instrument which yields a note of the proper height,
produces the same effect. Mounting a series of tubes, capable of
emitting all the notes of the gamut, over suitable flames, with an
instrument sufficiently powerful, and from a distance of 20 or 30
yards, a musician, by running over the scale, might call forth all the
notes in succession, the whole series of flames finally joining in the
song.

When a silent flame, capable of being excited in the manner here
described, is looked at in a moving mirror, it produces there a
continuous band of light. Nothing can be more beautiful than the sudden
breaking up of this band into a string of richly-luminous pearls at the
instant the voice is pitched to the proper note.

One singing flame may be caused to effect the musical ignition of
another. Before you are two small flames, _f′_ and _f_, Fig. 118 (p.
273), the tube over _f′_ being 10-1/2 inches, that over _f_ 12 inches
long. The shorter tube is clasped by a paper slider, _s_. The flame
_f′_ is now singing, but the flame _f_, in the longer tube, is silent.
I raise the paper slider which surrounds _f′_, so as to lengthen the
tube, and on approaching the pitch of the tube surrounding _f_, that
flame sings. The experiment may be varied by making _f_ the singing
flame and _f′_ the silent one at starting. Raising the telescopic
slider, a point is soon attained where the flame _f′_ commences its
song. In this way one flame may excite another through considerable
distances. It is also possible to silence the singing flame by the
proper management of the voice.


SENSITIVE NAKED FLAMES


§ 7. _Discovery of Sensitive Flames by Le Conte_

We have hitherto dealt with flames surrounded by resonant tubes; and
none of these flames, if naked, would respond in any way to such noise
or music as could be here applied. Still it is possible to make naked
flames thus sympathetic. This action of musical sounds upon naked
flames was first observed by Prof. Le Conte at a musical party in the
United States. His observation is thus described: “Soon after the
music commenced, I observed that the flame exhibited pulsations which
were _exactly synchronous_ with the audible beats. This phenomenon
was very striking to every one in the room, and especially so when the
strong notes of the violoncello came in. It was exceedingly interesting
to observe how perfectly even the _trills_ of this instrument were
reflected on the sheet of flame. _A deaf man might have seen the
harmony._ As the evening advanced, and the diminished consumption of
gas in the city _increased the pressure_, the phenomenon became more
conspicuous. The _jumping_ of the flame gradually increased, became
somewhat irregular, and, finally, it began to flare continuously,
emitting the characteristic sound, indicating the escape of a greater
amount of gas than could be properly consumed. I then ascertained,
by experiment, that the phenomenon _did not_ take place unless the
discharge of gas was so regulated that the flame approximated to
the condition of _flaring_. I likewise determined, by experiment,
that the effects _were not_ produced by jarring or shaking the floor
and walls of the room by means of repeated concussions. Hence it is
obvious that the pulsations of the flame _were not_ owing to _indirect_
vibrations propagated through the medium of the walls of the room to
the burning-apparatus, but must have been produced by the _direct_
influence of aërial sonorous pulses on the burning jet.”[52]

The significant remark, that the jumping of the flame was not observed
until it was near flaring, suggests the means of repeating the
experiments of Dr. Le Conte; while a more intimate knowledge of the
conditions of success enables us to vary and exalt them in a striking
degree. Before you burns a bright candle-flame, but no sound that can
be produced here has any effect upon it. Though sonorous waves of great
power be sent through the air, the candle-flame remains insensible.

[Illustration: FIG. 119.]

[Illustration: FIG. 120.]

But by proper precautions even a candle-flame may be rendered
sensitive. Urging from a small blow-pipe a narrow stream of air through
such a flame, an incipient flutter is produced. The flame then jumps
visibly to the sound of a whistle, or to a chirrup. The experiment may
be so arranged that, when the whistle sounds, the flame shall be either
restored almost to its pristine brightness, or that the small amount of
light it still possesses shall disappear.

The blow-pipe flame of our laboratory is totally unaffected by the
sound of the whistle as long as no air is urged through it. By properly
tempering the force of the blast, a flame is obtained of the shape
shown in Fig. 119. On sounding the whistle the erect portion of the
flame drops down, and while the sound continues the flame maintains
the form shown in Fig. 120.


§ 8. _Experiments on Fish-tail and Bat’s-wing Flames_

We now pass on to a thin sheet of flame, issuing from a common
fish-tail burner, Fig. 121. You might sing to this flame, varying the
pitch of your voice; no shiver of the flame would be visible. You might
employ pitch-pipes, tuning-forks, bells, and trumpets, with a like
absence of all effect. A barely perceptible motion of the interior of
the flame may be noticed when a shrill whistle is blown close to it.
But by turning the cock more fully on, the flame is brought to the
verge of flaring. And now, when the whistle is blown, the flame thrusts
suddenly out seven quivering tongues, Fig. 122. The moment the sound
ceases, the tongues disappear, and the flame becomes quiescent.

[Illustration: FIG. 121.]

[Illustration: FIG. 122.]

Passing from a fish-tail to a bat’s-wing burner, we obtain a broad,
steady flame, Fig. 123. It is quite insensible to the loudest
sound which would be tolerable here. The flame is fed from a small
gas-holder.[53] Increasing gradually the pressure, a slight flutter of
the edge of the flame at length answers to the sound of the whistle.
Turning on the gas until the flame is on the point of roaring, and
blowing the whistle, it roars, and suddenly assumes the form shown in
Fig. 124.

When a distant anvil is struck with a hammer, the flame instantly
responds by thrusting forth its tongues.

[Illustration: FIG. 123.]

[Illustration: FIG. 124.]

An essential condition to entire success in these experiments disclosed
itself in the following manner: I was operating on two fish-tail
flames, one of which jumped to a whistle while the other did not. The
gas of the non-sensitive flame was turned off, additional pressure
being thereby thrown upon the other flame. It flared, and its cock
was turned so as to lower the flame; but it now proved non-sensitive,
however close it might be brought to the point of flaring. The narrow
orifice of the half-turned cock interfered with the action of the
sound. When the gas was turned fully on, the flame being lowered by
opening the cock of the other burner, it became again sensitive. Up
to this time a great number of burners had been tried, but with many
of them the action was _nil_. Acting, however, upon the hint conveyed
by this observation, the cocks which fed the flames were more widely
opened, and our most refractory burners thus rendered sensitive.

In this way the observation of Dr. Le Conte is easily and strikingly
illustrated; in our subsequent, and far more delicate, experiments the
precaution just referred to is still more essential.


§ 9. _Experiments on Flames from Circular Apertures_

A long flame may be shortened and a short one lengthened, according
to circumstances, by sonorous vibrations. The flame shown in Fig. 125
is long, straight, and smoky; that in Fig. 126 is short, forked, and
brilliant. On sounding the whistle, the long flame becomes short,
forked, and brilliant, as in Fig. 127; while the forked flame becomes
long and smoky, as in Fig. 128. As regards, therefore, their response
to the sound of the whistle, one of these flames is the complement of
the other.

In Fig. 129 is represented another smoky flame which, when the whistle
sounds, breaks up into the form shown in Fig. 130.

When a brilliant sensitive flame illuminates an otherwise dark room,
in which a suitable bell is caused to strike, a series of periodic
quenchings of the light by the sound occurs. Every stroke of the bell
is accompanied by a momentary darkening of the room.

The foregoing experiments illustrate the lengthening and shortening of
flames by sonorous vibrations. They may also produce _rotation_. From
some of our homemade burners issue flat flames, about ten inches high,
and three inches across at their widest part. When the whistle sounds,
the plane of each flame turns ninety degrees round, and continues in
its new position as long as the sound continues.

[Illustration: FIG. 125.]

[Illustration: FIG. 126.]

[Illustration: FIG. 127.]

[Illustration: FIG. 128.]

[Illustration: FIG. 129.]

[Illustration: FIG. 130.]

A flame of admirable steadiness and brilliancy now burns before you.
It issues from a single circular orifice in a common iron nipple. This
burner, which requires great pressure to make its flame flare, has
been specially chosen for the purpose of enabling you to observe, with
distinctness, the gradual change from apathy to sensitiveness. The
flame, now 4 inches high, is quite indifferent to sound. On increasing
the pressure its height becomes 6 inches; but it is still indifferent.
When its length is 12 inches, a barely perceptible quiver responds to
the whistle. When 16 or 17 inches high, it jumps briskly the moment an
anvil is tapped or the whistle sounded. When the flame is 20 inches
long you observe a quivering at intervals, which announces that it is
near roaring. A slight increase of pressure causes it to roar, and
shorten at the same time to 8 inches.

Diminishing the pressure a little, the flame is again 20 inches long,
but it is on the point of roaring and shortening. Like the singing
flames which were started by the voice, it stands on the brink of
a precipice. The proper note pushes it over. It shortens when the
whistle sounds, exactly as it did when the pressure is in excess. The
action reminds one of the story of the Swiss muleteers, who are said
to tie up their bells at certain places lest the tinkle should bring
an avalanche down. The snow must be very delicately poised before this
could occur. It probably never did occur, but our flame illustrates the
principle. We bring it to the verge of falling, and the sonorous pulses
precipitate what was already imminent. This is the simple philosophy of
all these sensitive flames.

When the flame flares, the gas in the orifice of the burner is in a
state of vibration; conversely, when the gas in the orifice is thrown
into vibration, the flame, if sufficiently near the flaring point,
will flare. Thus the sonorous vibrations, by acting on the gas in the
passage of the burner, become equivalent to an augmentation of pressure
in the holder. In fact, we have here revealed to us the physical
cause of flaring through excess of pressure, which, common as it
is, has never been hitherto explained. The gas encounters friction
in the orifice of the burner, which, when the force of transfer is
sufficiently great, throws the issuing stream into the state of
vibration that produces flaring. It is because the flaring is thus
caused that an infinitesimal amount of energy in the form of vibrations
of the proper period can produce an effect equivalent to a considerable
increase of pressure.


§ 10. _Seat of Sensitiveness_

[Illustration: FIG. 131.]

That the external vibrations act upon the gas in the orifice of the
burner, and not first upon the burner itself, the tube leading to it,
or the flame above it, is thus proved. A glass funnel R, Fig. 131,
is attached to a tube 3 feet long and half an inch in diameter. A
sensitive flame _b_ is placed at the open end T of the tube, while a
small high-pitched reed is placed in the funnel at R. When the sound
is converged upon the root of the flame, as in Fig. 131, the action is
violent; when converged on a point half an inch above the burner, as in
Fig. 132, or at half an inch below the burner, as in Fig. 133, there is
no action. The glass tube may be dispensed with and the funnel alone
employed, if care be taken to screen off all sound, save that which
passes through the shank of the funnel.[54]


§ 11. _Influence of Pitch_

[Illustration: FIG. 132.]

[Illustration: FIG. 133.]

All sounds are not equally effective on the flame; waves of special
periods are required to produce the maximum effect. The effectual
periods are those which synchronize with the waves produced by the
friction of the gas itself against the sides of its orifice. With some
of these flames a low deep whistle is more effective than a shrill
one. With others the exciting tremors must be very rapid, and the
sound consequently shrill. Not one of these four tuning-forks, which
vibrate 256 times, 320 times, 384 times, and 512 times respectively
in a second, has any effect upon the flame from our iron nipple.
But, besides their fundamental tones, these forks, as you know, can
be caused to yield a series of overtones of very high pitch. The
vibrations of this series are 1,600, 2,000, 2,400, and 3,200 per
second, respectively. The flame jumps in response to each of these
sounds; the response to that of the highest pitch being the most prompt
and energetic of all.

To the tap of a hammer upon a board the flame responds; but to the tap
of the same hammer upon an anvil the response is much more brisk and
animated. The reason is, that the clang of the anvil is rich in the
higher tones to which the flame is most sensitive. The powerful tone
obtained when our inverted bell is reinforced by its resonant tube has
no power over this flame. But when a halfpenny is brought into contact
with the vibrating surface the flame instantly shortens, flutters, and
roars. I send an assistant with a smaller bell, worked by clockwork, to
the most distant part of the gallery. He there detaches the hammer; the
strokes follow each other in rhythmic succession, and at every stroke
the flame falls from a height of 20 to a height of 8 inches, roaring as
it falls.

[Illustration: FIG. 134.]

[Illustration: FIG. 135.]

The rapidity with which sound is propagated through air is well
illustrated by these experiments. There is no sensible interval between
the stroke of the bell and the ducking of the flame.

When the sound acting on the flame is of very short duration a curious
and instructive effect is observed. The sides of the flame half-way
down, and lower, are seen suddenly fringed by luminous tongues, the
central flame remaining apparently undisturbed in both height and
thickness. The flame in its normal state is shown in Fig. 134, and
with its fringes in Fig. 135. The effect is due to the retention of
the impression upon the retina. The flame actually falls as low as
the fringes, but its recovery is so quick that to the eye it does not
appear to shorten at all.[55]


§ 12. _The Vowel-flame_

A flame of astonishing sensitiveness now burns before you. It issues
from the single orifice of a steatite burner, and reaches a height of
24 inches. The slightest tap on a distant anvil reduces its height to 7
inches. When a bunch of keys is shaken the flame is violently agitated,
and emits a loud roar. The dropping of a sixpence into a hand already
containing coin, at a distance of 20 yards, knocks the flame down. It
is not possible to walk across the floor without agitating the flame.
The creaking of boots sets it in violent commotion. The crumpling, or
tearing of paper, or the rustle of a silk dress, does the same. It is
startled by the patter of a rain-drop. I hold a watch near the flame:
nobody hears its ticks; but you all see their effect upon the flame.
At every tick it falls and roars. The winding up of the watch also
produces tumult. The twitter of a distant sparrow shakes the flame; the
note of a cricket would do the same. A chirrup from a distance of 30
yards causes it to fall and roar. I repeat a passage from Spenser:

    “Her ivory forehead full of bounty brave,
      Like a broad table did itself dispread;
    For love his lofty triumphs to engrave,
      And write the battles of his great godhead.
    All truth and goodness might therein be read,
      For there their dwelling was, and when she spake,
    Sweet words, like dropping honey she did shed;
      And through the pearls and rubies softly brake
  A silver sound, which heavenly music seemed to make.”

The flame selects from the sounds those to which it can respond. It
notices some by the slightest nod, to others it bows more distinctly,
to some its obeisance is very profound, while to many sounds it turns
an entirely deaf ear.

[Illustration: FIG. 136.]

In Fig. 136, this wonderful flame is represented. On chirruping to
it, or on shaking a bunch of keys within a few yards of it, it falls
to the size shown in Fig. 137, the whole length, _a b_, of the flame
being suddenly abolished. The light at the same time is practically
destroyed, a pale and almost non-luminous residue of it alone
remaining. These figures are taken from photographs of the flame.

[Illustration: FIG. 137.]

To distinguish it from the others I have called this the “vowel-flame,”
because the different vowel-sounds affect it differently. A loud and
sonorous U does not move the flame; on changing the sound to O, the
flame quivers; when E is sounded, the flame is strongly affected. I
utter the words _boot_, _boat_, and _beat_, in succession. To the first
there is no response; to the second, the flame starts; by the third, is
thrown into greater commotion; the sound _Ah!_ is still more powerful.
Did we not know the constitution of vowel-sounds this deportment would
be an insoluble enigma. As it is, however, the flame illustrates the
theory of vowel-sounds. It is most sensitive to sounds of high pitch;
hence we should infer that the sound _Ah!_ contains higher notes than
the sound E; that E contains higher notes than O; and O higher notes
than U. I need not say that this agrees perfectly with the analysis of
Helmholtz.

[Illustration: FIG. 138.]

This flame is peculiarly sensitive to the utterance of the letter S.
A hiss contains the elements that most forcibly affect the flame.
The gas issues from its burner with a hiss, and an external sound of
this character is therefore exceedingly effective. From a metal box
containing compressed air I allow a puff to escape; the flame instantly
ducks down not by any transfer of _air_ from the box to the flame, for
the distance between both utterly excludes this idea—it is the _sound_
that affects the flame. From the most distant part of the gallery my
assistant permits the compressed air to issue in puffs from the box; at
every puff the flame suddenly falls. The hiss of the issuing air at the
one orifice precipitates the tumult of the flame at the other.

When a musical-box is placed on the table, and permitted to play, the
flame behaves like a sentient and motor creature—bowing slightly to
some tones, and courtesying deeply to others.

§ 13. _Mr. Philip Harry’s Sensitive Flame_

Mr. Philip Barry has discovered a new and very effective form of
sensitive flame, which he thus describes in a letter to myself: “It is
the most sensitive of all the flames that I am acquainted with, though
from its smaller size it is not so striking as your vowel-flame. It
possesses the advantage that the ordinary pressure in the gas-mains is
quite sufficient to produce it. The method of producing it consists in
igniting the gas (ordinary coal-gas) not at the burner but some inches
above it, by interposing between the burner and the flame a piece of
wire-gauze.”

I give a sketch of the arrangement adopted in Fig. 138. The space
between the burner and gauze was 2 inches. The gauze was about 7 inches
square, resting on the ring of a retort-stand. It had 32 meshes to the
lineal inch. The burner was Sugg’s steatite pinhole burner, the same as
used for the vowel-flame.

The flame is a slender cone about four inches high, the upper portion
giving a bright-yellow light, the base being a non-luminous blue flame.
At the least noise this flame roars, sinking down to the surface of the
gauze, becoming at the same time invisible. It is very active in its
responses, and, being rather a noisy flame, its sympathy is apparent to
the ear as well as the eye.

“To the vowel-sounds it does not appear to answer so discriminately as
the vowel-flame. It is extremely sensitive to A, very slightly to E,
more so to I, entirely non-sensitive to O, but slightly sensitive to U.

“It dances in the most perfect manner to a small musical snuff-box,
and is highly sensitive to most of the sonorous vibrations which affect
the vowel-flames.”


§ 14. _Sensitive Smoke-jets_

It is not to the flame, as such, that we owe the extraordinary
phenomena which have been just described. Effects substantially the
same are obtained when a jet of unignited gas, of carbonic acid,
hydrogen, or even air itself, issues from an orifice under proper
pressure. None of these gases, however, can be seen in its passage
through air, and, therefore, we must associate with them some substance
which, while sharing their motions, will reveal them to the eye. The
method employed from time to time in this place of rendering aërial
vortices visible is well known to many of you. By tapping a membrane
which closes the mouth of a large funnel filled with smoke, we
obtain beautiful smoke-rings, which reveal the motion of the air. By
associating smoke with our gas-jets, in the present instance, we can
also trace their course, and, when this is done, the unignited gas
proves as sensitive as the flames. The smoke-jets jump, shorten, split
into forks, or lengthen into columns, when the proper notes are sounded.

Underneath this gas-holder are placed two small basins, the one
containing hydrochloric acid, and the other ammonia. Fumes of
sal-ammoniac are thus copiously formed, and mingle with the gas
contained in the holder. We may, as already stated, operate with
coal-gas, carbonic acid, air, or hydrogen; each of them yields good
effects. From our excellent steatite burner now issues a thin column of
smoke. On sounding the whistle, which was so effective with the flames,
it is found ineffective. When, moreover, the highest notes of a series
of Pandean pipes are sounded, they are also ineffective. Nor will the
lowest notes answer. But when a certain pipe, which stands about the
middle of the series, is sounded, the smoke-column falls, forming a
short stem with a thick, bushy head. It is also pressed down, as if by
a vertical wind, by a knock upon the table. At every tap it drops. A
stroke on an anvil, on the contrary, produces little or no effect. In
fact, the notes here effective are of a much lower pitch than those
which were most efficient in the case of the flames.

[Illustration: FIG. 139.]

The amount of shrinkage exhibited by some of these smoke-columns, in
proportion to their length, is far greater than that of the flames. A
tap on the table causes a smoke-jet eighteen inches high to shorten
to a bushy bouquet, with a stem not more than an inch in height. The
smoke-column, moreover, responds to the voice. A cough knocks it down;
and it dances to the tune of a musical-box. Some notes cause the mere
top of the smoke-column to gather itself up into a bunch; at other
notes the bunch is formed midway down; while notes of more suitable
pitch cause the column to contract itself to a cumulus not much more
than an inch above the end of the burner. Various forms of the dancing
smoke-jet are shown in Fig. 139. As the music continues, the action of
the smoke-column consists of a series of rapid leaps from one of these
forms to another.

In a perfectly still atmosphere these slender smoke-columns rise
sometimes to a height of nearly two feet, apparently vanishing into
air at the summit. When this is the case, our most sensitive flames
fall far behind them in delicacy; and though less striking than the
flames, the smoke-wreaths are often more graceful. Not only special
words, but every word, and even every syllable, of the foregoing stanza
from Spenser, tumbles a really sensitive smoke-jet into confusion. To
produce such effects, a perfectly tranquil atmosphere is necessary.
Flame-experiments, in fact, are possible in an atmosphere where
smoke-jets are utterly unmanageable.[56]


§ 15. _Constitution of Liquid Veins: Sensitive Water-jets_

[Illustration: FIG. 140.]

[Illustration: FIG. 141.]

[Illustration: FIG. 142.]

We have thus far confined our attention to jets of ignited and
unignited coal-gas—of carbonic acid, hydrogen, and air. We will now
turn to jets of water. And here a series of experiments, remarkable
for their beauty, has long existed, which claim relationship to those
just described. These are the experiments of Felix Savart on liquid
veins. If the bottom of a vessel containing water be pierced by a
circular orifice, the descending liquid vein will exhibit two parts
unmistakably distinct. The part of the vein nearest the orifice is
steady and limpid, presenting the appearance of a solid glass rod.
It decreases in diameter as it descends, reaches a point of maximum
contraction, from which point downward it appears turbid and unsteady.
The course of the vein, moreover, is marked by periodic swellings and
contractions. Savart has represented these appearances as in Fig. 140.
The part _a n_ nearest the orifice is limpid and steady, while all the
part below _n_ is in a state of quivering motion. This lower part of
the vein appears continuous to the eye; but the finger can be sometimes
passed through it without being wetted. This, of course, could not be
the case if the vein were really continuous. The upper portion of the
vein, moreover, intercepts vision; the lower portion, even when the
liquid is mercury, does not. In fact, the vein resolves itself, at
_n_, into liquid spherules, its apparent continuity being due to the
retention of the impressions made by the falling drops upon the retina.
If, while looking at the disturbed portion of the vein, the head be
suddenly lowered, the descending column will be resolved for a moment
into separate drops. Perhaps the simplest way of reducing the vein to
its constituent spherules is to illuminate the vein, in a dark room, by
a succession of electric flashes. Every flash reveals the drops, as if
they were perfectly motionless in the air.

Could the appearance of the vein illuminated by a single flash be
rendered permanent, it would be that represented in Fig. 141. And here
we find revealed the cause of those swellings and contractions which
the disturbed portion of the vein exhibits. The drops, as they descend,
are continually changing their forms. When first detached from the
end of the limpid portion of the vein, the drop is a spheroid with
its longest axis vertical. But a liquid cannot retain this shape, if
abandoned to the forces of its own molecules. The spheroid seeks to
become a sphere—the longer diameter, therefore, shortens; but, like a
pendulum which seeks to return to its position of rest, the contraction
of the vertical diameter goes too far, and the drop becomes a flattened
spheroid. Now, the contractions of the jet are formed at those places
where the longest axis of the drop is vertical, while the swellings
appear where the longest axis is horizontal. It will be noticed that
between every two of the larger drops is a third one of much smaller
dimensions. According to Savart, their appearance is invariable.

I wish to make the constitution of a liquid vein evident to you by a
simple but beautiful experiment. The condensing lens has been removed
from our electric lamp, the light being permitted to pass through a
vertical slit directly from the carbon-points. The slice of light
thus obtained is so divergent that it illuminates, from top to bottom,
a liquid vein several feet long, and placed at some distance from
the lamp. Immediately in front of the camera is a large disk of zinc
with six radial slits, about ten inches long and an inch wide. By the
rotation of the disk the light is caused to fall in flashes upon the
jet; and, when the suitable speed of rotation has been attained, the
vein is resolved into its constituent spherules. Receiving the shadow
of the vein upon a white screen, its constitution is rendered clearly
visible to all here present.

This breaking-up of a liquid vein into drops has been a subject of
frequent experiment and much discussion. Savart traced the pulsations
to the orifice, but he did not think that they were produced by
friction. They are powerfully influenced by sonorous vibrations. In the
midst of a large city it is hardly possible to obtain the requisite
tranquillity for the full development of the continuous, portion of
the vein; still, Savart was so far able to withdraw his vein from the
influence of such irregular vibrations that its limpid portion became
elongated to the extent shown in Fig. 142. It will be understood that
Fig. 141 represents a vein exposed to the irregular vibrations of the
city of Paris, while Fig. 142 represents one produced under precisely
the same conditions, but withdrawn from those vibrations.

The drops into which the vein finally resolves itself are incipient
even in its limpid portion, announcing themselves there as annular
protuberances, which become more and more pronounced, until finally
they separate. Their birthplace is near the orifice itself, and under
even moderate pressure they succeed each other with sufficient
rapidity to produce a feeble musical note. By permitting the drops to
fall upon a membrane, the pitch of this note may be fixed; and now we
come to the point which connects the phenomena of liquid veins with
those of sensitive flames and smoke-jets. If a note in unison with that
of the vein be sounded near it, the limpid portion instantly shortens;
the pitch may vary to some extent, and still cause a shortening;
but the unisonant note is the most effectual. Savart’s experiments
on vertically-descending veins have been recently repeated in our
laboratory with striking effect. From a distance of thirty yards the
limpid portion of the vein has been shortened by the sound of an
organ-pipe of the proper pitch and of moderate intensity.

I have also recently gone carefully, not merely by reading, but by
experiment, over Plateau’s account of the resolution of a liquid
vein into drops. In his researches on the figures of equilibrium of
bodies withdrawn from the action of gravity, he finds that a liquid
cylinder is stable as long as its length does not exceed three times
its diameter; or, more accurately, as long as the ratio between them
does not exceed that of the diameter of a circle to its circumference,
or 3.1416. If this be a little exceeded the cylinder begins to narrow
at some point or other of its length; nips itself together, breaks,
and forms immediately two spheres. If the ratio of the length of the
cylinder to its diameter greatly exceed 3.1416, then, instead of
breaking up into two spheres, it breaks up into several.

A liquid cylinder may be obtained by introducing olive-oil into a
mixture of alcohol and water, of the same density as the oil. The
latter forms a sphere. Two disks of smaller diameter than the sphere
are brought into contact with it, and then drawn apart; the oil clings
to the disks, and the sphere is transformed into a cylinder. If the
quantity of oil be insufficient to produce the maximum length of
cylinder, more may be added by a pipette. In making this experiment it
will be noticed that, when the proper length is exceeded, the nipped
portion of the cylinder elongates, and exists for a moment as a very
thin liquid cylinder uniting the two incipient spheres; and that, when
rupture occurs, the thin cylinder, which has also exceeded _its_ proper
length, breaks so as to form a small spherule between the two larger
ones. This is a point of considerable significance in relation to our
present question.

Now, Plateau contends that the play of the molecular forces in a liquid
cylinder is not suspended by its motion of translation. The first
portion of a vein of water quitting an orifice is a cylinder, to which
the laws which he has established regarding motionless cylinders apply.
The moment the descending vein exceeds the proper length it begins to
pinch itself so as to form drops; but urged forward as it is by the
pressure above it, and by its own gravity, in the time required for the
rounding of the drop it reaches a certain distance from the orifice.
At this distance, the pressure remaining constant, and the vein being
withdrawn from external disturbance, rupture invariably occurs. And the
rupture is accompanied by the phenomenon which has just been called
significant. Between every two succeeding large drops a small spherule
is formed, as shown in Fig. 141.

Permitting a vein of oil to fall from an orifice, not through the
air, but through a mixture of alcohol and water of the proper density,
the continuous portion of the vein, its resolution into drops, and the
formation of the small spherule between each liberated drop and the
end of the liquid cylinder which it has just quitted, may be watched
with the utmost deliberation. The effect of this and other experiments
upon the mind will be to produce the conviction that the very beautiful
explanation offered by Plateau is also the true one. The various laws
established experimentally by Savart all follow immediately from
Plateau’s theory.

In a small paper published more than twenty years ago I drew attention
to the fact that when a descending vein intersects a liquid surface
above the point of rupture, if the pressure be not too great, it enters
the liquid _silently_; but when the surface intersects the vein below
the point of rupture a rattle is immediately heard, and bubbles are
copiously produced. In the former case, not only is there no violent
dashing aside of the liquid, but round the base of the vein, and in
opposition to its motion, the liquid collects in a heap, by its surface
tension or capillary attraction. This experiment can be combined with
two other observations of Savart’s, in a beautiful and instructive
manner. In addition to the shortening of the continuous portion by
sound, Savart found that, when he permitted his membrane to intersect
the vein at one of its protuberances, the sound was louder than when
the intersection occurred at the contracted portion.

I permitted a vein to descend, under scarcely any pressure, from a tube
three-quarters of an inch in diameter, and to enter silently a basin of
water placed nearly 20 inches below the orifice. On sounding vigorously
a Ut_{2} tuning-fork the pellucid jet was instantly broken, and as
many as three of its swellings were seen above the surface. The rattle
of air-bubbles was instantly heard, and the basin was seen to be filled
with them. The sound was allowed slowly to die out; the continuous
portion of the vein lengthened, and a series of alternations in the
production of the bubbles was observed. When the swellings of the vein
cut the surface of the water, the bubbles were copious and loud; when
the contracted portions crossed the surface, the bubbles were scanty
and scarcely audible.

Removing the basin, placing an iron tray in its place, and exciting the
fork, the vein, which at first struck silently upon the tray, commenced
a rattle which rose and sank with the dying out of the sound, according
as the swellings or contractions of the jet impinged upon the surface.
This is a simple and beautiful experiment.

[Illustration: FIG. 143.]

[Illustration: FIG. 144.]

[Illustration: FIG. 145.]

Savart also caused his vein to issue horizontally and at various
inclinations to the horizon, and found that, in certain cases,
sonorous vibrations were competent to cause a jet to divide into two
or three branches. In these experiments the liquid was permitted to
issue through an orifice in a thin plate. Instead of this, however,
we will resort to our favorite steatite burner; for with water also
it asserts the same mastery over its fellows that it exhibited with
flames and smoke-jets. It will, moreover, reveal to us some entirely
novel results. By means of an India-rubber tube the burner is connected
with the water-pipes of the Institution, and, by pointing it obliquely
upward, we obtain a fine parabolic jet (Fig. 143). At a certain
distance from the orifice, the vein resolves itself into beautiful
spherules, whose motions are not rapid enough to make the vein appear
continuous. At the vertex of the parabola the spray of pearls is more
than an inch in width, and, further on, the drops are still more widely
scattered. On sweeping a fiddle-bow across a tuning-fork which executes
512 vibrations in a second, the scattered drops, as if drawn together
by their mutual attractions, instantly close up, and form an apparently
continuous liquid arch several feet in height and span (shown in Fig.
144). As long as the proper note is maintained the vein looks like a
frozen band, so motionless does it appear. On stopping the fork the
arch is shaken asunder, and we have the same play of liquid pearls as
before. Every sweep of the bow, however, causes the drops to fall into
a common line of march.

A pitch-pipe, or an organ-pipe yielding the note of this tuning-fork,
also powerfully controls the vein. The voice does the same. On pitching
it to a note of moderate intensity, it causes the wandering drops to
gather themselves together. At a distance of twenty yards, the voice
is, to all appearance, as powerful in curbing the vein, and causing its
drops to close up, as it is when close to the issuing jet.

The effect of “beats” upon the vein is also beautiful and instructive.
They may be produced either by organ-pipes or by tuning-forks. When two
forks vibrate, the one 512 times and the other 508 times in a second,
you will learn in our next lecture that they produce four beats in a
second. When the forks are sounded the beats are heard, and the liquid
vein is seen to gather up its pearls, and scatter them in synchronism
with the beats. The sensitiveness of this vein is astounding; it
rivals that of the ear itself. Placing the two tuning-forks on a
distant table, and permitting the beats to die gradually out, the
vein continues its rhythm almost as long as hearing is possible.
A more sensitive vein might actually prove superior to the ear—a
very surprising result, considering the marvellous delicacy of this
organ.[57]

By introducing a Leyden-jar into the circuit of a powerful
induction-coil, a series of dense and dazzling flashes of light, each
of momentary duration, is obtained. Every such flash in a darkened room
renders the drops distinct, each drop being transformed into a little
star of intense brilliancy. If the vein be then acted on by a sound
of the proper pitch, it instantly gathers its drops together into a
necklace of inimitable beauty.

In these experiments the whole vein gathers itself into a single arched
band when the proper note is sounded; but, by varying the conditions,
it may be caused to divide into two or more such bands, as shown in
Fig. 145. Drawings, however, are ineffectual here; for the wonder of
these experiments depends mainly on the sudden transition of the vein
from one state to the other. In the _motion_ dwells the surprise, and
this no drawing can render.[58]


SUMMARY OF CHAPTER VI

When a gas-flame is placed in a tube, the air in passing over the flame
is thrown into vibration, musical sounds being the consequence.

Making allowance for the high temperature of the column of air
associated with the flame, the pitch of the note is that of an open
organ-pipe of the length of the tube surrounding the flame.

The vibrations of the flame, while the sound continues, consist of a
series of periodic extinctions, total or partial, between every two of
which the flame partially recovers its brightness.

The periodicity of the phenomenon may be demonstrated by means of a
concave mirror which forms an image of the vibrating flame upon a
screen. When the image is sharply defined, the rotation of the mirror
reduces the single image to a series of separate images of the flame.
The dark spaces between the images correspond to the extinctions of
the flame, while the images themselves correspond to its periods of
recovery.

Besides the fundamental note of the associated tube, the flame can also
be caused to excite the higher overtones of the tube. The successive
divisions of the column of air are those of an open organ-pipe when its
harmonic tones are sounded.

On sounding a note nearly in unison with a tube containing a silent
flame, the flame jumps; and if the position of the flame in the tube be
rightly chosen, the extraneous sound will cause the flame to sing.

While the flame is singing, a note nearly in unison with its own
produces beats, and the flame is seen to jump in synchronism with the
beats. The jumping is also observed when the position of the flame
within its tube is not such as to enable it to sing.


NAKED FLAMES

When the pressure of the gas which feeds a naked flame is augmented,
the flame, up to a certain point, increases in size. But if the
pressure be too great, the flame roars or flares.

The roaring or flaring of the flame is caused by the state of vibration
into which the gas is thrown in the orifice of the burner, when the
pressure which urges it through the orifice is excessive.

If the vibrations in the orifice of the burner be super-induced by an
extraneous sound, the flame will flare under a pressure less than that
which, of itself, would produce flaring.

The gas under excessive pressure has vibrations of a definite period
impressed upon it as it passes through the burner. To operate with
a maximum effect upon the flame the external sound must contain
vibrations synchronous with those of the issuing gas.

When such a sound is chosen, and when the flame is brought sufficiently
near its flaring-point, it furnishes an acoustic reagent of unexampled
delicacy.

At a distance of 30 yards, for example, the chirrup of a house-sparrow
would be competent to throw the flame into commotion.

It is not to the flame, as such, that we are to ascribe these effects.
Effects substantially similar are produced when we employ jets of
unignited coal-gas, carbonic acid, hydrogen, or air. These jets may be
rendered visible by smoke, and the smoke jets show a sensitiveness to
sonorous vibrations even greater than that of the flames.

When a brilliant sensitive flame illuminates an otherwise dark room,
in which a suitable bell is caused to strike, a series of periodic
quenchings of the light by the sound occurs. Every stroke of the bell
is accompanied by a momentary darkening of the room.

A jet of water descending from a circular orifice is composed of two
distinct portions, the one pellucid and calm; the other in commotion.
When properly analyzed the former portion is found continuous; the
latter being a succession of drops.

If these drops be received upon a membrane, a musical sound is
produced. When an extraneous sound of this particular pitch is produced
in the neighborhood of the vein, the continuous portion is seen to
shorten.

The continuous portion of the vein presents a series of swellings and
contractions, in the former of which the drops are flattened, and in
the latter elongated. The sound produced by the flattened drops on
striking the membrane is louder than that produced by the elongated
ones.

Above its point of rupture a vein of water may be caused to enter water
_silently_; but on sounding a suitable note, the rattle of bubbles
is immediately heard; the discontinuous part of the vein rises above
the surface, and as the sound dies out the successive swellings and
contractions produce alternations of the quantity and sound of the
bubbles.

In veins propelled obliquely, the scattered water-drops may be called
together by a suitable sound, so as to form an apparently continuous
liquid arch.

Liquid veins may be analyzed by the electric spark, or by a succession
of flashes illuminating the veins.




CHAPTER VII

_PART I_

RESEARCHES ON THE ACOUSTIC TRANSPARENCY OF THE ATMOSPHERE IN RELATION
TO THE QUESTION OF FOG-SIGNALLING

  Introduction—Instruments and Observations—Contradictory
  Results from the 19th of May to the 1st of July
  inclusive—Solution of Contradictions—Aërial Reflection
  and its Causes—Aërial Echoes—Acoustic Clouds—Experimental
  Demonstration of Stoppage of Sound by Aërial Reflection


§ 1. _Introduction_

We are now fully equipped for the investigation of an important
practical problem. The cloud produced by the puff of a locomotive can
quench the rays of the noonday sun; it is not, therefore, surprising
that in dense fogs our most powerful coast-lights, including even the
electric light, should become useless to the mariner.

Disastrous shipwrecks are the consequence. During the last ten years
no less than two hundred and seventy-three vessels have been reported
as totally lost on our own coasts in fog or thick weather. The loss,
I believe, has been far greater on the American seaboard, where trade
is more eager, and fogs more frequent, than they are here. No wonder,
then, that earnest efforts should be made to find a substitute for
light in sound-signals, powerful enough to give warning and guidance to
mariners while still at a safe distance from the shore.

Such signals have been established to some extent upon our own coasts,
and to a still greater extent along the coasts of Canada and the United
States. But the evidence as to their value and performance is of the
most conflicting character, and no investigation sufficiently thorough
to clear up the uncertainty has hitherto been made. In fact, while the
_velocity_ of sound has formed the subject of refined and repeated
experiment by the ablest philosophers, the publication of Dr. Derham’s
celebrated paper in the “Philosophical Transactions” for 1708 marks the
latest systematic inquiry into the causes which affect the _intensity_
of sound in the atmosphere.

Jointly with the Elder Brethren of the Trinity House, and as their
scientific adviser, I have recently had the honor of conducting an
inquiry designed to fill the blank here indicated.

One or two brief references will suffice to show the state of the
question when this investigation began. “Derham,” says Sir John
Herschel, “found that fogs and falling rain, but more especially snow,
tend powerfully to obstruct the propagation of sound, and that the same
effect was produced by a coating of fresh-fallen snow on the ground,
though when glazed and hardened at the surface by freezing it had no
such influence.”[59]

In a very clear and able letter, addressed to the President of the
Board of Trade in 1863,[60] Dr. Robinson, of Armagh, thus summarizes
our knowledge of fog-signals: “Nearly all that is known about
fog-signals is to be found in the ‘Report on Lights and Beacons’; and
of it much is little better than conjecture. Its substance is as
follows:

“‘Light is scarcely available for this purpose. Blue lights are used in
the Hooghly; but it is not stated at what distance they are visible in
fog; their glare may be seen further than their flame.[61] It might,
however, be desirable to ascertain how far the electric light, or its
flash, can be traced.[62]

“‘Sound is the only known means really effective; but about it
testimonies are conflicting, and there is scarcely one fact relating to
its use as a signal which can be considered as established. Even the
most important of all, the distance at which it ceases to be heard, is
undecided.

“‘Up to the present time all signal-sounds have been made in air,
though this medium has grave disadvantages: its own currents interfere
with the sound-waves, so that a gun or bell which is heard several
miles _down_ the wind is inaudible more than a few furlongs _up_ it. A
still greater evil is that it is least effective when most needed; for
fog is a powerful damper of sound.’”

Dr. Robinson here expresses the universally-prevalent opinion, and he
then assigns the theoretic cause. “Fog,” he says, “is a mixture of air
and globules of water, and at each of the innumerable surfaces where
these two touch, a portion of the vibration is reflected and lost.[63]
... Snow produces a similar effect, and one still more injurious.”

Reflection being thus considered to take place at the surfaces of
the suspended particles, it followed that the greater the number of
particles, or, in other words, the denser the fog, the more injurious
would be its action upon sound. Hence optic transparency came to be
considered a measure of acoustic transparency. On this point Dr.
Robinson, in the letter referred to, expresses himself thus: “At the
outset, it is obvious that, to make experiments _comparable_, we
must have some measure of the fog’s power of stopping sound, without
attending to which the most anomalous results may be expected. It
seems probable that this will bear some simple relation to its opacity
to light, and that the distance at which a given object, as a flag
or pole, disappears may be taken as the measure.” “Still, clear air”
was regarded in this letter as the best vehicle of sound, the alleged
action of fogs, rain, and snow being ascribed to their rendering the
atmosphere “a discontinuous medium.”

Prior to the investigation now to be described, the views here
enunciated were those universally entertained. That sound is unable
to penetrate fogs was taken to be “a matter of common observation.”
The bells and horns of ships were affirmed “not to be heard so far
in fogs as in clear weather.” In the fogs of London the noise of the
carriage-wheels was reported to be so much diminished that “they seem
to be at a distance where really close by.” My knowledge does not
inform me of the existence of any other source for these opinions
regarding the deadening power of fog than the paper of Derham,
published one hundred and sixty-seven years ago. In consequence of
their _à priori_ probability, his conclusions seem to have been
transmitted unquestioned from generation to generation of scientific
men.


§ 2. _Instruments and Observations_

On the 19th of May, 1873, this inquiry began. The South Foreland,
near Dover, was chosen as the signal-station, steam-power having
been already established there to work two powerful magneto-electro
lights. The observations for the most part were made afloat, one of
the yachts of the Trinity Corporation being usually employed for this
purpose. Two stations had been established, the one at the top, the
other at the bottom, of the South Foreland Cliff; and at each of them
trumpets, air-whistles, and steam-whistles of great size were mounted.
The whistles first employed were of English manufacture. To these was
afterward added a large United States whistle, and also a Canadian
whistle, of great reputed power.

On the 8th of October another instrument, which has played a specially
important part in these observations, was introduced. This was a
steam-siren, constructed and patented by Mr. Brown of New York, and
introduced by Prof. Henry into the lighthouse system of the United
States. As an example of international courtesy worthy of imitation, I
refer with pleasure to the fact that when informed by Major Elliot of
the United States Army that our experiments had begun, the Lighthouse
Board at Washington, of their own spontaneous kindness, forwarded to
us for trial a very noble instrument of this description, which was
immediately mounted at the South Foreland.

In the steam-siren, as in the ordinary one, described in Chapter II., a
fixed disk and a rotating disk are employed, but radial slits are used
instead of circular apertures. One disk is fixed vertically across the
throat of a conical trumpet 16-1/2 feet long, 5 inches in diameter
where the disk crosses it, and gradually opening out till at the other
extremity it reaches a diameter of 2 feet 3 inches. Behind the fixed
disk is the rotating one, which is driven by separate mechanism. The
trumpet is connected with a boiler. In our experiments steam of 70
lbs. pressure was for the most part employed. Just as in the ordinary
siren, when the radial slits of the two disks coincide, and then only,
a strong puff of steam escapes. Sound-waves of great intensity are
thus sent through the air, the pitch of the note depending on the
velocity of rotation. (A drawing of the steam-siren constitutes our
frontispiece.)

To the siren, trumpets, and whistles were added three guns—an
18-pounder, a 5-1/2-inch howitzer, and a 13-inch mortar. In our summer
experiments all three were fired; but the howitzer having shown itself
superior to the other guns it was chosen in our autumn experiments
as not only a fair but a favorable representative of this form of
signal. The charges fired were for the most part those now employed
at Holyhead, Lundy Island, and the Kish light-vessel; namely, 3 lbs.
of powder. Gongs and bells were not included in this inquiry, because
previous observations had clearly proved their inferiority to the
trumpets and whistles.

On the 19th of May the instruments tested were:

On the top of the cliff:

_a._ Two brass trumpets or horns, 11 feet 2 inches long, 2 inches in
diameter at the mouth-piece, and opening out at the other end to a
diameter of 22-1/2 inches. They were provided with vibrating steel
reeds 9 inches long, 2 inches wide, and 1/4 inch thick, and were
sounded by air of 18 lbs. pressure.

_b._ A whistle, shaped like that of a locomotive, 6 inches in diameter,
also sounded by air of 18 lbs. pressure.

_c._ A steam-whistle, 12 inches in diameter, attached to a boiler, and
sounded by steam of 64 lbs. pressure.

At the bottom of the cliff:

_d._ Two trumpets or horns, of the same size and arrangement as those
above, and sounded by air of the same pressure. They were mounted
vertically on the reservoir of compressed air; but within about two
feet of their extremities they were bent at a right angle, so as to
present their mouths to the sea.

_e._ A 6-inch air-whistle, similar to the one above, and sounded by the
same means.

The upper instruments were 235 feet above high-water mark, the lower
ones 40 feet. A vertical distance of 195 feet, therefore, separated the
instruments. A shaft, provided with a series of twelve ladders, led
from the one to the other.

Numerous comparative experiments made at the outset gave a slight
advantage to the upper instruments. They, therefore, were for the most
part employed throughout the subsequent inquiry.

Our first observations were a preliminary discipline rather than an
organized effort at discovery. On May 19th the maximum distance reached
by the sound was about three and a half miles.[64] The wind, however,
was high and the sea rough, so that local noises interfered to some
extent with our appreciation of the sound.

Mariners express the strength of the wind by a series of numbers
extending from 0 = calm to 12 = a hurricane, a little practice in
common producing a remarkable unanimity between different observers as
regards the force of the wind. Its force on May 19th was 6, and it blew
at right angles to the direction of the sound.

The same instruments on May 20th covered a greater range of sound;
but not much greater, though the disturbance due to local noises was
absent. At 4 miles’ distance in the axes of the horns they were barely
heard, the air at the time being calm, the sea smooth, and all other
circumstances exactly those which have been hitherto regarded as most
favorable to the transmission of sound. We crept a little further
away, and by stretched attention managed to hear at intervals, at a
distance of 6 miles, the faintest hum of the horns. A little further
out we again halted; but though local noises were absent, and though we
listened intently, we heard nothing.

This position, clearly beyond the range of whistles and trumpets, was
expressly chosen with the view of making what might be considered a
decisive comparative experiment between horns and guns as instruments
for fog-signalling. The distinct report of the 12 o’clock gun fired at
Dover on the 19th suggested this comparison, and through the prompt
courtesy of General Sir A. Horsford we were enabled to carry it out.
At 12.30 precisely the puff of an 18-pounder, with a 3-lb. charge, was
seen at Dover Castle, which was about a mile further off than the South
Foreland. Thirty-six seconds afterward the loud report of the gun was
heard, its complete superiority over the trumpets being thus, to all
appearance, demonstrated.

We clinched this observation by steaming out to a distance of 8-1/2
miles, where the report of a second gun was well heard by all of us.
At a distance of 10 miles the report of a third gun was heard by some,
and at 9·7 miles the report of a fourth gun was heard by all.

The result seemed perfectly decisive. Applying the law of inverse
squares, the sound of the gun at a distance of 6 miles from the
Foreland must have had more than two and a half times the intensity
of the sound of the trumpets. It would not have been rash under the
circumstances to have reported without qualification the superiority
of the gun as a fog-signal. No single experiment is, to my knowledge,
on record to prove that a sound once predominant would not be always
predominant, or that the atmosphere on different days would show
preferences to different sounds. On many subsequent occasions, however,
the sound of the horns proved distinctly superior to that of the
gun. This _selective_ power of the atmosphere revealed itself more
strikingly in our autumn experiments than in our summer ones; and it
was sometimes illustrated within a few hours of the same day: of two
sounds, for example, one might have the greatest range at 10 A.M., and
the other the greatest range at 2 P.M.

In the experiments on May 19th and 20th the superiority of the trumpets
over the whistles was decided; and indeed, with few exceptions, this
superiority was maintained throughout the inquiry. But there were
exceptions. On June 2d, for example, the whistles rose in several
instances to full equality with, and on rare occasions subsequently
even surpassed, the horns. The sounds were varied from day to day,
and various shiftings of the horns and reeds were resorted to, with a
view of bringing out their maximum power. On the date last mentioned
a single horn was sounded, two were sounded, and three were sounded
together; but the utmost range of the loudest sound, even with the
paddles stopped, did not exceed 6 miles. With the view of concentrating
their power, the axes of the horns had been pointed in the same
direction, and, unless stated to the contrary, this in all subsequent
experiments was the case.

On June 3d the three guns already referred to were permanently mounted
at the South Foreland. They were ably served by gunners from Dover
Castle.

On the same day dense clouds quite covered the firmament, some of them
particularly black and threatening, but a marked advance was observed
in the transmissive power of the air. At a distance of 6 miles the
horn-sounds were not quite quenched by the paddle-noises; at 8 miles
the whistles were heard, and the horns better heard; while at 9 miles,
with the paddles stopped, the horn-sounds alone were fairly audible.
During the day’s observations a remarkable and instructive phenomenon
was observed. Over us rapidly passed a torrential shower of rain,
which, according to Derham, is a potent damper of sound. We could,
however, notice no subsidence of intensity as the shower passed. It
is even probable that, had our minds been free from bias, we should
have noticed an augmentation of the sound, such as occurred with the
greatest distinctness on various subsequent occasions during violent
rain.

The influence of “beats” was tried on June 3d, by throwing the horns
slightly out of unison; but though the beats rendered the sound
characteristic, they did not seem to augment the range. At a distance
from the station curious fluctuations of intensity were noticed. Not
only did the different blasts vary in strength, but sudden swellings
and fallings off, even of the same blast, were observed. This was not
due to any variation on the part of the instruments, but purely to the
changes of the medium traversed by the sound. What these changes were
shall be indicated subsequently.

The range of our best horns on June 10th was 8-3/4 miles. The guns at
this distance were very feeble. That the loudness of the sound depends
on the shape of the gun was proved by the fact that thus far the
howitzer, with a 3-lb. charge, proved more effective than the other
guns.

On June 25th a gradual improvement in the transmissive power of the
air was observed from morning to evening; but at the last the maximum
range was only moderate. The fluctuations in the strength of the sound
were remarkable, sometimes sinking to inaudibility and then rising to
loudness. A similar effect, due to a similar cause, is often noticed
with church-bells. The acoustic transparency of the air was still
further augmented on the 26th: at a distance of 9-1/4 miles from the
station the whistles and horns were plainly heard against a wind with a
force of 4; while on the 25th, with a favoring wind, the maximum range
was only 6-1/2 miles. Plainly, therefore, something else than the wind
must be influential in determining the range of the sound.

On Tuesday, July 1st, observations were made on the decay of the sound
at various angular distances from the axis of the horn. As might be
expected, the sound in the axis was loudest, the decay being gradual on
both sides. In the case of the gun, however, the direction of pointing
has very little influence.

The day was acoustically clear; at a distance of 10 miles the horn
yielded a plain sound, while the American whistle seemed to surpass
the horn. Dense haze at this time quite hid the Foreland. At 10-1/2
miles occasional blasts of the horn came to us, but after a time all
sound ceased to be audible; it seemed as if the air, after having been
exceedingly transparent, had become gradually more opaque to the sound.

At 4.45 P.M. we took the master of the Varne light-ship on board the
“Irene.” He and his company had heard the sound at intervals during the
day, although he was dead to windward and distant 12-3/4 miles from the
source of sound.

Here a word of reflection on our observations may be fitly introduced.
It is, as already shown, an opinion entertained in high quarters that
the waves of sound are reflected at the limiting surfaces of the minute
particles which constitute haze and fog, the alleged waste of sound in
fog being thus explained. If, however, this be an efficient practical
cause of the stoppage of sound, and if clear calm air be, as alleged,
the best vehicle, it would be impossible to understand how to-day, in
a thick haze, the sound reached a distance of 12-3/4 miles, while on
May 20th, in a calm and hazeless atmosphere, the maximum range was only
from 5 to 6 miles. Such facts foreshadow a revolution in our notions
regarding the action of haze and fogs upon sound.

An interval of 12 hours sufficed to change in a surprising degree the
acoustic transparency of the air. On the 1st of July the sound had a
range of nearly 13 miles; on the 2d the range did not exceed 4 miles.


§ 3. _Contradictory Results_

Thus far the investigation proceeded with hardly a gleam of a principle
to connect the inconstant results. The distance reached by the sound
on the 19th of May was 3-1/2 miles; on the 20th it was 5-1/2 miles; on
the 2d of June 6 miles; on the 3d more than 9 miles; on the 10th it
was also 9 miles; on the 25th it fell to 6-1/2 miles; on the 26th it
rose again to more than 9-1/4 miles; on the 1st of July, as we have
just seen, it reached 12-3/4, whereas on the 2d the range shrunk to 4
miles. None of the meteorological agents observed could be singled out
as the cause of these fluctuations. The wind exerts an acknowledged
power over sound, but it could not account for these phenomena. On the
25th of June, for example, when the range was only 6-1/2 miles, the
wind was favorable; on the 26th, when the range exceeded 9-1/4 miles,
it was opposed to the sound. Nor could the varying optical clearness
of the atmosphere be invoked as an explanation; for on July 1st, when
the range was 12-3/4 miles, a thick haze hid the white cliffs of the
Foreland, while on many other days, when the acoustic range was not
half so great, the atmosphere was optically clear. Up to July 3d all
remained enigmatical; but on this date observations were made which
seemed to me to displace surmise and perplexity by the clearer light of
physical demonstration.


§ 4. _Solution of Contradictions_

On July 3d we first steamed to a point 2·9 miles S.W. by W. of the
signal-station. No sounds, not even the guns, were heard at this
distance. At 2 miles they were equally inaudible. But this being a
position at which the sounds, though strong in the axis of the horn,
invariably subsided, we steamed to the exact bearing from which our
observations had been made on July 1st. At 2.15 P.M., and at a distance
of 3-3/4 miles from the station, with calm, clear air and a smooth sea,
the horns and whistle (American) were sounded, but they were inaudible.
Surprised at this result, I signalled for the guns. They were all
fired, but, though the smoke seemed at hand, no sound whatever reached
us. On July 1st, in this bearing, the observed range of both horns and
guns was 10-1/2 miles, while on the bearing of the Varne light-vessel
it was nearly 13 miles. We steamed in to 3 miles, paused, and listened
with all attention; but neither horn nor whistle was heard. The guns
were again signalled for; five of them were fired in succession, but
not one of them was heard. We steamed on in the same bearing to 2
miles, and had the guns fired pointblank at us. The howitzer and the
mortar, with 3-lb. charges, yielded a feeble thud, while the 18-pounder
was wholly unheard. Applying the law of inverse squares, it follows
that, with the air and sea, according to accepted notions, in a far
worse condition, the sound at 2 miles’ distance on July 1st must have
had more than forty times the intensity which it possessed at the same
distance at 3 P.M. on the 3d.

“On smooth water,” says Sir John Herschel, “sound is propagated with
remarkable clearness and strength.” Here was the condition; still,
with the Foreland so close to us, the sea so smooth, and the air
so transparent, it was difficult to realize that the guns had been
fired or the trumpets blown at all. What could be the reason? Had
the sound been converted by internal friction into heat? or had
it been wasted in partial reflections at the limiting surfaces of
non-homogeneous masses of air? I ventured, two or three years ago, to
say something regarding the function of the Imagination in Science,
and, notwithstanding the care then taken, to define and illustrate
its real province, some persons, among whom were one or two able men,
deemed me loose and illogical. They misunderstood me. The faculty to
which I referred was that power of visualizing processes in space,
and the relations of space itself, which must be possessed by all
great physicists and geometers. Looking, for example, at two pieces
of polished steel, we have not a sense, or the rudiment of a sense,
to distinguish the inner condition of the one from that of the other.
And yet they may differ materially, for one may be a magnet, the other
not. What enabled Ampère to surround the atoms of such a magnet with
channels in which electric currents ceaselessly run, and to deduce from
these pictured currents all the phenomena of ordinary magnetism? What
enabled Faraday to visualize his lines of force, and make his mental
picture a guide to discoveries which have rendered his name immortal?
Assuredly it was the disciplined imagination. Figure the observers on
the deck of the “Irene,” with the invisible air stretching between
them and the South Foreland, knowing that it contained something which
stifled the sound, but not knowing what that something is. Their senses
are not of the least use to them; nor could all the philosophical
instruments in the world render them any assistance. They could not, in
fact, take a single step toward the solution without the formation of a
mental image—in other words, without the exercise of the imagination.

Sulphur, in homogeneous crystals, is exceedingly transparent to radiant
heat, whereas the ordinary brimstone of commerce is highly impervious
to it—the reason being that the brimstone does not possess the
molecular continuity of the crystal, but is a mere aggregate of minute
grains not in perfect optical contact with each other. Where this is
the case, a portion of the heat is always reflected on entering and on
quitting a grain; hence, when the grains are minute and numerous, this
reflection is so often repeated that the heat is entirely wasted before
it can plunge to any depth into the substance. The same remark applies
to snow, foam, clouds, and common salt, indeed, to all transparent
substances in powder; they are all impervious to light, not through the
immediate absorption or extinction of the light, but through repeated
internal reflection.

Humboldt, in his observations at the Falls of the Orinoco, is known to
have applied these principles to sound. He found the noise of the falls
far louder by night than by day, though in that region the night is far
noisier than the day. The plain between him and the falls consisted
of spaces of grass and rock intermingled. In the heat of the day he
found the temperature of the rock to be considerably higher than that
of the grass. Over every heated rock, he concluded, rose a column of
air rarefied by the heat; its place being supplied by the descent of
heavier air. He ascribed the deadening of the sound to the reflections
which it endured at the limiting surfaces of the rarer and denser
air. This philosophical explanation made it generally known that a
non-homogeneous atmosphere is unfavorable to the transmission of sound.

But what on July 3d, not with the variously-heated plain of Antures,
but with a calm sea as a basis for the atmosphere, could so destroy its
homogeneity as to enable it to quench in so short a distance so vast a
body of sound? My course of thought at the time was thus determined:
As I stood upon the deck of the “Irene” pondering the question, I
became conscious of the exceeding power of the sun beating against
my back and heating the objects near me. Beams of equal power were
falling on the sea, and must have produced copious evaporation. That
the vapor generated should so rise and mingle with the air as to form
an absolutely homogeneous medium, was in the highest degree improbable.
It would be sure, I thought, to rise in invisible streams, breaking
through the superincumbent air now at one point, now at another, thus
rendering the air _flocculent_ with wreaths and striæ, charged in
different degrees with the buoyant vapor. At the limiting surfaces of
these spaces, though invisible, we should have the conditions necessary
to the production of partial echoes and the consequent waste of sound.
Ascending and descending air-currents, of different temperatures, as
far as they existed, would also contribute to the effect.

Curiously enough, the conditions necessary for the testing of this
explanation immediately set in. At 3.15 P.M. a solitary cloud threw
itself athwart the sun, and shaded the entire space between us and the
South Foreland. The heating of the water and the production of vapor-
and air-currents were checked by the interposition of this screen;
hence the probability of suddenly-improved transmission. To test
this inference, the steamer was immediately turned and urged back to
our last position of inaudibility. The sounds, as I expected, were
distinctly though faintly heard. This was at 3 miles’ distance. At
3-3/4 miles, the guns were fired, both pointblank and elevated. The
faintest pop was all that we heard; but we did hear a pop, whereas
we had previously heard nothing, either here or three-quarters of a
mile nearer. We steamed out to 4-1/4 miles, where the sounds were
for a moment faintly heard; but they fell away as we waited; and
though the greatest quietness reigned on board, and though the sea
was without a ripple, we could hear nothing. We could plainly see the
steam-puffs which announced the beginning and the end of a series of
trumpet-blasts, but the blasts themselves were quite inaudible.

It was now 4 P.M., and my intention at first was to halt at this
distance, which was beyond the sound-range, but not far beyond it,
and see whether the lowering of the sun would not restore the power
of the atmosphere to transmit the sound. But after waiting a little
the anchoring of a boat was suggested, so as to liberate the steamer
for other work; and though loth to lose the anticipated revival of the
sounds myself, I agreed to this arrangement. Two men were placed in the
boat and requested to give all attention, so as to hear the sound if
possible. With perfect stillness around them they heard nothing. They
were then instructed to hoist a signal if they should hear the sounds,
and to keep it hoisted as long as the sounds continued.

At 4.45 we quitted them and steamed toward the South Sand Head
light-ship. Precisely 15 minutes after we had separated from them the
flag was hoisted; the sound had at length succeeded in piercing the
body of air between the boat and the shore.

We continued our journey to the light-ship, went on board, heard the
report of the lightsmen, and returned to our anchored boat. We then
learned that when the flag was hoisted the horn-sounds were heard, that
they were succeeded after a little time by the whistle-sounds, and that
both increased in intensity as the evening advanced. On our arrival, of
course, we heard the sounds ourselves.

We pushed the test further by steaming further out. At 5-3/4 miles we
halted and heard the sounds: at 6 miles we heard them distinctly, but
so feebly that we thought we had reached the limit of the sound-range;
but while we waited the sounds rose in power. We steamed to the Varne
buoy, which is 7-3/4 miles from the signal-station, and heard the
sounds there better than at 6 miles’ distance. We continued our course
outward to 10 miles, halted there for a brief interval, but heard
nothing.

Steaming, however, on to the Varne light-ship, which is situated at
the other end of the Varne shoal, we hailed the master, and were
informed by him that up to 5 P.M. nothing had been heard, but that at
that hour the sounds began to be audible. He described one of them as
“very gross, resembling the bellowing of a bull,” which very accurately
characterizes the sound of the large American steam-whistle. At the
Varne light-ship, therefore, the sounds had been heard toward the close
of the day; though it is 12-3/4 miles from the signal-station. I think
it probable that, at a point 2 miles from the Foreland, the sound at 5
P.M. possessed fifty times the intensity which it possessed at 2 P.M.
To such undreamed-of fluctuations is the atmosphere liable. On our
return to Dover Bay, at 10 P.M., we heard the sounds, not only distinct
but loud, where nothing could be heard in the morning.


§ 5. _Other Remarkable Instances of Acoustic Opacity_

In his excellent lecture entitled “Wirkungen aus der Ferne,” Dove has
collected some striking cases of the interception of sound. The Duke of
Argyll has also favored me with some highly-interesting illustrations;
but nothing of this description that I have read equals in point of
interest the following account of the battle of Gaines’s Farm, for
which I am indebted to the Rector of the University of Virginia:

  “LYNCHBURG, VIRGINIA, _March 19, 1874_.

  “SIR—I have just read with great interest your lecture of
  January 16th, on the acoustic transparency and opacity of the
  atmosphere. The remarkable observations you mention induce me
  to state to you a fact which I have occasionally mentioned,
  but always, where I am not well known, with the apprehension
  that my veracity would be questioned. It made a strong
  impression on me at the time, but was an insoluble mystery
  until your discourse gave me a possible solution.

  “On the afternoon of June 28, 1862, I rode, in company
  with General G. W. Randolph, then Secretary of War of the
  Confederate States, to Price’s house, about nine miles from
  Richmond; the evening before General Lee had begun his attack
  on McClellan’s army, by crossing the Chickahominy about four
  miles above Price’s, and driving in McClellan’s right wing.
  The battle of Gaines’s Farm was fought the afternoon to
  which I refer. The valley of the Chickahominy is about one
  mile and a half wide from hilltop to hilltop. Price’s is on
  one hilltop, that nearest to Richmond; Gaines’s farm, just
  opposite, is on the other, reaching back in a plateau to Cold
  Harbor.

  “Looking across the valley I saw a good deal of the battle,
  Lee’s right resting in the valley, the Federal left wing the
  same. My line of vision was nearly in the line of the lines of
  battle. I saw the advance of the Confederates, their repulse
  two or three times, and in the gray of the evening the final
  retreat of the Federal forces.

  “I distinctly saw the musket-fire of both lines, the smoke,
  individual discharges, the flash of the guns. I saw batteries
  of artillery on both sides come into action and fire rapidly.
  Several field-batteries on each side were plainly in sight.
  Many more were hid by the timber which bounded the range of
  vision.

  “Yet looking for nearly two hours, from about 5 to 7 P.M. on
  a midsummer afternoon, at a battle in which at least 50,000
  men were actually engaged, and doubtless at least 100 pieces
  of field-artillery, through an atmosphere optically as limpid
  as possible, _not a single sound of the battle_ was audible to
  General Randolph and myself. I remarked it to him at the time
  as astonishing.

  “Between me and the battle was the deep broad valley of the
  Chickahominy, partly a swamp shaded from the declining sun by
  the hills and forest in the west (my side). Part of the valley
  on each side of the swamp was cleared; some in cultivation,
  some not. Here were conditions capable of providing several
  belts of air, varying in the amount of watery vapor (and
  probably in temperature), arranged like laminæ at right angles
  to the acoustic waves as they came from the battlefield to me.

  “Respectfully,

  “Your obedient servant,

  “R. G. H. KEAN.

  “PROF. JOHN TYNDALL.”

I learn from a subsequent letter that during the battle the air was
still.—J. T.


§ 6. _Echoes from Invisible Acoustic Clouds_

But both the argument and the phenomena have a complementary side,
which we have now to consider. A stratum of air less than 3 miles thick
on a calm day has been proved competent to stifle both the cannonade
and the horn-sounds employed at the South Foreland; while, according
to the foregoing explanation, this result was due to the reflection of
the sound from invisible _acoustic clouds_ which filled the atmosphere
on a day of perfect _optical_ transparency. But, granting this, it is
incredible that so great a body of sound could utterly disappear in so
short a distance without rendering some account of itself. Supposing,
then, instead of placing ourselves behind the acoustic cloud, we were
to place ourselves in front of it, might we not, in accordance with the
law of conservation, expect to receive by reflection the sound which
had failed to reach us by transmission? The case would then be strictly
analogous to the reflection of light from an ordinary cloud to an
observer between it and the sun.

My first care in the early part of the day in question was to assure
myself that our inability to hear the sound did not arise from any
derangement of the instruments on shore. Accompanied by the private
secretary of the Deputy Master of the Trinity House, at 1 P.M. I was
rowed to the shore, and landed at the base of the South Foreland
Cliff. The body of air which had already shown such extraordinary
power to intercept the sound, and which manifested this power still
more impressively later in the day, was now in front of us. On it
the sonorous waves impinged, and from it they were sent back with
astonishing intensity. The instruments, hidden from view, were on the
summit of a cliff 235 feet above us, the sea was smooth and clear of
ships, the atmosphere was without a cloud, and there was no object
in sight which could possibly produce the observed effect. From the
perfectly transparent air the echoes came, at first with a strength
apparently little less than that of the direct sound, and then dying
away. A remark made by my talented companion in his notebook at the
time shows how the phenomenon affected him: “Beyond saying that the
echoes seemed to come from the expanse of ocean, it did not appear
possible to indicate any more definite point of reflection.” Indeed
no such point was to be seen; the echoes reached us, as if by magic,
from the invisible acoustic clouds with which the optically transparent
atmosphere was filled. The existence of such clouds in all weathers,
whether optically cloudy or serene, is one of the most important points
established by this inquiry.

Here, in my opinion, we have the key to many of the mysteries and
discrepancies of evidence which beset this question. The foregoing
observations show that there is no need to doubt either the veracity
or the ability of the conflicting witnesses, for the variations of
the atmosphere are more than sufficient to account for theirs. The
mistake, indeed, hitherto has been, not in reporting incorrectly, but
in neglecting the monotonous operation of repeating the observations
during a sufficient time. I shall have occasion to remark subsequently
on the mischief likely to arise from giving instructions to mariners
founded on observations of this incomplete character.


It required, however, long pondering and repeated observation before
this conclusion took firm root in my mind; for it was opposed to
the results of great observers, and to the statements of celebrated
writers. In science as elsewhere, a mind of any depth which accepts
a doctrine undoubtingly, discards it unwillingly. The question of
aërial echoes has a historic interest. While cloud-echoes have been
accepted as demonstrated by observation, it has been hitherto held as
established that audible echoes never occur in optically clear air. We
owe this opinion to the admirable report of Arago on the experiments
made to determine the velocity of sound at Montlhéry and Villejuif in
1822.[65] Arago’s account of the phenomenon observed by him and his
colleagues is as follows: “Before ending this note we will only add
that the shots fired at Montlhéry were accompanied by a rumbling like
that of thunder, which lasted from 20 to 25 seconds. Nothing of this
kind occurred at Villejuif. Once we heard two distinct reports, a
second apart, of the Montlhéry cannon. In two other cases the report
of the same gun was followed by a prolonged rumbling. These phenomena
never occurred without clouds. Under a clear sky the sounds were
single and instantaneous. May we not, therefore, conclude that the
multiple reports of the Montlhéry gun heard at Villejuif were echoes
from the clouds, and may we not accept this fact as favorable to the
explanation given by certain physicists of the rolling of thunder?”

This explanation of the Montlhéry echoes is an inference from
observations made at Villejuif. The inference requires qualification.
Some hundreds of cannon-shots have been fired at the South Foreland,
many of them when the heavens were completely free from clouds, and
never in a single case has a _roulement_ similar to that noticed at
Montlhéry been absent. It follows, moreover, so hot upon the direct
sound as to present hardly a sensible breach of continuity between the
sound and the echo. This could not be the case if the clouds were its
origin. A reflecting cloud, at the distance of a mile; would leave
a silent interval of nearly ten seconds between sound and echo; and
had such an interval been observed at Montlhéry, it could hardly have
escaped record by the philosophers stationed there; but they have not
recorded it.

I think both the fact and the inference need reconsideration. For
our observations prove to demonstration that air of perfect visual
transparency is competent to produce echoes of great intensity and
long duration. The subject is worthy of additional illustration. On
the 8th of October, as already stated, the siren was established at
the South Foreland. I visited the station on that day, and listened
to its echoes. They were far more powerful than those of the horn.
Like the others, they were perfectly continuous, and faded, as if into
distance, gradually away. The direct sound seemed rendered complex and
multitudinous by its echoes, which resembled a band of trumpeters,
first responding close at hand, and then retreating rapidly toward the
coast of France. The siren echoes on that day had 11 seconds’, those
of the horn 8 seconds’, duration.

In the case of the siren, moreover, the reinforcement of the direct
sound by its echo was distinct. About a second after the commencement
of the siren-blast the echo struck in as a new sound. This first echo,
therefore, must have been flung back by a body of air not more than 600
or 700 feet in thickness. The few detached clouds visible at the time
were many miles away, and could clearly have had nothing to do with the
effect.

On the 10th of October I was again at the Foreland listening to the
echoes, with results similar to those just described. On the 15th I had
an opportunity of remarking something new concerning them at Dungeness,
where a horn similar to, but not so powerful as, those at the South
Foreland, has been mounted. It rotates automatically through an arc of
210°, halting at four different points on the arc and emitting a blast
of 6 seconds’ duration, these blasts being separated from each other by
intervals of silence of 20 seconds.

The new point observed was this: as the horn rotated the echoes were
always returned along the line in which the axis of the horn pointed.
Standing either behind or in front of the lighthouse tower, or closing
the eyes so as to exclude all knowledge of the position of the horn,
the direction of its axis when sounded could always be inferred from
the direction in which the aërial echoes reached the shore. Not only,
therefore, is knowledge of _direction_ given by a sound, but it may
also be given by the aërial echoes of the sound.

On the 17th of October, at about 5 P.M., the air being perfectly free
from clouds, we rowed toward the Foreland, landed, and passed over the
seaweed to the base of the cliff. As I reached the base the position of
the “Galatea” was such that an echo of astonishing intensity was sent
back from her side; it came as if from an independent source of sound
established on board the steamer. This echo ceased suddenly, leaving
the aërial echoes to die gradually into silence.

At the base of the cliff a series of concurrent observations made the
duration of the aërial siren-echoes from 13 to 14 seconds.

Lying on the shingle under a projecting roof of chalk, the somewhat
enfeebled diffracted sound reached me, and I was able to hear
with great distinctness, about a second after the starting of the
siren-blast, the echoes striking in and reinforcing the direct sound.
The first rush of echoed sound was very powerful, and it came, as
usual, from a stratum of air 600 or 700 feet in thickness. On again
testing the duration of the echoes, it was found to be from 14 to 15
seconds. The perfect clearness of the afternoon caused me to choose
it for the examination of the echoes. It is worth remarking that this
was our day of longest echoes, and it was also our day of greatest
acoustic transparency, this association suggesting that the duration
of the echo is a measure of the atmospheric _depths_ from which it
comes. On no day, it is to be remembered, was the atmosphere free from
invisible acoustic clouds; and on this day, and when their presence
did not prevent the direct sound from reaching to a distance of 15 or
16 nautical miles, they were able to send us echoes of 15 seconds’
duration.

On various occasions, when fully three miles from the shore, the
Foreland bearing north, we have had the distinct echoes of the siren
sent back to us from the cloudless _southern_ air.

To sum up this question of aërial echoes. The siren sounded three
blasts a minute, each of 5 seconds’ duration. From the number of days
and the number of hours per day during which the instrument was in
action we can infer the number of blasts. They reached nearly twenty
thousand. The blasts of the horns exceeded this number, while hundreds
of shots were fired from the guns. Whatever might be the state of the
weather, cloudy or serene, stormy or calm, the aërial echoes, though
varying in strength and duration from day to day, were never absent;
and on many days, “under a perfectly clear sky,” they reached, in the
case of the siren, an astonishing intensity. It is doubtless to these
air-echoes, and not to cloud-echoes, that the rolling of thunder is to
be ascribed.


§ 7. _Experimental Demonstration of Reflection from Gases_

Thus far we have dealt in inference merely, for the interception
of sound through aërial reflection has never been experimentally
demonstrated; and, indeed, according to Arago’s observation, which
has hitherto held undisputed possession of the scientific field, it
does not sensibly exist. But the strength of science consists in
verification, and I was anxious to submit the question of aërial
reflection to an experimental test. The knowledge gained in the last
lecture enables us to apply such a test; but, as in most similar cases,
it was not the simplest combinations that were first adopted. Two gases
of different densities were to be chosen, and I chose carbonic acid and
coal-gas. With the aid of my skillful assistant, Mr. John Cottrell,
a tunnel was formed, across which five-and-twenty layers of carbonic
acid were permitted to fall, and five-and-twenty alternate layers of
coal-gas to rise. Sound was sent through this tunnel, making fifty
passages from medium to medium in its course. These, I thought, would
waste in aërial echoes a sensible portion of sound.

To indicate this waste an objective test was found in one of the
sensitive flames described in the last chapter. Acquainted with it, we
are prepared to understand a drawing and description of the apparatus
first employed in the demonstration of aërial reflection. The following
clear account of the apparatus was given by a writer in “Nature,”
February 5, 1874:

“A tunnel _t t′_ (Fig. 146), 2 inches square, 4 feet 8 inches long,
open at both ends, and having a glass front, runs through the
box _a b c d_. The spaces above and below are divided into cells
opening into the tunnel by transverse orifices exactly corresponding
vertically. Each alternate cell of the upper series—the 1st, 3d,
5th, etc.—communicates by a bent tube (_e e e_) with a common upper
reservoir (_g_), its counterpart cell in the lower series having a
free outlet into the air. In like manner the 2d, 4th, 6th, etc., of
the lower series of cells are connected by bent tubes (_n n n_) with
the lower reservoir (_i_), each having its direct passage into the
air through the cell immediately above it. The gas-distributors (_g_
and _i_) are filled from both ends at the same time, the upper with
carbonic-acid gas, the lower with coal-gas, by branches from their
respective supply-pipes (_f_ and _h_). A well-padded box (P) open to
the end of the tunnel forms a little cavern, whence the sound-waves are
sent forth by an electric bell (dotted in the figure). A few feet
from the other end of the tunnel, and in a direct line with it, is a
sensitive flame (_k_), provided with a funnel as sound-collector, and
guarded from chance currents by a shade.

[Illustration: FIG. 146.]

“The bell was set ringing. The flame, with quick response to each
blow of the hammer, emitted a sort of musical roar, shortening and
lengthening as the successive sound-pulses reached it. The gases were
then admitted. Twenty-five flat jets of coal-gas ascended from the
tubes below, and twenty-five cascades of carbonic acid fell from the
tubes above. That which was a homogeneous medium had now fifty limiting
surfaces, from each of which a portion of the sound was thrown back.
In a few moments these successive reflections became so effective that
no sound having sufficient power to affect the flame could pierce the
clear, optically-transparent, but acoustically-opaque, atmosphere in
the tunnel. So long as the gases continued to flow the flame remained
perfectly tranquil. When the supply was cut off, the gases rapidly
diffused into the air. The atmosphere of the tunnel became again
homogeneous, and therefore acoustically transparent, and the flame
responded to each sound-pulse as before.”

Not only do gases of different densities act thus upon sound, but
atmospheric air in layers of different temperatures does the same.
Across a tunnel resembling _t t′_, Fig. 146, sixty-six platinum wires
were stretched, all of them being in metallic connection. The bell, in
its padded box, was placed at one end of the tunnel, and the sensitive
flame _k_, near its flaring-point, at the other. When the bell rang
the flame flared. A current from a strong voltaic battery being sent
through the platinum wires, they became heated: layers of warm air rose
from them through the tunnel, and immediately the agitation of the
flame was stilled. On stopping the current, the agitation recommenced.
In this experiment the platinum wires had not reached a red heat.
Employing half the number and the same battery, they were raised to
a red heat, the action in this case upon the sound-waves being also
energetic. Employing one-third of the number of wires, and the same
strength of battery, the wires were raised to a white heat. Here also
the flame was immediately rendered tranquil by the stoppage of the
sound.


§ 8. _Reflection from Vapors_

But not only do gases of different densities, and air of different
temperatures, act thus upon sound, but air saturated, in different
degrees, with the vapors of volatile liquids can be shown by experiment
to produce the same effect. Into the path pursued by the carbonic acid
in our first experiment a flask, which I have frequently employed to
charge air with vapor, was introduced. Through a volatile liquid,
partially filling the flask, air was forced into the tunnel _t t_′,
which was thus divided into spaces of air saturated with the vapor, and
other spaces in their ordinary condition. The action of such a medium
upon the sound-waves issuing from the bell is very energetic, instantly
reducing the violently-agitated flame to stillness and steadiness.
The removal of the heterogeneous medium instantly restores the noisy
flaring of the flame.

A few illustrations of the action of non-homogeneous atmospheres,
produced by the saturation of layers of air with the vapors of volatile
liquids, may follow here:

_Bisulphide of Carbon._—Flame very sensitive, and noisily responsive to
the sound. The action of the non-homogeneous atmosphere was prompt and
strong, stilling the agitated flame.

_Chloroform._—Flame still very sensitive; action similar to the last.

_Iodide of Methyl._—Action prompt and energetic.

_Amylene._—Very fine action; a short and violently-agitated flame was
immediately rendered tall and quiescent.

_Sulphuric Ether._—Action prompt and energetic.

The vapor of water at ordinary temperatures is so small in quantity and
so attenuated that it requires special precautions to bring out its
action. But with such precautions it was found competent to reduce to
quiescence the sensitive flame.

As the skill and knowledge of the experimenter augment he is often
able to simplify his experimental combinations. Thus, in the present
instance, by the suitable arrangement of the source of sound and the
sensitive flame, it was found that not only twenty-five layers, but
three or four layers of coal-gas and carbonic acid sufficed to still
the agitated flame. Nay, with improved manipulation, the action of a
single layer of either gas was rendered perfectly sensible. So also as
regards heated layers of air, not only were sixty-six or twenty-two
heated platinum wires found sufficient, but the heated air from two or
three candle-flames, or even from a single flame, or a heated poker,
was found perfectly competent to stop the flame’s agitation. The same
remark applies to vapors. Three or four heated layers of air, saturated
with the vapor of a volatile liquid, stilled the flame; and, by
improved manipulation, the action of a single saturated layer could be
rendered sensible. In all these cases, moreover, a small, high-pitched
reed might be substituted for the bell.

My assistant has devised the simple apparatus sketched in Fig. 147, for
showing reflection by gases, vapors, and heated air. At the end A of
the square pipe A B is a small vibrating reed of high pitch, the sound
of which violently agitates the sensitive flame _f_. To the horizontal
tube _g g′_ are attached four small burners, and above them four
chimneys, through which the heated gases from the flames can ascend
into A B. When the coverings of the chimneys are removed and the gas is
ignited, the air within A B is rendered rapidly non-homogeneous, and
immediately stills the agitated flame.

[Illustration: FIG. 147.]

The pipe A B may be turned upside down, an orifice seen between A and
B fitting on to the stand which supports the tube. The conduit _t_
leads into a shallow rectangular box, which communicates by a series of
transverse apertures with A B. When air, saturated with the vapor of a
volatile liquid, is forced through these apertures, the atmosphere in
A B is immediately rendered heterogeneous, the agitated flame being as
rapidly stilled.

[Illustration: FIG. 148.]

In the experiments at the South Foreland, not only was it proved that
the acoustic clouds stopped the sound; but, in the proper position, the
sounds which had been refused transmission were received by reflection.
I wished very much to render this echoed sound evident experimentally;
and stated to my assistant that we ought to be able to accomplish this.
Mr. Cottrell met my desire by the following beautiful experiment, which
has been thus described before the Royal Society:

A vibrating reed B (Fig. 148) was placed so as to send sound-waves
through a tin tube, 38 inches long, and 1-3/4 inch diameter, in the
direction B A, the action of the sound being rendered manifest by its
causing a sensitive flame placed at F′ to become violently agitated.

“The invisible heated layer immediately above the luminous portion
of an ignited coal-gas flame issuing from an ordinary bat’s-wing
burner was allowed to stream upward across the end A of the tin tube.
A portion of the sound issuing from the tube was reflected at the
limiting surfaces of the heated layer; the part transmitted being now
only competent to slightly agitate the sensitive flame at F′.

“The heated layer was then placed at such an angle that the reflected
portion of the sound was sent through a second tin tube, A F (of the
same dimensions as B A). Its action was rendered visible by causing a
second sensitive flame placed at the end of the tube at F to become
violently affected. This _echo_ continued active as long as the heated
layer intervened; but upon its withdrawal the sensitive flame placed
at F′, receiving the whole of the direct pulse, became again violently
agitated, and at the same moment the sensitive flame at F, ceasing to
be affected by the echo, resumed its former tranquillity.

“Exactly the same action takes place when the luminous portion of a
gas-flame is made the reflecting layer; but in the experiments above
described the invisible layer above the flame only was used. By proper
adjustment of the pressure of the gas the flame at F′ can be rendered
so moderately sensitive to the direct sound-wave that the portion
transmitted through the reflecting layer shall be incompetent to affect
the flame. Then by the introduction and withdrawal of the bat’s-wing
flame the two sensitive flames can be rendered alternately quiescent
and strongly agitated.

“An illustration is here afforded of the perfect analogy between
light and sound; for if a beam of light be projected from B to F′,
and a plate of glass be introduced at A in the exact position of the
reflecting layer of gas, the beam will be divided, one portion being
reflected in the direction A F, and the other portion transmitted
through the glass toward F′, exactly as the sound-wave is divided into
a reflected and transmitted portion by the layer of heated gas or
flame.”

Thus far, therefore, we have placed our subject in the firm grasp of
experiment; nor shall we find this test failing us further on.


_PART II_

INVESTIGATION OF THE CAUSES WHICH HAVE HITHERTO BEEN SUPPOSED EFFECTIVE
IN PREVENTING THE TRANSMISSION OF SOUND THROUGH THE ATMOSPHERE

  Action of Hail and Rain—Action of Snow—Action of Fog;
  Observations in London—Experiments on Artificial
  Fogs—Observations on Fogs at the South Foreland—Action of
  Wind—Atmospheric Selection—Influence of Sound-Shadow


§ 1. _Action of Hail and Rain_

In the first part of this chapter it was demonstrated that the optic
transparency and acoustic transparency of our atmosphere were by no
means necessarily coincident; that on days of marvellous optical
clearness the atmosphere may be filled with impervious acoustic clouds,
while days optically turbid may be acoustically clear. We have now
to consider, in detail, the influence of various agents which have
hitherto been considered potent in reference to the transmission of
sound through the atmosphere.

Derham, and after him all other writers, considered that falling rain
tended powerfully to obstruct sound. An observation on June 3d has
been already referred to as tending to throw doubt on this conclusion.
Two other crucial instances will suffice to show its untenability. On
the morning of October 8th, at 7.45 A.M., a thunderstorm accompanied by
heavy rain broke over Dover. But the clouds subsequently cleared away,
and the sun shone strongly on the sea. For a time the optical clearness
of the atmosphere was extraordinary, but it was acoustically opaque.
At 2.30 P.M. a densely-black scowl again overspread the heavens to the
W.S.W. The distance being 6 miles, and all hushed on board, the horn
was heard very feebly, the siren more distinctly, while the howitzer
was better than either, though not much superior to the siren.

A squall approached us from the west. In the Alps or elsewhere I have
rarely seen the heavens blacker. Vast cumuli floated to the N.E. and
S.E.; vast streamers of rain descended in the W.N.W.; huge scrolls of
cloud hung in the N.; but spaces of blue were to be seen to the N.N.E.

At 7 miles’ distance the siren and horn were both feeble, while the gun
sent us a very faint report. A dense shower now enveloped the Foreland.

The rain at length reached us, falling heavily all the way between
us and the Foreland; but the sound, instead of being deadened, rose
perceptibly in power. Hail was now added to the rain, and the shower
reached a tropical violence, the hailstones floating thickly on the
flooded deck. In the midst of this furious squall both the horns and
the siren were distinctly heard; and as the shower lightened, thus
lessening the local pattering, the sounds so rose in power that we
heard them at a distance of 7-1/2 miles distinctly louder than they
had been heard through the rainless atmosphere at 5 miles.

At 4 P.M. the rain had ceased and the sun shone clearly through the
calm air. At 9 miles’ distance the horn was heard feebly, the siren
clearly, while the howitzer sent us a loud report. All the sounds were
better heard at this distance than they had previously been at 5-1/2
miles; from which, by the law of inverse squares, it follows that
the intensity of the sound at 5-1/2 miles’ distance must have been
augmented at least threefold by the descent of the rain.

On the 23d of October our steamer had forsaken us for shelter, and I
sought to turn the weather to account by making other observations on
both sides of the fog-signal station. Mr. Douglass, the chief engineer
of the Trinity House, was good enough to undertake the observations
N.E. of the Foreland; while Mr. Ayers, the assistant engineer, walked
in the other direction. At 12.50 P.M. the wind blew a gale, and broke
into a thunderstorm with violent rain. Inside and outside the Cornhill
Coast-guard Station, a mile from the instruments in the direction of
Dover, Mr. Ayers heard the sound of the siren through the storm; and
after the rain had ceased, all sounds were heard distinctly louder
than before. Mr. Douglass had sent a fly before him to Kingsdown, and
the driver had been waiting for fifteen minutes before he arrived.
During this time no sound had been heard, though 40 blasts had
been blown in the interval; nor had the coast-guard man on duty, a
practiced observer, heard any of them throughout the day. During the
thunderstorm, and while the rain was actually falling with a violence
which Mr. Douglass describes as perfectly torrential, the sounds
became audible and were heard by all.

To rain, in short, I have never been able to trace the slightest
deadening influence upon sound. The reputed barrier offered by “thick
weather” to the passage of sound was one of the causes which tended to
produce hesitation in establishing sound-signals on our coasts. It is
to be hoped that the removal of this error may redound to the advantage
of coming generations of seafaring men.


§ 2. _Action of Snow_

Falling snow, according to Derham, is the most serious obstacle of all
to the transmission of sound. We did not extend our observations at the
South Foreland into snowy weather; but a previous observation of my own
bears directly upon this point. On Christmas night, 1859, I arrived at
Chamouni, through snow so deep as to obliterate the road-fences, and to
render the labor of reaching the village arduous in the extreme. On the
26th and 27th it fell heavily. On the 27th, during a lull in the storm,
I reached the Montavert, sometimes breast deep in snow. On the 28th,
with great difficulty, two lines of stakes were set out across the
glacier, with the view of determining its winter motion. On the 29th
the entry in my journal, written in the morning, is: “Snow, heavy snow;
it must have descended through the entire night, the quantity freshly
fallen is so great.”

Under these circumstances I planted my theodolite beside the Mer de
Glace, having waded to my position through snow, which, being dry,
reached nearly to my breast. Assistants were sent across the glacier
with instructions to measure the displacement of a transverse line of
stakes planted previously in the snow. A storm drifted up the valley,
darkening the air as it approached. It reached us, the snow falling
more heavily than I had ever seen it elsewhere. It soon formed a heap
on the theodolite, and thickly covered my own clothes. Here, then,
was a combination of snow in the air, and of soft fresh snow on the
ground, such as Derham could hardly have enjoyed; still through such
an atmosphere I was able to make my instructions audible quite across
the glacier, the distance being half a mile, while the experiment was
rendered reciprocal by one of my assistants making his voice audible to
me.


§ 3. _Passage of Sound through Textile Fabrics, and through Artificial
Showers_

The flakes here were so thick that it was only at intervals that I
was able to pick up the retreating forms of the men. Still the air
through which the flakes fell was continuous. Did the flakes merely
yield passively to the sonorous waves, swinging like the particles
of air themselves to and fro as the sound-waves passed them? Or did
the waves bend by diffraction round the flakes, and emerge from them
without sensible loss? Experiment will aid us here by showing the
astonishing facility with which sound makes its way among obstacles,
and passes through tissues, so long as the continuity of the air in
their interstices is preserved.

A piece of millboard or of glass, a plank of wood, or the hand, placed
across the open end _t′_ of the tunnel _a b c d_, Fig. 146 (page 334),
intercepts the sound of the bell, placed in the padded box P, and
stills the sensitive flame _k_.

An ordinary cambric pocket-handkerchief, on the other hand, placed
across the tunnel-end produced hardly an appreciable effect upon the
sound. Through two layers of the handkerchief the flame was strongly
agitated; through four layers it was still agitated; while through six
layers, though nearly stilled, it was not entirely so.

Dipping the same handkerchief into water, and stretching a single
wetted layer across the tunnel-end, it stilled the flame as effectually
as the millboard or the wood. Hence the conclusion that the sound-waves
in the first instance passed through the interstices of the cambric.

Through a single layer of thin silk the sound passed without sensible
interruption; through six layers the flame was strongly agitated; while
through twelve layers the agitation was quite perceptible.

A single layer of this silk, when wetted, stilled the flame.

A layer of soft lint produced but little effect upon the sound; a layer
of thick flannel was almost equally ineffectual. Through four layers of
flannel the flame was perceptibly agitated. Through a single layer of
green baize the sound passed almost as freely as through air; through
four layers of the baize the action was still sensible. Through a layer
of close hard felt, half an inch thick, the sound-waves passed with
sufficient energy to sensibly agitate the flame. Through 200 layers of
cotton-net the sound passed freely. I did not witness these effects
without astonishment.

A single layer of thin oiled silk stopped the sound and stilled the
flame. A leaf of common note-paper, or a five-pound note, also stopped
the sound.

The sensitive flame is not absolutely necessary to these experiments.
Let a ticking watch be hung six inches from the ear, a cambric
handkerchief dropped between it and the ear scarcely sensibly affects
the ticking; a sheet of oil-skin or an intensely heated gas-column cuts
it almost wholly off.

But though oiled silk, foreign post, or a banknote, can stop the
sound, a film sufficiently thin to yield freely to the aërial pulses
transmits it. A thick soap-film produces an obvious effect upon the
sensitive flame; a very thin one does not. The augmentation of the
transmitted sound may be observed simultaneously with the generation
and brightening of the colors which indicate the increasing thinness of
the film. A very thin collodion-film acts in the same way.

Acquainted with the foregoing facts regarding the passage of sound
through cambric, silk, lint, flannel, baize, felt, and cotton-net,
you are prepared for the statement that the sound-waves pass without
sensible impediment through heavy artificial showers of rain, hail,
and snow. Water-drops, seeds, sand, bran, and flocculi of various
kinds, have been employed to form such showers; through all of these,
as through the actual rain and hail already described, and through the
snow on the Mer de Glace, the sound passes without sensible obstruction.


§ 4. _Action of Fog. Observations in London_

But the mariner’s greatest enemy, fog, is still to be dealt with;
and here for a long time the proper conditions of experiment were
absent. Up to the end of November we had had frequent days of haze,
sufficiently thick to obscure the white cliffs of the Foreland, but
no real fog. Still those cases furnished demonstrative evidence that
the notions entertained regarding the reflection of sound by suspended
particles were wrong; for on many days of the thickest haze the sound
covered twice the range attained on other days of perfect optical
transparency. Such instances dissolved the association hitherto assumed
to exist between acoustic transparency and optic transparency, but they
left the action of dense fogs undetermined.

On December 9th a memorable fog settled down on London. I addressed a
telegram to the Trinity House suggesting some gun observations. With
characteristic promptness came the reply that they would be made in the
afternoon at Blackwall. I went to Greenwich in the hope of hearing the
guns across the river; but the delay of the train by the fog rendered
my arrival too late. Over the river the fog was very dense, and through
it came various sounds with great distinctness. The signal-bell of an
unseen barge rang clearly out at intervals, and I could plainly hear
the hammering at Cubitt’s Town, half a mile away, on the opposite side
of the river. No deadening of the sound by the fog was apparent.

Through this fog and various local noises, Captain Atkins and Mr.
Edwards heard the report of a 12-pounder carronade with a 1-lb. charge
distinctly better than the 18-pounder with a 3-lb. charge, an optically
clear atmosphere, and all noises absent, on July 3d.

Anxious to turn to the best account a phenomenon for which we had
waited so long, I tried to grapple with the problem by experiments on
a small scale. On the 10th, I stationed my assistant with a whistle
and organ-pipe on the walk below the southwest end of the bridge
dividing Hyde Park from Kensington Gardens. From the eastern end of
the Serpentine I heard distinctly both the whistle and the pipe, which
produced 380 waves a second. On changing places with my assistant, I
heard for a time the distinct blast of the whistle only. The deeper
note of the organ-pipe at length reached me, rising sometimes to great
distinctness, and sometimes falling to inaudibility. The whistle showed
the same intermittence as to period, but in an opposite sense; for,
when the whistle was faint, the pipe was strong, and _vice versa_. To
obtain the fundamental note of the pipe, it had to be blown gently, and
on the whole the whistle proved the most efficient in piercing the fog.

An extraordinary amount of sound filled the air during these
experiments. The resonant roar of the Bayswater and Knightsbridge
roads; the clangor of the great bell of Westminster; the
railway-whistles, which were frequently blown, and the fog-signals
exploded at the various metropolitan stations, were all heard with
extraordinary intensity. This could by no means be reconciled with the
statements so categorically made regarding the acoustic impenetrability
of a London fog.

On the 11th of December, the fog being denser than before, I heard
every blast of the whistle, and occasional blasts of the pipe, over
the distance between the bridge and the eastern end of the Serpentine.
On joining my assistant at the bridge, the loud concussion of a gun
was heard by both of us. A police-inspector affirmed that it came
from Woolwich, and that he had heard several shots about 2 P.M. and
previously. The fact, if a fact, was of the highest importance; so I
immediately telegraphed to Woolwich for information. Prof. Abel kindly
furnished me with the following particulars:

“The firing took place at 1.40 P.M. The guns proved were of
comparatively small size—64-pounders, with 10-lb. charges of powder.

“The concussion experienced at my house and office, about
three-quarters of a mile from the butt, was decidedly more severe than
that experienced when the heaviest guns are proved with charges of
110 to 120 lbs. of powder. There was a dense fog here at the time of
firing.”

These were the guns heard by the police-inspector; on subsequent
inquiry it was ascertained that two guns were fired about 3 P.M. These
were the guns heard by myself.

Prof. Abel also communicated to me the following fact: “Our workman’s
bell at the Arsenal Gate, which is of moderate size and anything but
clear in tone, is pretty distinctly heard by Prof. Bloxam _only_ when
the wind is _northeast_. During the whole of last week the bell was
heard with great distinctness, the wind being _southwesterly_ (opposed
to the sound). The distance of the bell from Bloxam’s house is about
three-quarters of a mile as the crow flies.”

Assuredly no question of science ever stood so much in need of revision
as this of the transmission of sound through the atmosphere. Slowly,
but surely, we mastered the question; and the further we advanced, the
more plainly it appeared that our reputed knowledge regarding it was
erroneous from beginning to end.

On the morning of the 12th the fog attained its maximum density. It
was not possible to read at my window, which fronted the open western
sky. At 10.30 I sent an assistant to the bridge, and listened for his
whistle and pipe at the eastern end of the Serpentine. The whistle
rose to a shrillness far surpassing anything previously heard, but it
sank sometimes almost to inaudibility; proving that, though the air
was on the whole highly homogeneous, acoustic clouds still drifted
through the fog. A second pipe, which was quite inaudible yesterday,
was plainly heard this morning. We were able to discourse across the
Serpentine to-day with much greater ease than yesterday.

During our summer observations I had once or twice been able to fix the
position of the Foreland in thick haze by the direction of the sound.
To-day my assistant, hidden by the fog, walked up to the Watermen’s
Boathouse sounding his whistle; and I walked along the opposite side of
the Serpentine, clearly appreciating for a time that the line joining
us was oblique to the axis of the river. Coming to a point which seemed
to be exactly abreast of him, I marked it; and on the following day,
when the fog had cleared away, the marked position was found to be
perfectly exact. When undisturbed by echoes, the ear, with a little
practice, becomes capable of fixing with great precision the direction
of a sound.

On reaching the Serpentine this morning, a peal of bells, which then
began to ring, seemed so close at hand that it required some reflection
to convince me that they were ringing to the north of Hyde Park. The
sounds fluctuated wonderfully in power. Prior to the striking of
eleven by the great bell of Westminster, a nearer bell struck with
loud clangor. The first five strokes of the Westminster bell were
afterward heard, one of them being extremely loud; but the last six
strokes were inaudible. An assistant was stationed to attend to the 12
o’clock bells. The clock which had struck so loudly at 11 was unheard
at 12, while of the Westminster bell eight strokes out of twelve were
inaudible. To such astonishing changes is the atmosphere liable.

At 7 P.M. the Westminster bell, striking seven, was not at all heard
from the Serpentine, while the nearer bell already alluded to was heard
distinctly. The fog had cleared away, and the lamps on the bridge
could be seen from the eastern end of the Serpentine burning brightly;
but, instead of the sound sharing the improvement of the light, what
might be properly called an acoustic fog took the place of its optical
predecessor. Several series of the whistle and organ-pipe were sounded
in succession; one series only of the whistle-sounds was heard, all
the others being quite inaudible. Three series of the organ-pipe were
heard, but very faintly. On reversing the positions and sounding as
before, nothing whatever was heard.

At 8 o’clock the chimes and hour-bell of the Westminster clock were
both very loud. The “acoustic fog” had shifted its position, or
temporarily melted away.

Extraordinary fluctuations were also observed in the case of the
church-bells heard in the morning: in a few seconds they would sink
from a loudly ringing-peal into utter silence, from which they would
rapidly return to loud-tongued audibility. The intermittent drifting of
fog over the sun’s disk (by which his light is at times obscured, at
times revealed) is the optical analogue of these effects. As regards
such changes, the acoustic deportment of the atmosphere is a true
transcript of its optical deportment.

At 9 P.M. three strokes only of the Westminster clock were heard; the
others were inaudible. The air had relapsed in part into its condition
at 7 P.M., when all the strokes were unheard. The quiet of the park
this evening, as contrasted with the resonant roar which filled the air
on the two preceding days, was very remarkable. The sound, in fact, was
stifled in the optically clear but acoustically flocculent atmosphere.

On the 13th, the fog being displaced by thin haze, I went again to
the Serpentine. The carriage-sounds were damped to an extraordinary
degree. The roar of the Knightsbridge and Bayswater roads had subsided,
the tread of troops which passed us a little way off was unheard,
while at 11 A.M. both the chimes and the hour-bell of the Westminster
clock were stifled. Subjectively considered, all was favorable to
auditory impressions; but the very cause that damped the local noises
extinguished our experimental sounds. The voice across the Serpentine
to-day, with my assistant plainly visible in front of me, was
distinctly feebler than it had been when each of us was hidden from the
other in the densest fog.

Placing the source of sound at the eastern end of the Serpentine I
walked along its edge from the bridge toward the end. The distance
between these two points is about 1,000 paces. After 500 of them had
been stepped, the sound was not so distinct as it had been at the
bridge on the day of densest fog; hence, by the law of inverse squares,
the optical cleansing of the air through the melting away of the fog
had so darkened it acoustically that a sound generated at the eastern
end of the Serpentine was lowered to one-fourth of its intensity at a
point midway between the end and the bridge.

To these demonstrative observations one or two subsequent ones may
be added. On several of the moist and warm days, at the beginning of
1874, I stood at noon beside the railing of St. James’s Park, near
Buckingham Palace, three-quarters of a mile from the clock-tower, which
was clearly visible. Not a single stroke of “Big Ben” was heard. On
January 19th fog and drizzling rain obscured the tower; still from the
same position I not only heard the strokes of the great bell, but also
the chimes of the quarter-bells.

During the exceedingly dense and “dripping” fog of January 22d, from
the same railings, I heard every stroke of the bell. At the end of the
Serpentine, when the fog was densest, the Westminster bell was heard
striking loudly eleven. Toward evening this fog began to melt away,
and at 6 o’clock I went to the end of the Serpentine to observe the
effect of the optical clearing upon the sound. Not one of the strokes
reached me. At 9 o’clock and at 10 o’clock my assistant was in the same
position, and on both occasions he failed to hear a single stroke of
the bell. It was a case precisely similar to that of December 13th,
when the dissolution of the fog was accompanied by a decided acoustic
thickening of the air.[66]


§ 5. _Observations at the South Foreland_

Satisfactory, and indeed conclusive, as these results seemed, I desired
exceedingly to confirm them by experiments with the instruments
actually employed at the South Foreland. On the 10th of February I had
the gratification of receiving the following note and inclosure from
the Deputy Master of Trinity House:

  “MY DEAR TYNDALL—The inclosed will show how accurately
  your views have been verified, and I send them on at once
  without waiting for the details. I think you will be glad
  to have them, and as soon as I get the report it shall be
  sent to you. I made up my mind ten days ago that there
  would be a chance in the light foggy-disposed weather at
  home, and therefore sent the ‘Argus’ off at an hour’s
  notice, and requested the Fog Committee to keep one
  member on board. On Friday I was so satisfied that the
  fog would occur that I sent Edwards down to record the
  observations.

  “Very truly yours,

  “FRED. ARROW.”

The inclosure referred to was notes from Captain Atkins and Mr.
Edwards. Captain Atkins writes thus:

“As arranged, I came down here by the mail express, meeting Mr. Edwards
at Cannon Street. We put up at the Dover Castle, and next morning at
7 I was awoke by sounds of the siren. On jumping up I discovered that
the long-looked-for fog had arrived, and that the ‘Argus’ had left her
moorings.

“However, had I been on board, the instructions I left with Troughton
(the master of the ‘Argus’) could not have been better carried out.
About noon the fog cleared up, and the ‘Argus’ returned to her
moorings, when I learned that they had taken both siren and horn
sounds to a distance of 11 miles from the station, where they dropped
a buoy. This I knew to be correct, as I have this morning recovered
the buoy, and the distances both in and out agree with Troughton’s
statement. I have also been to the Varne light-ship (12-3/4 miles from
the Foreland), and ascertained that during the fog of Saturday forenoon
they ‘distinctly’ heard the sounds.”


Mr. Edwards, who was constantly at my side during our summer and autumn
observations, and who is thoroughly competent to form a comparative
estimate of the strength of the sounds, states that those of the 7th
were “extraordinarily loud,” both Captain Atkins and himself being
awoke by them. He does not remember ever before hearing the sounds so
loud in Dover; it seemed as though the observers were close to the
instruments.

Other days of fog preceded this one, and they were all days of acoustic
transparency, the day of densest fog being acoustically the clearest of
all.

The results here recorded are of the highest importance, for they bring
us face to face with a dense fog and an actual fog-signal, and confirm
in the most conclusive manner the previous observations. The fact of
Captain Atkins and Mr. Edwards being awakened by the siren proves,
beyond all our previous experience, its power during this dense fog.

It is exceedingly interesting to compare the transmission of sound on
February 7th with its transmission on October 14th. The wind on both
days had the same strength and direction. My notes of the observations
show the latter to have been throughout a day of extreme optical
clearness. The range was 10 miles. During the fog of February 7th the
“Argus” heard the sound at 11 miles; and it was also heard at the Varne
light-vessel, which is 12-3/4 miles from the Foreland.

It is also worthy of note that through the same fog the sounds were
well heard at the South Sand Head light-vessel, which is in the
opposite direction from the South Foreland, and was actually behind
the siren. For this important circumstance is to be borne in mind: on
February 7th the siren happened to be pointed, not toward the “Argus,”
but toward Dover. Had the yacht been in the axis of the instrument it
is highly probable that the sound would have been heard all the way
across to the coast of France.

It is hardly necessary for me to say a word to guard myself against the
misconception that I consider sound to be assisted by the fog itself.
The fog-particles have no more influence upon the waves of sound than
the suspended particles stirred up over the banks of Newfoundland
have upon the waves of the Atlantic. A homogeneous air is the usual
associate of fog, and hence the acoustic clearness of foggy weather.


§ 6. _Experiments on Artificial Fogs_

These observations are clinched and finished by being brought within
the range of laboratory experiment. Here we shall learn incidentally a
lesson as to the caution required from an experimenter.

The smoke from smouldering brown paper was allowed to stream upward
through its rectangular apertures, into the tunnel _a b c d_ (Fig.
146); the action upon the sound-waves was strong, rendering the short
and agitated sensitive flame _k_ tall and quiescent.

Air first passed through ammonia, then through hydrochloric acid, and,
thus loaded with thick fumes, was sent into the tunnel; the agitated
flame was rendered immediately quiescent, indicating a very decided
action on the part of the artificial fog.

Air passed through perchloride of tin and sent into the tunnel produced
exceedingly dense fumes. The action upon the sound-waves was very
strong.

The dense smoke of resin, burned before the open end of the tunnel, and
blown into it with a pair of bellows, had also the effect of stopping
the sound-waves, so as to still the agitated flame.

The conclusion seems clear, and its perfect harmony with the prevalent
_à priori_ notions as to the action of fog upon sound makes it almost
irresistible. But caution is here necessary. The smoke of the brown
paper was _hot_; the flask containing the hydrochloric acid was _hot_;
that containing the perchloride of tin was _hot_; while the resin fumes
produced by a red-hot poker were also obviously hot. Were the results,
then, due to the fumes or to the differences of temperature? The
observations might well have proved a trap to an incautious reasoner.

Instead of the smoke and heated air, the heated air alone from four
red-hot pokers was permitted to stream upward into the tunnel; the
action on the sound-waves was very decided, though the tunnel was
optically empty. The flame of a candle was placed at the upper end, and
the hot air just above its tip was blown into the tunnel; the action on
the sensitive flame was decided. A similar effect was produced when the
air, ascending from a red-hot iron, was blown into the tunnel.

In these latter cases the tunnel remained optically clear, while
the same effect as that produced by the resin, smoke, and fumes was
observed. Clearly, then, we are not entitled to ascribe, without
further investigation, to the artificial fog an effect which may have
been due to the air which accompanied it.

Having eliminated the fog and proved the non-homogeneous air effective,
our reasoning will be completed by eliminating the heat, and proving
the fog ineffective.

Instead of the tunnel _a b c d_, Fig. 146, a cupboard with glass sides,
3 feet long, 2 feet wide, and about 5 feet high, was filled with fumes
of various kinds. Here it was thought the fumes might remain long
enough for differences of temperature to disappear. Two apertures were
made in two opposite panes of glass 3 feet asunder. In front of one
aperture was placed the bell in its padded box, and behind the other
aperture, and at some distance from it, the sensitive flame.

Phosphorus placed in a cup floating on water was ignited within the
closed cupboard. The fumes were so dense that considerably less than
the three feet traversed by the sound extinguished totally a bright
candle-flame.

At first there was a slight action upon the sound; but this rapidly
vanished, the flame being no more affected than if the sound had passed
through pure air. The first action was manifestly due to differences of
temperature, and it disappeared when the temperature was equalized.

The cupboard was next filled with the dense fumes of gunpowder. At
first there was a slight action; but this disappeared even more rapidly
than in the case of the phosphorus, the sound passing as if no fumes
were there. It required less than half a minute to abolish the action
in the case of the phosphorus, but a few seconds sufficed in the case
of the gunpowder. These fumes were far more than sufficient to quench
the candle-flame.

The dense smoke of resin, when the temperature had become equable,
exerted no action on the sound.

The fumes of gum-mastic were equally ineffectual.

The fumes of the perchloride of tin, though of extraordinary density,
exerted no sensible effect upon the sound.

Exceedingly dense fumes of chloride of ammonium next filled the
cupboard. A fraction of the length of the 3-foot tube sufficed to
quench the candle-flame. Soon after the cupboard was filled, the sound
passed without the least sensible deterioration. An aperture at the top
of the cupboard was opened; but though a dense smoke-column ascended
through it, many minutes elapsed before the candle-flame could be seen
through the attenuated fog.

Steam from a copper boiler was so copiously admitted into the
cupboard as to fill it with a dense cloud. No real cloud was ever so
dense; still the sound passed through it without the least sensible
diminution. This being the case, cloud-echoes are not a likely
phenomenon.

In all of these cases, when a couple of Bunsen’s burners were ignited
within the cupboard containing the fumes, less than a minute’s action
rendered the air so heterogeneous that the sensitive flame was
completely stilled.

These acoustically inactive fogs were subsequently proved competent to
cut off the electric light.

Experiment and observation go, therefore, hand in hand in demonstrating
that fogs have no sensible action upon sound. The notion of their
impenetrability, which so powerfully retarded the introduction of
phonic coast-signals, being thus abolished, we have solid ground for
the hope that disasters due to fogs and thick weather will in the
future be materially mitigated.


§ 7. _Action of Wind_

In stormy weather we were frequently forsaken by our steamer, which had
to seek shelter in the Downs or Margate Roads, and on such occasions
the opportunity was turned to account to determine the effect of the
wind. On October 11th, accompanied by Mr. Douglass and Mr. Edwards, I
walked along the cliffs from Dover Castle toward the Foreland, the wind
blowing strongly against the sound. About a mile and a half from the
Foreland, we first heard the faint but distinct sound of the siren. The
horn-sound was inaudible. A gun fired during our halt was also unheard.

As we approached the Foreland we saw the smoke of a gun. Mr. Edwards
heard a faint crack, but neither Mr. Douglass nor I heard anything.
The sound of the siren was at the same time of piercing intensity. We
waited for ten minutes, when another gun was fired. The smoke was at
hand, and I thought I heard a faint thud, but could not be certain.
My companions heard nothing. On pacing the distance afterward we were
found to be only 550 yards from the gun. We were shaded at the time
by a slight eminence from both the siren and the gun, but this could
not account for the utter extinction of the gun-sound at so short a
distance, and at a time when the siren sent to us a note of great power.

Mr. Ayres at my request walked windward along the cliff, while Mr.
Douglass proceeded to St. Margaret’s Bay. During their absence I
had three guns fired. Mr. Ayres heard only one of them. Favored by
the wind, Mr. Douglass, at twice the distance, and far more deeply
immersed in the sound-shadow, heard all three reports with the utmost
distinctness.

Joining Mr. Douglass, we continued our walk to a distance of
three-quarters of a mile beyond St. Margaret’s Bay. Here, being dead to
leeward, though the wind blew with unabated violence, the sound of the
siren was borne to us with extraordinary power.[67] In this position
we also heard the gun loudly, and two other loud reports at the proper
interval of ten minutes, as we returned to the Foreland.

It is within the mark to say that the gun on October 11th was heard
five times, and might have been heard fifteen times, as far to leeward
as to windward.

In windy weather the shortness of its sound is a serious drawback to
the use of the gun as a signal. In the case of the horn and siren, time
is given for the attention to be fixed upon the sound; and a single
puff, while cutting out a portion of the blast, does not obliterate it
wholly. Such a puff, however, may be fatal to the momentary gun-sound.

On the leeward side of the Foreland, on the 23d of October, the sounds
were heard at least four times as far as on the windward side, while in
both directions the siren possessed the greatest penetrative power.

On the 24th the wind shifted to E.S.E., and the sounds, which, when the
wind was W.S.W., failed to reach Dover, were now heard in the streets
through thick rain. On the 27th the wind was E.N.E. In our writing-room
in the Lord Warden Hotel, in the bedrooms, and on the staircase, the
sound of the siren reached us with surprising power, piercing through
the whistling and moaning of the wind, which blew through Dover toward
Folkestone. The sounds were heard by Mr. Edwards and myself at 6 miles
from the Foreland on the Folkestone road; and had the instruments not
then ceased sounding, they might have been heard much further. At the
South Sand Head light-vessel, 3-3/4 miles on the opposite side, no
sound had been heard throughout the day. On the 28th, the wind being
N. by E., the sounds were heard in the middle of Folkestone, 8 miles
off, while in the opposite direction they failed to reach 3-3/4 miles.
On the 29th the limits of range were Eastware Bay on the one side, and
Kingsdown on the other; on the 30th the limits were Kingsdown on the
one hand, and Folkestone Pier on the other. With a wind having a force
of 4 or 5 it was a very common observation to hear the sound in one
direction three times as far as in the other.

This well-known effect of the wind is exceedingly difficult to explain.
Indeed, the only explanation worthy of the name is one offered by Prof.
Stokes, and suggested by some remarkable observations of De la Roche.
In Vol. I. of “Annales de Chimie” for 1816, p. 176, Arago introduces De
la Roche’s memoir in these words: “L’auteur arrive à des conclusions,
qui d’abord pourront paraître paradoxales, mais ceux qui savent combien
il mettait de soins et d’exactitude dans toutes ses recherches se
garderont sans doute d’opposer une opinion populaire à des expériences
positives.” The strangeness of De la Roche’s results consisted in his
establishing, by quantitative measurements, not only that sound has
a greater range in the direction of the wind than in the opposite
direction, but that the range at right angles to the wind is the
maximum.

In a short but exceedingly able communication, presented to the British
Association in 1857, the eminent physicist above mentioned points out
a cause which, _if sufficient_, would account for the results referred
to. The lower atmospheric strata are retarded by friction against
the earth, and the upper ones by those immediately below them; the
velocity of transition, therefore, in the case of wind, increases from
the ground upward. It may be proved that this difference of velocity
tilts the sound-wave upward in a direction opposed to, and downward
in a direction coincident with, the wind. In this latter case the
direct wave is reinforced by the wave reflected from the earth. Now
the reinforcement is greatest in the direction in which the direct
and reflected waves inclose the smallest angle; and this is at right
angles to the direction of the wind. Hence the greater range in this
direction. It is not, therefore, according to Prof. Stokes, a stifling
of the sound to windward, but a tilting of the sound-wave over the
heads of the observers, that defeats the propagation in that direction.

This explanation calls for verification, and I wished much to test it
by means of a captive balloon rising high enough to catch the deflected
wave; but on communicating with Mr. Coxwell, who has earned for himself
so high a reputation as an aeronaut, and who has always shown himself
so willing to promote a scientific object, I learned with regret that
the experiment was too dangerous to be carried out.[68]


§ 8. _Atmospheric Selection_

It has been stated that the atmosphere on different days shows
preferences to different sounds. This point is worthy of further
illustration.

After the violent shower which passed over us on October 18th, the
sounds of all the instruments, as already stated, rose in power; but
it was noticed that the horn-sound, which was of lower pitch than
that of the siren, improved most, at times not only equalling, but
surpassing, the sound of its rival. From this it might be inferred that
the atmospheric change produced by the rain favored more especially the
transmission of the longer sonorous waves.

But our programme enabled us to go further than mere inference. It
had been arranged on the day mentioned that up to 3.30 P.M. the siren
should perform 2,400 revolutions a minute, generating 480 waves a
second. As long as this rate continued, the horn, after the shower, had
the advantage. The rate of rotation was then changed to 2,000 a minute,
or 400 waves a second, when the siren-sound immediately surpassed that
of the horn. A clear connection was thus established between aërial
reflection and the length of the sonorous waves.

The 10-inch Canadian whistle being capable of adjustment so as to
produce sounds of different pitch, on the 10th of October I ran through
a series of its sounds. The shrillest appeared to possess great
intensity and penetrative power. The belief is common that a note of
this character (which affects so powerfully, and even painfully, an
observer close at hand) has also the greatest range. Mr. A. Gordon,
in his examination before the Committee on Lighthouses, in 1845,
expressed himself thus: “When you get a shrill sound, high in the
scale, that sound is carried much further than a lower note in the
scale.” I have heard the same opinion expressed by other scientific men.

On the 14th of October the point was submitted to an experimental test.
It had been arranged that up to 11.30 A.M. the Canadian whistle, which
had been heard with such piercing intensity on the 10th, should sound
its shrillest note. At the hour just mentioned we were beside the Varne
buoy, 7-3/4 miles from the Foreland. The siren, as we approached the
buoy, was heard through the paddle-noises; the horns were also heard,
but more feebly than the siren. We paused at the buoy and listened for
the 11.30 gun. Its boom was heard by all. Neither before nor during
the pause was the shrill-sounding Canadian whistle once heard. At the
appointed time it was adjusted to produce its ordinary low-pitched
note, which was immediately heard. Further out the low boom of the
cannon continued audible after all the other sounds had ceased.

But it was only during the early part of the day that this preference
for the longer wave was manifested. At 3 P.M. the case was completely
altered, for then the high-pitched siren was heard when all the other
sounds were inaudible. On many other days we had illustrations of the
varying comparative power of the siren and the gun. On the 9th of
October sometimes the one, sometimes the other, was predominant. On the
morning of the 13th the siren was clearly heard on Shakespeare’s Cliff,
while two guns with their puffs perfectly visible were unheard. On
October 16th, 2 miles from the signal-station, the gun at 11 o’clock
was inferior to the siren, but both were heard. At 12.30, the distance
being 6 miles, the gun was quite unheard, while the siren continued
faintly audible. Later on in the day the experiment was twice repeated.
The puff of the gun was in each case seen, but nothing was heard. In
the last experiment, when the gun was quenched, the siren sent forth a
sound so strong as to maintain itself through the paddle-noises. The
day was clearly hostile to the passage of the longer sonorous waves.

October 17th began with a preference for the shorter waves. At 11.30
A.M. the mastery of the siren over the gun was pronounced; at 12.30
the gun slightly surpassed the siren; at 1, 2, and 2.30 P.M. the gun
also asserted its mastery. This preference for the longer waves was
continued on October 18th. On October 20th the day began in favor
of the gun, then both became equal, and finally the siren gained
the mastery; but the day had become stormy, and a storm is always
unfavorable to the momentary gun-sound. The same remark applies to
the experiments of October 21st. At 11 A.M., distance 6-1/2 miles,
when the siren made itself heard through the noises of wind, sea,
and paddles, the gun was fired; but, though listened for with all
attention, no sound was heard. Half an hour later the result was the
same. On October 24th five observers saw the flash of the gun at a
distance of 5 miles, but heard nothing; all of them at this distance
heard the siren distinctly; a second experiment on the same day yielded
the same result. On the 27th also the siren was triumphant; and on
three distinct occasions on the 29th its mastery over the gun was very
decided.

Such experiments yield new conceptions as to the scattering of sound
in the atmosphere. No sound here employed is a simple sound; in every
case the fundamental note is accompanied by others, and the action
of the atmosphere on these different groups of waves has its optical
analogue in that scattering of the waves of the luminiferous ether
which produces the various shades and colors of the sky.


§ 9. _Concluding Remarks_

A few additional remarks and suggestions will fitly wind up this
chapter. It has been proved that in some states of the weather the
howitzer firing a 3-lb. charge commands a larger range than the
whistles, trumpets, or siren. This was the case, for example, on the
particular day, October 17th, when the ranges of all the sounds reached
their maximum.

On many other days, however, the inferiority of the gun to the siren
was demonstrated in the clearest manner. The gun-puffs were seen with
the utmost distinctness at the Foreland, but no sound was heard, the
note of the siren at the same time reaching us with distinct and
considerable power.

The disadvantages of the gun are these:

_a._ The duration of the sound is so short that, unless the observer is
prepared beforehand, the sound, through lack of attention rather than
through its own powerlessness, is liable to be unheard.

_b._ Its liability to be quenched by a local sound is so great that it
is sometimes obliterated by a puff of wind taking possession of the
ears at the time of its arrival. This point was alluded to by Arago,
in his report on the celebrated experiments of 1822. By such a puff a
momentary gap is produced in the case of a continuous sound, but not
entire extinction.

_c._ Its liability to be quenched or deflected by an opposing wind, so
as to be practically useless at a very short distance to windward, is
very remarkable. A case has been cited in which the gun failed to be
heard against a violent wind at a distance of 550 yards from the place
of firing, the sound of the siren at the same time reaching us with
great intensity.

Still, notwithstanding these drawbacks, I think the gun is entitled to
rank as a first-class signal. I have had occasion myself to observe its
extreme utility at Holyhead and the Kish light-vessel near Kingstown.
The commanders of the Holyhead boats, moreover, are unanimous in their
commendation of the gun. An important addition in its favor is the fact
that in a fog the flash or glare often comes to the aid of the sound.
On this point, the evidence is quite conclusive.

There may be cases in which the combination of the gun with one of
the other signals may be desirable. Where it is wished to confer an
unmistakable individuality on a fog-signal station, such a combination
might with advantage be resorted to.

If the gun be retained as one form of fog-signal (and I should be sorry
at present to recommend its total abolition), it ought to be of the
most suitable description. Our experiments prove the sound of the gun
to be dependent on its shape; but we do not know that we have employed
the best shape. This suggests the desirability of constructing a gun
with special reference to the production of sound.[69]

An absolutely uniform superiority on all days cannot be conceded
to any one of the instruments subjected to examination; still, our
observations have been so numerous and long-continued as to enable us
to come to the sure conclusion, that, on the whole, the steam-siren is
the most powerful fog-signal which has hitherto been tried in England.
It is specially powerful when local noises, such as those of wind,
rigging, breaking waves, shore-surf, and the rattle of pebbles, have to
be overcome. Its density, quality, pitch, and penetration, render it
dominant over such noises after all other signal-sounds have succumbed.

I have not, therefore, hesitated to recommend the introduction of the
siren as a coast-signal.

It will be desirable in each case to confer upon the instrument a
power of rotation, so as to enable the person in charge of it to point
its trumpet against the wind or in any other required direction.
This arrangement was made at the South Foreland, and it presents no
mechanical difficulty. It is also desirable to mount the siren, so as
to permit of the depression of its trumpet fifteen or twenty degrees
below the horizon.

In selecting the position at which a fog-signal is to be mounted, the
possible influence of a sound-shadow, and the possible extinction of
the sound by the interference of the direct waves with waves reflected
from the shore, must form the subject of the gravest consideration.
Preliminary trials may, in most cases, be necessary before fixing on
the precise point at which the instrument is to be placed.

The siren which has been long known to scientific men is worked with
air; and it would be worth while to try how the fog-siren would behave
supposing compressed air to be substituted for steam. Compressed air
might also be tried with the whistles.

No fog-signal hitherto tried is able to fulfil the condition laid
down in a very able letter already referred to, namely, “_that all
fog-signals should be distinctly audible for at least 4 miles, under
every circumstance_.” Circumstances may exist to prevent the most
powerful sound from being heard at half this distance. What may with
certainty be affirmed is, that in almost all cases the siren may
certainly be relied on at a distance of 2 miles; in the great majority
of cases it may be relied upon at a distance of 3 miles, and in the
majority of cases to a distance greater than 3 miles.

Happily the experiments thus far made are perfectly concurrent in
indicating that at the particular time when fog-signals are needed, the
air holding the fog in suspension is in a highly-homogeneous condition;
hence it is in the highest degree probable that in the case of fog we
may rely upon the signals being effective at far greater distances than
those just mentioned.

I am cautious not to inspire the mariner with a confidence which may
prove delusive. When he hears a fog-signal he ought, as a general rule
(at all events until extended experience justifies the contrary), to
assume the source of sound to be not more than 2 or 3 miles distant,
and to heave his lead or to take other necessary precautions. If he
errs at all in his estimate of distance, it ought to be on the side of
safety.

With the instruments now at our disposal wisely established along our
coasts, I venture to think that the saving of property in ten years
will be an exceedingly large multiple of the outlay necessary for the
establishment of such signals. The saving of life appeals to the higher
motives of humanity.

In a report written for the Trinity House on the subject of
fog-signals, my excellent predecessor, Prof. Faraday, expresses the
opinion that a false promise to the mariner would be worse than no
promise at all. Casting our eyes back upon the observations here
recorded, we find the sound-range on clear, calm days varying from
2-1/2 miles to 16-1/2 miles. It must be evident that an instruction
founded on the latter observation would be fraught with peril in
weather corresponding to the former. Not the maximum but the minimum
sound-range should be impressed upon the mariner. Want of attention to
this point may be followed by disastrous consequences.

This remark is not made without cause. I have before me a “Notice to
Mariners” regarding a fog-whistle recently mounted at Cape Race, which
is reputed to have a range of 20 miles in calm weather, 30 miles with
the wind, and in stormy weather or against the wind 7 to 10 miles.
Now, considering the distance reached by sound in our observations, I
should be willing to concede the possibility, in a more homogeneous
atmosphere than ours, of a sound-range on _some_ calm days of 20 miles,
and on _some_ light windy days of 30 miles, to a powerful whistle; but
I entertain a strong belief that the stating of these distances, or of
the distance 7 to 10 miles against a storm, without any qualification,
is calculated to inspire the mariner with false confidence. I would
venture to affirm that at Cape Race calm days might be found in which
the range of the sound will be less than one-fourth of what this
notice states it to be. Such publications ought to be without a trace
of exaggeration, and furnish only data on which the mariner may with
perfect confidence rely. My object in extending these observations over
so long a period was to make evident to all how fallacious it would
be, and how mischievous it might be, to draw general conclusions from
observations made in weather of great acoustic transparency.

Thus ends, for the present at all events, an inquiry which I trust
will prove of some importance, scientific as well as practical. In
conducting it I have had to congratulate myself on the unfailing aid
and co-operation of the Elder Brethren of the Trinity House. Captain
Drew, Captain Close, Captain Were, Captain Atkins, and the Deputy
Master, have all from time to time taken part in the inquiry. To the
eminent arctic navigator, Admiral Collinson, who showed throughout
unflagging and, I would add, philosophic interest in the investigation,
I am indebted for most important practical aid. He was almost always
at my side, comparing opinions with me, placing the steamer in the
required positions, and making with consummate skill and promptness
the necessary sextant observations. I am also deeply sensible of the
important services rendered by Mr. Douglass, the able and indefatigable
engineer, by Mr. Ayres, the assistant engineer, and by Mr. Price
Edwards, the private secretary of the Deputy Master of the Trinity
House.

The officers and gunners at the South Foreland also merit my best
thanks, as also Mr. Holmes and Mr. Laidlaw, who had charge of the
trumpets, whistles, and siren.

In the subsequent experimental treatment of the subject I have been
most ably aided by my excellent assistant, Mr. John Cottrell.


NOTE

  In the Appendix will be found a brief paper on “Acoustic
  Reversibility,” in which I offer a solution of a difficulty
  encountered by the French philosophers in their experiments
  on the velocity of sound in 1822. The solution is based on
  the experiments and observations recorded in the foregoing
  chapter.—J. T.


SUMMARY OF CHAPTER VII

The paper of Dr. Derham, published in the “Philosophical Transactions”
for 1708, has been hitherto the almost exclusive source of our
knowledge of the causes which affect the transmission of sound through
the atmosphere.

Derham found that fog obstructed sound, that rain and hail obstructed
sound, but that above all things falling snow, or a coating of fresh
snow upon the ground, tended to check the propagation of sound through
the atmosphere.

With a view to the protection of life and property at sea in the
years 1873 and 1874, this subject received an exhaustive examination,
observational and experimental. The investigation was conducted at the
expense of the Government and under the auspices of the Elder Brethren
of the Trinity House.

The most conflicting results were at first obtained. On the 19th of
May, 1873, the sound range was 3-1/3 miles; on the 20th it was 5-1/2
miles; on the 2d of June, 6 miles; on the 3d, more than 9 miles; on the
10th, 9 miles; on the 25th, 6 miles; on the 26th, 9-1/4 miles; on the
1st of July, 12-3/4 miles; on the 2d, 4 miles; while on the 3d, with a
clear calm atmosphere and smooth sea, it was less than 3 miles.

These discrepancies were proved to be due to a state of the air which
bears the same relation to sound that cloudiness does to light. By
streams of air differently heated, or saturated in different degrees
with aqueous vapors, the atmosphere is rendered _flocculent_ to sound.

_Acoustic clouds_, in fact, are incessantly floating or flying through
the air. They have nothing whatever to do with ordinary clouds, fogs,
or haze. The most transparent atmosphere may be filled with them;
converting days of extraordinary optical transparency into days of
equally extraordinary acoustic opacity.

The connection hitherto supposed to exist between a clear atmosphere
and the transmission of sound is therefore dissolved.

The intercepted sound is wasted by repeated reflections in the acoustic
cloud, as light is wasted by repeated reflections in an ordinary cloud.
And as from the ordinary cloud the light reflected reaches the eye, so
from the perfectly invisible acoustic cloud the reflected sound reaches
the ear.

Aërial echoes of extraordinary intensity and of long duration are thus
produced. They occur, contrary to the opinion hitherto entertained, in
the clearest air.

It is to the wafting of such acoustic clouds through the atmosphere
that the fluctuations in the sounds of our public clocks and of
church-bells are due.

The existence of these aërial echoes has been proved both by
observation and experiment. They may arise either from air-currents
differently heated, or from air-currents differently saturated with
vapor.

Rain has no sensible power to obstruct sound.

Hail has no sensible power to obstruct sound.

Snow has no sensible power to obstruct sound.

Fog has no sensible power to obstruct sound.

The air associated with fog is, as a general rule, highly homogeneous
and favorable to the transmission of sound. The notions hitherto
entertained regarding the action of fog are untenable.

Experiments on artificial showers of rain, hail, and snow, and on
artificial fogs of extraordinary density, confirm the results of
observation.

As long as the air forms a continuous medium the amount of sound
scattered by small bodies suspended in it is astonishingly small.

This is illustrated by the ease with which sound traverses layers of
calico, cambric, silk, flannel, baize, and felt. It freely passes
through all these substances in thicknesses sufficient to intercept the
light of the sun.

Through six layers of thin silk, for example, it passes with little
obstruction; it finds its way through a layer of close felt half
an inch thick, and it is not wholly intercepted by 200 layers of
cotton-net.

The atmosphere exercises a selective choice upon the waves of sound
which varies from day to day, and even from hour to hour. It is
sometimes favorable to the transmission of the longer, and at other
times favorable to the transmission of the shorter, sonorous waves.

The recognized action of the wind has been confirmed by this
investigation.




CHAPTER VIII

  Law of Vibratory Motions in Water and Air—Superposition of
  Vibrations—Interference of Sonorous Waves—Destruction of
  Sound by Sound—Combined Action of Two Sounds nearly in Unison
  with each other—Theory of Beats—Optical Illustration of the
  Principle of Interference—Augmentation of Intensity by Partial
  Extinction of Vibrations—Resultant Tones—Conditions of their
  Production—Experimental Illustrations—Difference-Tones and
  Summation-Tones—Theories of Young and Helmholtz


§ 1. _Interference of Water-Waves_

From a boat in Cowes Harbor, in moderate weather, I have often watched
the masts and ropes of the ships, as mirrored in the water. The images
of the ropes revealed the condition of the surface, indicating by
long and wide protuberances the passage of the larger rollers, and,
by smaller indentations, the ripples which crept like parasites over
the sides of the larger waves. The sea was able to accommodate itself
to the requirements of all its undulations, great and small. When the
surface was touched with an oar, or when drops were permitted to fall
from the oar into the water, there was also room for the tiny wavelets
thus generated. This carving of the surface by waves and ripples had
its limit only in my powers of observation; every wave and every ripple
asserted its right of place, and retained its individual existence,
amid the crowd of other motions which agitated the water.

The law that rules this chasing of the sea, this crossing and
intermingling of innumerable small waves, is _that the resultant
motion of every particle of water is the sum of the individual motions
imparted to it_. If a particle be acted on at the same moment by two
impulses, both of which tend to raise it, it will be lifted by a force
equal to the sum of both. If acted upon by two impulses, one of which
tends to raise it, and the other to depress it, it will be acted upon
by a force equal to the difference of both. When, therefore, the sum
of the motions is spoken of, the _algebraic sum_ is meant—the motions
which tend to raise the particle being regarded as positive, and those
which tend to depress it as negative.

When two stones are cast into smooth water, 20 or 30 feet apart,
round each stone is formed a series of expanding circular waves,
every one of which consists of a ridge and a furrow. The waves touch,
cross each other, and carve the surface into little eminences and
depressions. Where ridge coincides with ridge, we have the water raised
to a double height; where furrow coincides with furrow, we have it
depressed to a double depth; where ridge coincides with furrow, we
have the water reduced to its average level. The resultant motion of
the water at every point is, as above stated, the algebraic sum of the
motions impressed upon that point. And if, instead of two sources of
disturbance, we had ten, or a hundred, or a thousand, the consequence
would be the same; the actual result might transcend our powers of
observation, but the law above enunciated would still hold good.

Instead of the intersection of waves from two distinct centres of
disturbance, we may cause direct and reflected waves, from the same
centre, to cross each other. Many of you know the beauty of the effects
produced when light is reflected from ripples of water. When mercury
is employed the effect is more brilliant still. Here, by a proper
mode of agitation, direct and reflected waves may be caused to cross
and interlace, and by the most wonderful self-analysis to untie their
knotted scrolls. The adjacent figure (Fig. 149), which is copied from
the excellent “Wellenlehre” of the brothers Weber, will give some idea
of the beauty of these effects. It represents the chasing produced by
the intersection of direct and reflected water-waves in a circular
vessel, the point of disturbance (marked by the smallest circle in the
figure) being midway between the centre and the circumference.

[Illustration: FIG. 149.]

This power of water to accept and transmit multitudinous impulses is
shared by air, which concedes the right of space and motion to any
number of sonorous waves. The same air is competent to accept and
transmit the vibrations of a thousand instruments at the same time.
When we try to visualize the motion of that air—to present to the eye
of the mind the battling of the pulses direct and reverberated—the
imagination retires baffled from the attempt. Still, amid all the
complexity, the law above enunciated holds good, every particle of air
being animated by a resultant motion, which is the algebraic sum of all
the individual motions imparted to it. And the most wonderful thing of
all is, that the human ear, though acted on only by a cylinder of that
air, which does not exceed the thickness of a quill, can detect the
components of the motion, and, by an act of attention, can even isolate
from the aërial entanglement any particular sound.


§ 2. _Interference of Sound_

When two unisonant tuning-forks are sounded together, it is easy to see
that the forks may so vibrate that the condensations of the one shall
coincide with the condensations of the other, and the rarefactions
of the one with the rarefactions of the other. If this be the case,
the two forks will assist each other. The condensations will, in
fact, become more condensed, the rarefactions more rarefied; and as
it is upon the difference of density between the condensations and
rarefactions that _loudness_ depends, the two vibrating forks, thus
supporting each other, will produce a sound of greater intensity than
that of either of them vibrating alone.

It is, however, also easy to see that the two forks may be so related
to each other that one of them shall require a condensation at the
place where the other requires a rarefaction; that the one fork shall
urge the air-particles forward, while the other urges them backward.
If the opposing forces be equal, particles so solicited will move
neither backward nor forward, the aërial rest which corresponds to
silence being the result. Thus it is possible, by adding the sound of
one fork to that of another, to abolish the sounds of both. We have
here a phenomenon which, above all others, characterizes wave-motion.
It was this phenomenon, as manifested in optics, that led to the
undulatory theory of light, the most cogent proof of that theory being
based upon the fact that, by adding light to light, we may produce
darkness, just as we can produce silence by adding sound to sound.

[Illustration: FIG. 150.]

During the vibration of a tuning-fork the distance between the two
prongs is alternately increased and diminished. Let us call the motion
which increases the distance the _outward swing_, and that which
diminishes the distance the _inward swing_ of the fork. And let us
suppose that our two forks, A and B, Fig. 150, reach the limits of
their outward swing and their inward swing at the same moment. In this
case the _phases_ of their motion, to use the technical term, are
the same. For the sake of simplicity we will confine our attention
to the right-hand prongs, A and B, of the two forks, neglecting the
other two prongs; and now let us ask what must be the distance between
the prongs A and B, when the condensations and rarefactions of both,
indicated respectively by the dark and light shading, coincide? A
little reflection will make it clear that if the distance from B to A
be equal to the length of a whole sonorous wave, coincidence between
the two systems of waves must follow. The same would evidently occur
were the distance between A and B two wave-lengths, three wave-lengths,
four wave-lengths—in short, any number of whole wavelengths. In all
such cases we shall have coincidence of the two systems of waves, and
consequently a reinforcement of the sound of the one fork by that of
the other. Both the condensations and rarefactions between A and C are,
in this case, more pronounced than they would be if either of the forks
were suppressed.

[Illustration: FIG. 151.]

But if the prong B be only half the length of a wave behind A, what
must occur? Manifestly the rarefactions of one of the systems of waves
will then coincide with the condensations of the other system, the
air to the right of A being reduced to quiescence. This is shown in
Fig. 151, where the uniformity of shading indicates an absence both
of condensations and rarefactions. When B is two half wave-lengths
behind A, the waves, as already explained, support each other; when
they are three half wave-lengths apart, they destroy each other. Or
expressed generally, we have augmentation or destruction according as
the distance between the two prongs amounts to an even or an odd number
of semi-undulations. Precisely the same is true of the waves of light.
If through any cause one system of ethereal waves be any _even_ number
of semi-undulations behind another system, the two systems support each
other when they coalesce, and we have more light. If the one system be
any _odd_ number of semi-undulations behind the other, they oppose each
other, and a destruction of light is the result of their coalescence.

The action here referred to, both as regards sound and light is called
_Interference_.


§ 3. _Experimental Illustrations_

[Illustration: FIG. 152.]

Sir John Herschel was the first to propose to divide a stream of
sound into two branches, of different lengths, causing the branches
afterward to reunite, and interfere with each other. This idea has
been recently followed out with success by M. Quincke; and it has
been still further improved upon by M. König. The principle of these
experiments will be at once evident from Fig. 152. The tube _o f_
divides into two branches at _f_, the one branch being carried round
_n_, and the other round _m_. The two branches are caused to reunite
at _g_, and to end in a common canal, _g p_. The portion _b n_ of
the tube which slides over _a b_ can be drawn out as shown in the
figure, and thus the sound-waves can be caused to pass over different
distances in the two branches. Placing a vibrating tuning-fork at _o_,
and the ear at _p_, when the two branches are of the same length, the
waves through both reach the ear together, and the sound of the fork
is heard. Drawing _n b_ out, a point is at length obtained where the
sound of the fork is extinguished. This occurs when the distance _a
b_ is one-fourth of a wave-length; or, in other words, when the whole
right-hand branch is half a wave-length longer than the left-hand one.
Drawing _b n_ still further out, the sound is again heard; and when
twice the distance _a b_ amounts to a whole wave-length, it reaches a
maximum. Thus, according as the difference of both branches amounts to
half a wave-length, or to a whole wave-length, we have reinforcement
or destruction of the two series of sonorous waves. In practice, the
tube _o f_ ought to be prolonged until the direct sound of the fork is
unheard, the attention of the ear being then wholly concentrated on the
sounds that reach it through the tube.

It is quite plain that the wave-length of any simple tone may be
readily found by this instrument. It is only necessary to ascertain the
difference of path which produces complete interference. Twice this
difference is the wave-length; and if the rate of vibration be at the
same time known, we can immediately calculate the velocity of sound in
air.

Each of the two forks now before you executes exactly 256 vibrations
in a second. Sounded together, they are in unison. Loading one of
them with a bit of wax, it vibrates a little more slowly than its
neighbor. The wax, say, reduces the number of vibrations to 255
in a second; how must their waves affect each other? If they start
at the same moment, condensation coinciding with condensation, and
rarefaction with rarefaction, it is quite manifest that this state of
things cannot continue. At the 128th vibration their phases are in
complete opposition, one of them having gained half a vibration on
the other. Here the one fork generates a condensation where the other
generates a rarefaction; and the consequence is, that the two forks,
at this particular point, completely neutralize each other. From this
point onward, however, the forks support each other more and more,
until, at the end of a second, when the one has completed its 255th,
and the other its 256th vibration, condensation again coincides with
condensation, and rarefaction with rarefaction, the full effect of both
sounds being produced upon the ear.

It is quite manifest that under these circumstances we cannot have the
continuous flow of perfect unison. We have, on the contrary, alternate
reinforcements and diminutions of the sound. We obtain, in fact, the
effect known to musicians by the name of _beats_, which, as here
explained, are a result of interference.

I now load this fork still more heavily, by attaching a fourpenny-piece
to the wax; the coincidences and interferences follow each other more
rapidly than before; we have a quicker succession of beats. In our
last experiment, the one fork accomplished one vibration more than the
other in a second, and we had a single beat in the same time. In the
present case, one fork vibrates 250 times, while the other vibrates
256 times in a second, and the number of beats per second is 6. A
little reflection will make it plain that in the interval required
by the one fork to execute one vibration more than the other, a beat
must occur; and inasmuch as, in the case now before us, there are
six such intervals in a second, there must be six beats in the same
time. In short, _the number of beats per second is always equal to the
difference between the two rates of vibration_.


§ 4. _Interference of Waves from Organ-pipes_

[Illustration: FIG. 153.]

Beats may be produced by all sonorous bodies. These two tall
organ-pipes, for example, when sounded together, give powerful
beats, one of them being slightly longer than the other. Here are
two other pipes, which are now in perfect unison, being exactly of
the same length. But it is only necessary to bring the finger near
the embouchure of one of the pipes, Fig. 153, to lower its rate of
vibration, and produce loud and rapid beats. The placing of the
hand over the open top of one of the pipes also lowers its rate of
vibration, and produces beats, which follow each other with augmented
rapidity as the top of the pipe is closed more and more. By a stronger
blast the first two harmonics of the pipes are brought out. These
higher notes also interfere, and you have these quicker beats.

[Illustration: FIG. 154.]

No more beautiful illustration of this phenomenon can be adduced than
that furnished by two sounding-flames. Two such flames are now before
you, the tube surrounding one of them being provided with a telescopic
slider, Fig. 154. There are, at present, no beats, because the tubes
are not sufficiently near unison. I gradually lengthen the shorter tube
by raising its slider. Rapid beats are now heard; now they are slower;
now slower still; and now both flames sing together in perfect unison.
Continuing the upward motion of the slider, I make the tube too long;
the beats begin again, and quicken, until finally their sequence is
so rapid as to appeal only as roughness to the ear. The flames, you
observe, dance within their tubes in time to the beats. As already
stated, these beats cause a silent flame within a tube to quiver when
the voice is thrown to a proper pitch, and when the position of the
flame is rightly chosen, the beats set it singing. With the flames of
large rose-burners, and with tin tubes from 3 to 9 feet long, we obtain
beats of exceeding power.

[Illustration: FIG. 155.]

You have just heard the beats produced by two tall organ-pipes nearly
in unison with each other. Two other pipes are now mounted on our
wind-chest, Fig. 155, each of which, however, is provided at its centre
with a membrane intended to act upon a flame.[70] Two small tubes lead
from the spaces closed by the membranes, and unite afterward, the
membranes of both the organ-pipes being thus connected with the same
flame. By means of the sliders, _s s′_, near the summits of the pipes,
they are either brought into unison or thrown out of it at pleasure.
They are not at present in unison, and the beats they produce follow
each other with great rapidity. The flame connected with the central
membranes dances in time to the beats. When brought nearer to unison,
the beats are slower, and the flame at successive intervals withdraws
its light and appears to exhale it. A process which reminds you of the
inspiration and expiration of the breath is thus carried on by the
flame. If the mirror, M, be now turned, the flame produces a luminous
band—continuous at certain places, but for the most part broken into
distinct images of the flame. The continuous parts correspond to the
intervals of interference where the two sets of vibrations abolish each
other.

Instead of permitting both pipes to act upon the same flame, we may
associate a flame with each of them. The deportment of the flames is
then very instructive. Imagine both flames to be in the same vertical
line, the one of them being exactly under the other. Bringing the
pipes into unison, and turning the mirror, we resolve each flame into
a chain of images, but we notice that the images of the one occupy the
spaces _between_ the images of the other. The periods of extinction of
the one flame, therefore, correspond to the periods of kindling of the
other. The experiment proves that, when two unisonant pipes are placed
thus close to each other, their vibrations are in opposite phases. The
consequence of this is, that the two sets of vibrations permanently
neutralize each other, so that at a little distance from the pipes you
fail to hear the fundamental tone of either. For this reason we cannot,
with any advantage, place close to each other in an organ several pipes
of the same pitch.


§ 5. _Lissajous’s Illustration of Beats of Two Tuning-forks_

[Illustration: FIG. 156.]

In the case of beats, the amplitude of the oscillating air reaches
a maximum and a minimum periodically. By the beautiful method of M.
Lissajous we can illustrate optically this alternate augmentation and
diminution of amplitude. Placing a large tuning-fork, T′, Fig. 156,
in front of the lamp L, a luminous beam is received upon the mirror
attached to the fork. This is reflected back to the mirror of a second
fork, T, and by it thrown on to the screen, where it forms a luminous
disk. When the bow is drawn over the fork T′, the beam, as in the
experiments described in the second chapter, is tilted up and down, the
disk upon the screen stretching to a luminous band three feet long.
If, in drawing the bow over this second fork, the vibrations of both
coincide in phase, the band will be lengthened; if the phases are in
opposition, total or partial neutralization of the one fork by the
other will be the result. It so happens that in the present instance
the second fork adds something to the action of the first, the band
of light being now four feet long. These forks have been tuned as
perfectly as possible. Each of them executes exactly 64 vibrations in
a second; the initial relation of their phases remains, therefore,
constant, and hence you notice a gradual shortening of the luminous
band, like that observed during the subsidence of the vibrations of a
single fork. The band at length dwindles to the original disk, which
remains motionless upon the screen.

By attaching, with wax, a threepenny-piece to the prong of one of
these forks, its rate of vibration is lowered. The phases of the
two forks cannot now retain a constant relation to each other. One
fork incessantly gains upon the other, and the consequence is that
sometimes the phases of both coincide, and at other times they are in
opposition. Observe the result. At the present moment the two forks
conspire, and we have a luminous band four feet long upon the screen.
This slowly contracts, drawing itself up to a mere disk; but the action
halts here only during the moment of opposition. That passed, the
forks begin again to assist each other, and the disk once more slowly
stretches into a band. The action here is very slow; but it may be
quickened by attaching a sixpence to the loaded fork. The band of light
now stretches and contracts in perfect rhythm. The action, rendered
thus optically evident, is impressed upon the air of this room; its
particles alternately vibrate and come to rest, and, as a consequence,
beats are heard in synchronism with the changes of the figure upon the
screen.

The time which elapses from maximum to maximum, or from minimum to
minimum, is that required for the one fork to perform one vibration
more than the other. At present this time is about two seconds. In two
seconds, therefore, one beat occurs. When we augment the dissonance
by increasing the load, the rhythmic lengthening and shortening of
the band is more rapid, while the intermittent hum of the forks is
more audible. There are now six elongations and shortenings in the
interval taken up a moment ago by one; the beats at the same time being
heard at the rate of three a second. By loading the forks still more,
the alternations may be caused to succeed each other so rapidly that
they can no longer be followed by the eye, while the beats, at the
same time, cease to be individually distinct, and appeal as a kind of
roughness to the ear.

[Illustration: FIG. 157.]

In the experiments with a single tuning-fork, already described (Fig.
22, Chapter II.), the beam reflected from the fork was received on
a looking-glass, and, by turning the glass, the band of light on
the screen was caused to stretch out into a long wavy line. It was
explained at the time that the loudness of the sound depended on the
depth of the indentations. Hence, if the band of light of varying
length now before us on the screen be drawn out in a sinuous line,
the indentations ought to be at some places deep, while at others
they ought to vanish altogether. This is the case. By a little tact
the mirror of the fork T (Fig. 156) is caused to turn through a small
angle, a sinuous line composed of swellings and contractions (Fig. 157)
being drawn upon the screen. The swellings correspond to the periods
of sound, and the contractions to those of silence.[71]

Two vibrating bodies, then, each of which separately produces a musical
sound, can, when acting together, neutralize each other. Hence, by
quenching the vibrations of one of them, we may give sonorous effect to
the other. It often happens, for instance, that when two tuning-forks,
on their resonant cases, are vibrating in unison, the stoppage of one
of them is accompanied by an augmentation of the sound. This point
may be further illustrated by the vibrating bell, already described
(Fig. 78, Chapter IV.) Placing its resonant tube in front of one of
its nodes, a sound is heard, but nothing like what is heard when the
tube is opposed to a ventral segment. The reason of this is that the
vibrations of a bell on the opposite sides of a nodal line are in
opposite directions, and they therefore interfere with each other.
By introducing a glass plate between the bell and the tube, the
vibrations on one side of the nodal line may be intercepted; an instant
augmentation of the sound is the consequence.


§ 6. _Interference of Waves from a Vibrating Disk. Hopkins’s and
Lissajous’s Illustrations_

In a vibrating disk every two adjacent sectors move at the same time
in opposite directions. When the one sector rises the other falls, the
nodal line marking the limit between them. Hence, at the moment when
any sector produces a condensation in the air above it, the adjacent
sector produces a rarefaction in the same air. A partial destruction
of the sound of one sector by the other is the result. You will now
understand the instrument by which the late William Hopkins illustrated
the principle of interference. The tube A B, Fig. 158, divides at B
into two branches. The end A of the tube is closed by a membrane.
Scattering sand upon this membrane, and holding the ends of the
branches over _adjacent_ sectors of a vibrating disk, no motion (or, at
least, an extremely feeble motion) of the sand is perceived. Placing
the ends of the two branches over _alternate_ sectors of the disk, the
sand is tossed from the membrane, proving that in this case we have
coincidence of vibration on the part of the two sectors.

[Illustration: FIG. 158.]

[Illustration: FIG. 159.]

We are now prepared for a very instructive experiment, which we owe to
M. Lissajous. Drawing a bow over the edge of a brass disk, I divide it
into six vibrating sectors. When the palm of the hand is brought over
any one of them, the sound, instead of being diminished, is augmented.
When two hands are placed over two _adjacent_ sectors, you notice
no increase of the sound; but when they are placed over _alternate_
sectors, as in Fig. 159, a striking augmentation of the sound is
the consequence. By simply lowering and raising the hands, marked
variations of intensity are produced. By the approach of the hands
the vibrations of the two sectors are intercepted; their interference
right and left being thus abolished, the remaining sectors sound more
loudly. Passing the single hand to and fro along the surface, you also
hear a rise and fall of the sound. It rises when the hand is over a
vibrating sector; it falls when the hand is over a nodal line. Thus,
by sacrificing a portion of the vibrations, we make the residue more
effectual. Experiments similar to these may be made with light and
radiant heat. If of two beams of the former, which destroy each other
by interference, one be removed, light takes the place of darkness;
and if of two interfering beams of the latter one be intercepted, heat
takes the place of cold.


§ 7. _Quenching the Sound of one Prong of a Tuning-fork by that of the
other_

You have remarked the almost total absence of sound on the part of a
vibrating tuning-fork when held free in the hand. The feebleness of the
fork as a sounding body arises in part from interference. The prongs
always vibrate in opposite directions, one producing a condensation
where the other produces a rarefaction, a destruction of sound being
the consequence. By simply passing a pasteboard tube over one of the
prongs of the fork, its vibrations are in part intercepted, and an
augmentation of the sound is the result. The single prong is thus
proved to be more effectual than the two prongs. There are positions in
which the destruction of the sound of one prong by that of the other
is _total_. These positions are easily found by striking the fork and
turning it round before the ear. When the back of the prong is parallel
to the ear, the sound is heard; when the side surfaces of both prongs
are parallel to the ear, the sound is also heard; but when the _corner_
of a prong is carefully presented to the ear, the sound is utterly
destroyed. During one complete rotation of the fork we find four
positions where the sound is thus obliterated.

[Illustration: FIG. 160.]

Let _s s_ (Fig. 160) represent the two ends of the tuning-fork, looked
down upon as it stands upright. When the ear is placed at _a_ or _b_,
or at _c_ or _d_, the sound is heard. Along the four dotted lines,
on the contrary, the waves generated by the two prongs completely
neutralize each other, and nothing is there heard. These lines have
been proved by Weber to be hyperbolic curves; and this must be their
character according to the principle of interference.

This remarkable case of interference, which was first noticed by Dr.
Thomas Young, and thoroughly investigated by the brothers Weber, may
be rendered audible by means of resonance. Bringing a vibrating fork
over a jar which resounds to it, and causing the fork to rotate slowly,
in four positions we have a loud resonance; in four others absolute
silence, alternate risings and fallings of the sound accompanying
the fork’s rotation. While the fork is over the jar with its corner
downward and the sound entirely extinguished, let a pasteboard tube
be passed over one of its prongs, as in Fig. 161, a loud resonance
announces the withdrawal of the vibrations of that prong. To obtain
this effect, the fork must be held over the centre of the jar, so that
the air shall be symmetrically distributed on both sides of it. Moving
the fork from the centre toward one of the sides, without altering its
inclination in the least, we obtain a forcible sound. Interference,
however, is also possible near the side of the jar. Holding the fork,
not with its corner downward, but with both its prongs in the same
horizontal plane, a position is soon found near the side of the jar
where the sound is extinguished. In passing completely from side to
side over the mouth of the jar, two such places of interference are
discoverable.

[Illustration: FIG. 161.]

[Illustration: FIG. 162.]

A variety of experiments will suggest themselves to the reflecting
mind, by which the effect of interference may be illustrated. It is
easy, for example, to find a jar which resounds to a vibrating plate.
Such a jar, placed over a vibrating segment of the plate, produces a
powerful resonance. Placed over a nodal line, the resonance is entirely
absent; but if a piece of pasteboard be interposed between the jar and
plate, so as to cut off the vibrations on one side of the nodal line,
the jar instantly resounds to the vibrations of the other. Again,
holding two forks, which vibrate with the same rapidity, over two
resonant jars, the sound of both flows forth in unison. When a bit of
wax is attached to one of the forks, powerful beats are heard. Removing
the wax, the unison is restored. When one of these unisonant forks is
placed in the flame of a spirit-lamp its elasticity is changed, and it
produces long loud beats with its unwarmed fellow.[72] If while one
of the forks is sounding on its resonant case the other be excited
and brought near the mouth of the case, as in Fig. 162, loud beats
declare the absence of unison. Dividing a jar by a vertical diaphragm,
and bringing one of the forks over one of its halves, and the other
fork over the other; the two semi-cylinders of air produce beats by
their interference. But the diaphragm is not necessary; on removing
it, the beats continue as before, one-half of the same column of air
interfering with the other.[73]

The intermittent sound of certain bells, heard more especially when
their tones are subsiding, is an effect of interference. The bell,
through lack of symmetry, as explained in the fourth chapter, vibrates
in one direction a little more rapidly than in the other, and beats
are the consequence of the coalescence of the two different rates of
vibration.


RESULTANT TONES

We have now to turn from this question of interference to the
consideration of a new class of musical sounds, of which the beats were
long considered to be the progenitors. The sounds here referred to
require for their production the union of two distinct musical tones.
Where such union is effected, under the proper conditions, _resultant
tones_ are generated, which are quite distinct from the primaries
concerned in their production. They were discovered, in 1745, by a
German organist named Sorge, but the publication of the fact attracted
little attention. They were discovered independently, in 1754, by the
celebrated Italian violinist Tartini, and after him have been called
Tartini’s tones.

To produce them it is desirable, if not necessary, to have the two
primary tones of considerable intensity. Helmholtz prefers the siren
to all other means of exciting them, and with this instrument they
are very readily obtained. It requires some attention at first, on
the part of the listener, to single out the resultant tone from the
general mass of sound; but, with a little practice, this is readily
accomplished; and though the unpracticed ear may fail, in the first
instance, thus to analyze the sound, the clang-tint is influenced in an
unmistakable manner by the admixture of resultant tones. I set Dove’s
siren in rotation, and open two series of holes at the same time; with
the utmost strain of attention, I am as yet unable to hear the least
symptom of a resultant tone. Urging the instrument to greater rapidity,
a dull, low droning mingles with the two primary sounds. Raising the
speed of rotation, the low, resultant tone rises rapidly in pitch, and
now, to those who stand close to the instrument, it is very audible.
The two series of holes here open number 8 and 12 respectively. The
resultant tone is in this case an octave below the deepest of the two
primaries. Opening two other series of orifices, numbering 12 and
16 respectively, the resultant tone is quite audible. Its rate of
vibration is one-third of the rate of the deepest of the two primaries.
In all cases, _the resultant tone is that which corresponds to a rate
of vibration equal to the difference of the rates of the two primaries_.

The resultant tone here spoken of is that actually heard in the
experiment. But with finer methods of experiment other resultant tones
are proved to exist. Those on which we have now fixed our attention
are, however, the most important. They are called _difference-tones_ by
Helmholtz, in consequence of the law just mentioned.

To bring these resultant tones audibly forth, the primaries must,
as already stated, be forcible. When they are feeble the resultants
are unheard. I am acquainted with no method of exciting these tones
more simple and effectual than a pair of suitable singing-flames.
Two such flames may be caused to emit powerful notes—self-created,
self-sustained, and requiring no muscular effort on the part of the
observer to keep them going. Here are two of them. The length of the
shorter of the two tubes surrounding these flames is 10-3/8 inches,
that of the other is 11·4 inches. I hearken to the sound, and in the
midst of the shrillness detect a very deep resultant tone. The reason
of its depth is manifest: the two tubes being so nearly alike in
length, the difference between their vibrations is small, and the note
corresponding to this difference, therefore, low in pitch. Lengthening
one of the tubes by means of its slider, the resultant tone rises
gradually, and now it swells surprisingly. When the tube is shortened
the resultant tone falls, and thus, by alternately raising and lowering
the slider, the resultant tone is caused to rise and sink in accordance
with the law which makes the number of its vibrations the difference
between the number of its two primaries.

We can determine, with ease, the actual number of vibrations
corresponding to any one of those resultant tones. The sound of
the flame is that of the open tube which surrounds it, and we have
already learned (Chapter III.) that the length of such a tube is half
that of the sonorous wave it produces. The wave-length, therefore,
corresponding to our 10-3/8-inch tube is 20-3/4 inches. The velocity
of sound in air of the present temperature is 1,120 feet a second.
Bringing these feet to inches, and dividing by 20-3/4, we find the
number of vibrations corresponding to a length of 10-3/8 inches to be
648 per second.

But it must not be forgotten here that the air in which the vibrations
are actually executed is much more elastic than the surrounding
air. The flame heats the air of the tube, and the vibrations must,
therefore, be executed more rapidly than they would be in an ordinary
organ-pipe of the same length. To determine the actual number of
vibrations, we must fall back upon our siren; and with this instrument
it is found that the air within the 10-3/8 inch tube executes 717
vibrations in a second. The difference of 69 vibrations a second is
due to the heating of the aërial column. Carbonic acid and aqueous
vapor are, moreover, the product of the flame’s combustion, and their
presence must also affect the rapidity of the vibration.

Determining in the same way the rate of vibration of the 11·4-inch
tube, we find it to be 667 per second; the difference between
this number and 717 is 50, which expresses the rate of vibration
corresponding to the first deep resultant tone.

But this number does not mark the limit of audibility. Permitting the
11·4-inch tube to remain as before, and lengthening its neighbor, the
resultant tone sinks near the limit of hearing. When the shorter tube
measures 11 inches, the deep sound of the resultant tone is still
heard. The number of vibrations per second executed in this 11-inch
tube is 700. We have already found the number executed in the 11·4-inch
tube to be 667; hence 700-667=33, which is the number of vibrations
corresponding to the resultant tone now plainly heard when the
attention is converged upon it. We here come very near the limit which
Helmholtz has fixed as that of musical audibility. Combining the sound
of a tube 17-3/8 inches in length with that of a 10-3/8-inch tube, we
obtain a resultant tone of higher pitch than any previously heard. Now
the actual number of vibrations executed in the longer tube is 459; and
we have already found the vibrations of our 10-3/8-inch tube to be 717;
hence 717-459=258, which is the number corresponding to the resultant
tone now audible. This note is almost exactly that of one of our series
of tuning-forks, which vibrates 256 times in a second.

And now we will avail ourselves of a beautiful check which this result
suggests to us. The well-known fork which vibrates at the rate just
mentioned is here, mounted on its case, and I touch it with the bow so
lightly that the sound alone could hardly be heard; but it instantly
coalesces with the resultant tone, and the beats produced by their
combination are clearly audible. By loading the fork, and thus altering
its pitch, or by drawing up the paper slider, and thus altering the
pitch of the flame, the rate of these beats can be altered, exactly
as when we compare two primary tones together. By slightly varying
the size of the flame, the same effect is produced. We cannot fail to
observe how beautifully these results harmonize with each other.

Standing midway between the siren and a shrill singing-flame, and
gradually raising the pitch of the siren, the resultant tone soon makes
itself heard, sometimes swelling out with extraordinary power. When a
pitch-pipe is blown near the flame, the resultant tone is also heard,
seeming, in this case, to originate in the ear itself, or rather in the
brain. By gradually drawing out the stopper of the pipe, the pitch of
the resultant tone is caused to vary in accordance with the law already
enunciated.

The resultant tones produced by the combination of the ordinary
harmonic intervals[74] are given in the following table:

  Interval        Ratio of       Difference       The resultant tone is
                  vibrations                      deeper than the lowest
                                                  primary tone by

  Octave            1 : 2            1             0
  Fifth             2 : 3            1             an octave
  Fourth            3 : 4            1             a twelfth
  Major third       4 : 5            1             two octaves
  Minor third       5 : 6            1             two octaves and a
                                                     major third
  Major sixth       3 : 5            2             a fifth
  Minor sixth       5 : 8            3             major sixth

The celebrated Thomas Young thought that these resultant tones were
due to the coalescence of rapid beats, which linked themselves
together like the periodic impulses of an ordinary musical note. This
explanation harmonized with the fact that the number of the beats,
like that of the vibrations of the resultant tone, is equal to the
difference between the two sets of vibrations. This explanation,
however, is insufficient. The beats tell more forcibly upon the ear
than any continuous sound. They can be plainly heard when each of the
two sounds that produce them has ceased to be audible. This depends
in part upon the sense of hearing, but it also depends upon the fact
that when two notes of the same intensity produce beats, the amplitude
of the vibrating air-particles is at times destroyed, and at times
doubled. But by doubling the amplitude we quadruple the intensity of
the sound. Hence, when two notes of the same intensity produce beats,
_the sound incessantly varies between silence and a tone of four times
the intensity of either of the interfering ones_.

If, therefore, the resultant tones were due to the beats of their
primaries, they ought to be heard, even when the primaries are feeble.
But they are not heard under these circumstances. When several sounds
traverse the same air, each particular sound passes through the air as
if it alone were present, each particular element of a composite sound
asserting its own individuality. Now, this is in strictness true only
when the amplitudes of the oscillating particles are infinitely small.
Guided by pure reasoning, the mathematician arrives at this result.
The law is also practically true when the disturbances are _extremely_
small; but it is _not_ true after they have passed a certain limit.
Vibrations which produce a large amount of disturbance give birth to
secondary waves, which appeal to the ear as resultant tones. This
has been proved by Helmholtz, and, having proved this, he inferred
further that there are also resultant tones formed by the _sum_ of
the primaries, as well as by their difference. He thus discovered the
_summation-tones_ before he had heard them; and bringing his result to
the test of experiment, he found that these tones had a real physical
existence. They are not at all to be explained by Young’s theory.

Another consequence of this departure from the law of superposition is,
that a single sounding body, which disturbs the air beyond the limits
of the law of superposition, also produces secondary waves, which
correspond to the harmonic tones of the vibrating body. For example,
the rate of vibration of the first overtone of a tuning-fork, as stated
in the fourth chapter, is 6-1/4 times the rate of the fundamental
tone. But Helmholtz shows that a tuning-fork, not excited by a bow, but
vigorously struck against a pad, emits the _octave_ of its fundamental
note, this octave being due to the secondary waves set up when the
limits of the law of superposition have been exceeded.

These considerations make it probably evident to you that a coalescence
of musical sounds is a far more complicated dynamical condition than
you have hitherto supposed it to be. In the music of an orchestra,
not only have we the fundamental tones of every pipe and of every
string, but we have the overtones of each, sometimes audible as far
as the sixteenth in the series. We have also resultant tones; both
difference-tones and summation-tones; all trembling through the
same air, all knocking at the self-same tympanic membrane. We have
fundamental tone interfering with fundamental tone; overtone with
overtone; resultant tone with resultant tone. And, besides this,
we have the members of each class interfering with the members of
every other class. The imagination retires baffled from any attempt
to realize the physical condition of the atmosphere through which
these sounds are passing. And, as we shall immediately learn, the
aim of music, through the centuries during which it has ministered
to the pleasure of man, has been to arrange matters empirically, so
that the ear shall not suffer from the discordance produced by this
multitudinous interference. The musicians engaged in this work knew
nothing of the physical facts and principles involved in their efforts;
they knew no more about it than the inventors of gunpowder knew about
the law of atomic proportions. They tried and tried till they obtained
a satisfactory result; and now, when the scientific mind is brought to
bear upon the subject, order is seen rising through the confusion, and
the results of pure empiricism are found to be in harmony with natural
law.


SUMMARY OF CHAPTER VIII

When several systems of waves proceeding from distinct centres of
disturbance pass through water or air, the motion of every particle is
the algebraic sum of the several motions impressed upon it.

In the case of water, when the crests of one system of waves coincide
with the crests of another system, higher waves will be the result of
the coalescence of the two systems. But when the crests of one system
coincide with the sinuses, or furrows, of the other system, the two
systems, in whole or in part, destroy each other.

This coalescence and destruction of two systems of waves is called
_interference_.

Similar remarks apply to sonorous waves. If in two systems of sonorous
waves condensation coincides with condensation, and rarefaction with
rarefaction, the sound produced by such coincidence is louder than that
produced by either system taken singly. But if the condensations of the
one system coincide with the rarefactions of the other, a destruction,
total or partial, of both systems is the consequence.

Thus, when two organ-pipes of the same pitch are placed near each other
on the same wind-chest and thrown into vibration, they so influence
each other that as the air enters the embouchure of the one it quits
that of the other. At the moment, therefore, the one pipe produces a
condensation the other produces a rarefaction. The sounds of two such
pipes mutually destroy each other.

When two musical sounds of nearly the same pitch are sounded together
the flow of the sound is disturbed by _beats_.

These beats are due to the alternate coincidence and interference
of the two systems of sonorous waves. If the two sounds be of the
same intensity, their coincidence produces a sound of four times the
intensity of either; while their opposition produces absolute silence.

The effect, then, of two such sounds, in combination, is a series of
shocks, which we have called “beats,” separated from each other by a
series of “pauses.”

The rate at which the beats succeed each other is equal to the
difference between the two rates of vibration.

When a bell or disk sounds, the vibrations on opposite sides of the
same nodal line partially neutralize each other; when a tuning-fork
sounds, the vibrations of its two prongs in part neutralize each other.
By cutting off a portion of the vibrations in these cases the sound may
be intensified.

When a luminous beam, reflected on to a screen from two tuning-forks
producing beats, is acted upon by the vibrations of both, the
intermittence of the sound is announced by the alternate lengthening
and shortening of the band of light upon the screen.

The law of the superposition of vibrations above enunciated is
strictly true only when the amplitudes are exceedingly small. When the
disturbance of the air by a sounding body is so violent that the law no
longer holds good, secondary waves are formed, which correspond to the
harmonic tones of the sounding body.

When two tones are rendered so intense as to exceed the limits of
the law of superposition, their secondary waves combine to produce
_resultant tones_.

Resultant tones are of two kinds; the one class corresponding to rates
of vibration equal to the difference of the rates of the two primaries;
the other class corresponding to rates of vibration equal to the sum of
the two primaries. The former are called _difference-tones_, the latter
_summation-tones_.




CHAPTER IX

  Combination of Musical Sounds—The smaller the Two Numbers
  which express the Ratio of their Rates of Vibration, the
  more perfect is the Harmony of Two Sounds—Notions of
  the Pythagoreans regarding Musical Consonance—Euler’s
  Theory of Consonance—Theory of Helmholtz—Dissonance
  due to Beats—Interference of Primary Tones and of
  Over-tones—Mechanism of Hearing—Schultze’s Bristles—The
  Otoliths—Corti’s Fibres—Graphic Representation of
  Consonance and Dissonance—Musical Chords—The Diatonic
  Scale—Optical Illustration of Musical Intervals—Lissajous’s
  Figures—Sympathetic Vibrations—Various Modes of illustrating
  the Composition of Vibrations


§ 1. _The Facts of Musical Consonance_

The subject of this day’s lecture has two sides, a physical and
an æsthetical. We have to-day to study the question of musical
consonance—to examine musical sounds in definite combination with each
other, and to unfold the reason why some combinations are pleasant and
others unpleasant to the ear.

Pythagoras made the first step toward the physical explanation of the
musical intervals. This great philosopher stretched a string, and then
divided it into three equal parts. At one of its points of division he
fixed it firmly, thus converting it into two, one of which was twice
the length of the other. He sounded the two sections of the string
simultaneously, and found the note emitted by the short section to be
the higher octave of that emitted by the long one. He then divided
his string into two parts, bearing to each other the proportion of
2:3, and found that the notes were separated by an interval of a
fifth. Thus, dividing his string at different points, Pythagoras
found the so-called consonant intervals in music to correspond with
certain lengths of his string; and he made the extremely important
discovery that _the simpler the ratio of the two parts into which
the string was divided, the more perfect was the harmony of the two
sounds_. Pythagoras went no further than this, and it remained for
the investigators of a subsequent age to show that the strings act in
this way in virtue of the relation of their lengths to the number of
their vibrations. Why simplicity should give pleasure remained long an
enigma, the only pretence of a solution being that of Euler, which,
briefly expressed, is, that the human soul takes a constitutional
delight in simple calculations.

The double siren (Fig. 163) enables us to obtain a great variety of
combinations of musical sounds. And this instrument possesses over all
others the advantage that, by simply counting the number of orifices
corresponding respectively to any two notes, we obtain immediately
the ratio of their rates of vibration. Before proceeding to these
combinations I will enter a little more fully into the action of the
double siren than has been hitherto deemed necessary or desirable.

[Illustration: FIG. 163.]

The instrument, as already stated, consists of two of Dove’s sirens, C′
and C, connected by a common axis, the upper one being turned upside
down. Each siren is provided with four series of apertures, numbering
as follows:

                    Upper siren             Lower siren
                Number of apertures     Number of apertures

  1st Series            16                      18
  2d Series             15                      12
  3d Series             12                      10
  4th Series             9                       8

The number 12, it will be observed, is common to both sirens. I
open the two series of 12 orifices each, and urge air through the
instrument; both sounds flow together in perfect unison; the unison
being maintained, however the pitch may be exalted. We have, however,
already learned (Chapter II.) that by turning the handle of the upper
siren the orifices in its wind-chest C′ are caused either to meet those
of its rotating disk, or to retreat from them, the pitch of the upper
siren being thereby raised or lowered. This change of pitch instantly
announces itself by beats. The more rapidly the handle is turned, the
more is the tone of the upper siren raised above or depressed below
that of the lower one, and, as a consequence, the more rapid are the
beats.

Now the rotation of the handle is so related to the rotation of the
wind-chest C′ that when the handle turns through half a right angle
the wind-chest turns through one-sixth of a right angle, or through
the one-twenty-fourth of its whole circumference. But in the case
now before us, where the circle is perforated by 12 orifices, the
rotation through one-twenty-fourth of its circumference causes the
apertures of the upper wind-chest to be closed at the precise moments
when those of the lower one are opened, and _vice versa_. It is plain,
therefore, that the intervals between the puffs of the lower siren,
which correspond to the rarefactions of its sonorous waves, are here
filled by the puffs, or condensations, of the upper siren. In fact,
the condensations of the one coincide with the rarefactions of the
other, and the absolute extinction of the sounds of both sirens is the
consequence.

I may seem to you to have exceeded the truth here; for when the handle
is placed in the position which corresponds to absolute extinction,
you still hear a distinct sound. And, when the handle is turned
continuously, though alternate swellings and sinkings of the tone
occur, the sinkings by no means amount to absolute silence. The reason
is this: The sound of the siren is a highly composite one. By the
suddenness and violence of its shocks, not only does it produce waves
corresponding to the number of its orifices, but the aërial disturbance
breaks up into secondary waves, which associate themselves with the
primary waves of the instrument, exactly as the harmonics of a string,
or of an open organ-pipe, mix with their fundamental tone. When the
siren sounds, therefore, it emits, besides the fundamental tone, its
octave, its twelfth, its double octave, and so on. That is to say,
it breaks the air up into vibrations which have twice, three times,
four times, etc., the rapidity of the fundamental one. Now, by turning
the upper siren through one-twenty-fourth of its circumference, we
extinguish utterly the fundamental tone. But we do not extinguish its
octave.[75] Hence, when the handle is in the position which corresponds
to the extinction of the fundamental tone, instead of silence we have
the full first harmonic of the instrument.

Helmholtz has surrounded both his upper and his lower siren with
circular brass boxes, B, B′, each composed of two halves, which can
be readily separated (one-half of each box is removed in the figure).
These boxes exalt by their resonance the fundamental tone of the
instrument, and enable us to follow its variations much more easily
than if it were not thus reinforced. It requires a certain rapidity of
rotation to reach the maximum resonance of the brass boxes; but when
this speed is attained, the fundamental tone swells out with greatly
augmented force, and, if the handle be then turned, the beats succeed
each other with extraordinary power.

Still, as already stated, the pauses between the beats of the
fundamental tone are not intervals of absolute silence, but are
filled by the higher octave; and this renders caution necessary when
the instrument is employed to determine rates of vibration. It is
not without reason that I say so. Wishing to determine the rate of
vibration of a small singing-flame, I once placed a siren at some
distance from it, sounded the instrument, and after a little time
observed the flame dancing in synchronism with audible beats. I took
it for granted that unison was nearly attained, and, under this
assumption, determined the rate of vibration. The number obtained was
surprisingly low—indeed not more than half what it ought to be. What
was the reason? Simply this: I was dealing, not with the fundamental
tone of the siren, but with its higher octave. This octave and the
flame produced beats by their coalescence; and hence the counter of
the instrument, which recorded the rate, not of the octave, but of
the fundamental, gave a number which was only half the true one. The
fundamental tone was afterward raised to unison with the flame. On
approaching unison beats were again heard, and the jumping of the flame
proceeded with an energy greater than that observed in the case of the
octave. The counter of the instrument then recorded the accurate rate
of the flame’s vibration.

The tones first heard in the case of the siren are always overtones.
These attain sonorous continuity sooner than the fundamental, flowing
as smooth musical sounds while the fundamental tone is still in
a state of intermittence. The siren is, however, so delicately
constructed that a rate of rotation which raises the fundamental tone
above its fellows is almost immediately attained. And if we seek, by
making the blast feeble, to keep the speed of rotation low, it is at
the expense of intensity. Hence the desirability, if we wish to examine
the overtones, of devising some means by which a strong blast and slow
rotation shall be possible.

Helmholtz caused a spring to press as a light brake against the disk
of the siren. Thus raising by slow degrees the speed of rotation, he
was able deliberately to notice the predominance of the overtones at
the commencement, and the final triumph of the fundamental tone. He did
not trust to the direct observation of pitch, but determined the tone
by the number of beats corresponding to one revolution of the handle
of the upper siren. Supposing 12 orifices to be opened above and 12
below, the motion of the handle through 45° produces interference, and
extinguishes the fundamental tone. The coincidences of that tone occur
at the end of every rotation of 90°. Hence, for the fundamental tone,
there must be _four_ beats for every complete rotation of the handle.
Now Helmholtz, when he made the arrangement just described, found that
the first beats numbered, not 4, but 12, for every revolution. They
were, in fact, the beats, not of the fundamental tone, not even of the
first overtone, but of the second overtone, whose rate of vibration is
three times that of the fundamental. These beats continued as long as
the number of air-shocks did not exceed 30 or 40 per second. When the
shocks were between 40 and 80 per second, the beats fell from 12 to
8 for every revolution of the handle. Within this interval the first
overtone, or the octave of the fundamental tone, was the most powerful,
and made the beats its own. Not until the impulses exceeded 80 per
second did the beats sink to 4 per revolution. In other words, not
until the speed of rotation had passed this limit was the fundamental
tone able to assert its superiority over its companions.

This premised, we will combine the tones in definite order, while the
cultivated ears here present shall judge of their musical relationship.
The flow of perfect unison when the two series of 12 orifices each are
opened has been already heard. I now open a series of 8 holes in the
upper and of 16 in the lower siren. The interval you judge at once to
be an octave. If a series of 9 holes in the upper and of 18 holes in
the lower siren be opened, the interval is still an octave. This proves
that the interval is not disturbed by altering the absolute rates of
vibration, so long as the _ratio_ of the two rates remains the same.
The same truth is more strikingly illustrated by commencing with a
low speed of rotation, and urging the siren to its highest pitch; as
long as the orifices are in the ratio of 1:2, we retain the constant
interval of an octave. Opening a series of 10 holes in the upper and of
15 in the lower siren, the ratio is as 2:3, and every musician present
knows that this is the interval of a fifth. Opening 12 holes in the
upper and 18 in the lower siren does not change the interval. Opening
two series of 9 and 12, or of 12 and 16, we obtain an interval of a
fourth; the ratio in both these cases being as 3:4. In like manner two
series of 8 and 10, or of 12 and 15, give us the interval of a major
third; the ratio in this case being as 4:5. Finally, two series of 10
and 12, or of 15 and 18, yield the interval of a minor third, which
corresponds to the ratio 5:6.

These experiments amply illustrate two things: First, that a musical
interval is determined, not by the absolute number of vibrations of the
two combining notes, but by the ratio of their vibrations. Secondly,
and this is of the utmost significance, that the smaller the two
numbers which express the ratio of the two rates of vibration, the
more perfect is the consonance of the two sounds. The most perfect
consonance is the unison 1:1; next comes the octave 1:2; after that
the fifth 2:3; then the fourth 3:4; then the major third 4:5; and
finally the minor third 5:6. We can also open two series numbering,
respectively, 8 and 9 orifices: this interval corresponds to _a tone_
in music. It is a dissonant combination. Two series which number
respectively 15 and 16 orifices make the interval of a _semi-tone_: it
is a very sharp and grating dissonance.


§ 2. _The Theory of Musical Consonance. Pythagoras and Euler_

Whence, then, does this arise? Why should the smaller ratio express the
more perfect consonance? The ancients attempted to solve this question.
The Pythagoreans found intellectual repose in the answer “All is number
and harmony.” The numerical relations of the seven notes of the musical
scale were also thought by them to express the distances of the planets
from their central fire; hence the choral dance of the worlds, the
“music of the spheres,” which, according to his followers, Pythagoras
alone was privileged to hear. And might we not in passing contrast this
glorious superstition with the grovelling delusion which has taken
hold of the fantasy of our day? Were the character which superstition
assumes in different ages an indication of man’s advance or
retrogression, assuredly the nineteenth century would have no reason to
plume itself, in comparison with the sixth B.C. A more earnest attempt
to account for the more perfect consonance of the smaller ratios was
made by the celebrated mathematician, Euler, and his explanation,
if such it could be called, long silenced, if it did not satisfy,
inquirers. Euler analyzes the cause of pleasure. We take delight in
_order_; it is pleasant to us to observe means “co-operant to an end.”
But then, the effort to discern order must not be so great as to weary
us. If the relations to be disentangled are too complicated, though we
may see the order, we cannot enjoy it. The simpler the terms in which
the order expresses itself, the greater is our delight. Hence the
superiority of the simpler ratios in music over the more complex ones.
Consonance, then, according to Euler, was the satisfaction derived from
the perception of order without weariness of mind.

But in this theory it was overlooked that Pythagoras himself, who first
experimented on the musical intervals, knew nothing about rates of
vibration. It was forgotten that the vast majority of those who take
delight in music, and who have the sharpest ears for the detection
of a dissonance, are in the condition of Pythagoras, knowing nothing
whatever about rates or ratios. And it may also be added that the
scientific man, who is fully informed upon these points, has his
pleasure in no way enhanced by his knowledge. Euler’s explanation,
therefore, does not satisfy the mind, and it was reserved for an
eminent German investigator of our own day, after a profound analysis
of the entire question, to assign the physical cause of consonance and
dissonance—a cause which, when once clearly stated, is so simple and
satisfactory as to excite surprise that it remained so long without a
discoverer.

Various expressions employed in our previous lectures have already, in
part, forestalled Helmholtz’s explanation of consonance and dissonance.
Let me here repeat an experiment which will, almost of itself, force
this explanation upon your attention. Before you are two jets of
burning gas, which can be converted into singing-flames by inclosing
them within two tubes (represented in Fig. 118). The tubes are of the
same length, and the flames are now singing in unison. By means of
a telescopic slider I lengthen slightly one of the tubes; you hear
deliberate beats, which succeed each other so slowly that they can
readily be counted. I augment still further the length of the tube.
The beats are now more rapid than before: they can barely be counted.
It is perfectly manifest that the shocks of which you are now sensible
differ only in point of rapidity from the slow beats which you heard
a moment ago. There is no breach of continuity here. We begin slowly,
we gradually increase the rapidity, until finally the succession of
the beats is so rapid as to produce that particular grating effect
which every musician that hears it would call _dissonance_. Let us now
reverse the process, and pass from these quick beats to slow ones. The
same continuity of the phenomenon is noticed. By degrees the beats
separate from each other more and more, until finally they are slow
enough to be counted. Thus these singing-flames enable us to follow the
beats with certainty, until they cease to be beats and are converted
into dissonance.

This experiment proves conclusively that dissonance _may_ be produced
by a rapid succession of beats; and I imagine this cause of dissonance
would have been pointed out earlier, had not men’s minds been thrown
off the proper track by the theory of “resultant tones” enunciated by
Thomas Young. Young imagined that, when they were quick enough, the
beats ran together to form a resultant tone. He imagined the linking
together of the beats to be precisely analogous to the linking together
of simple musical impulses; and he was strengthened in this notion by
the fact already adverted to, that the first difference-tone, that is
to say, the loudest resultant tone, corresponded, as the beats do, to
a rate of vibration equal to the difference of the rates of the two
primaries. The fact, however, is that the effect of beats upon the ear
is altogether different from that of the successive impulses of an
ordinary musical tone.


§ 3. _Sympathetic Vibrations_

But to grasp, in all its fulness, the new theory of musical consonance
some preliminary studies will be necessary. And here I would ask you to
call to mind the experiments (in Chapter III.) by which the division of
a string into its harmonic segments was illustrated. This was done by
means of little paper riders, which were unhorsed, or not, according
as they occupied a ventral segment or a node upon the string. Before
you at present is the sonometer, employed in the experiments just
referred to. Along it, instead of one, are stretched two strings, about
three inches asunder. By means of a key these strings are brought into
unison. And now I place a little paper rider upon the middle of one
of them, and agitate the other. What occurs? The vibrations of the
sounding string are communicated to the bridges on which it rests, and
through the bridges to the other string. The individual impulses are
very feeble, but, because the two strings are in unison, the impulses
can so accumulate as finally to toss the rider off the untouched string.

Every experiment executed with the riders and a single string may be
repeated with these two unisonant strings. Damping, for instance, one
of the strings, at a point one-fourth of its length from one of its
ends, and placing the red and blue riders formerly employed, not on the
nodes and ventral segments of the damped string, but at points upon the
second string exactly opposite to those nodes and segments, when the
bow is passed across the shorter segment of the damped string, the five
red riders on the adjacent string are unhorsed, while the four blue
ones remain tranquilly in their places. By relaxing one of the strings,
it is thrown out of unison with the other, and then all efforts to
unhorse the riders are unavailing. That accumulation of impulses,
which unison alone renders possible, cannot here take place, and the
consequence is that, however great the agitation of the one string may
be, it fails to produce any sensible effect upon the other.

The influence of synchronism may be illustrated in a still more
striking manner, by means of two tuning-forks which sound the same
note. Two such forks mounted on their resonant supports are placed upon
the table. I draw the bow vigorously across one of them, permitting
the other fork to remain untouched. On stopping the agitated fork, the
sound is enfeebled, but by no means quenched. Through the air and
through the wood the vibrations have been conveyed from fork to fork,
and the untouched fork is the one you now hear. When, by means of a
morsel of wax, a small coin is attached to one of the forks, its power
of influencing the other ceases; the change in the rate of vibration,
if not very small, so destroys the sympathy between the two forks as to
render a response impossible. On removing the coin the untouched fork
responds as before.

This communication of vibrations through wood and air may be obtained
when the forks, mounted on their cases, stand several feet apart. But
the vibrations may also be communicated through the air alone. Holding
the resonant case of a vigorously vibrating fork in my hand, I bring
one of its prongs near an unvibrating one, placing the prongs back
to back, but allowing a space of air to exist between them. Light as
is the vehicle, the accumulation of impulses, secured by the perfect
unison of the two forks, enables the one to set the other in vibration.
Extinguishing the sound of the agitated fork, that which a moment
ago was silent continues sounding, having taken up the vibrations of
its neighbor. Removing one of the forks from its resonant case, and
striking it against a pad, it is thrown into strong vibration. Held
free in the air, its sound is audible. But, on bringing it close
to the silent mounted fork, out of the silence rises a full mellow
sound, which is due, not to the fork originally agitated, but to its
sympathetic neighbor.

Various other examples of the influence of synchronism, already
brought forward, will occur to you here; and cases of the kind might
be indefinitely multiplied. If two clocks, for example, with pendulums
of the same period of vibration, be placed against the same wall, and
if one of the clocks is set going and the other not, the ticks of the
moving clock, transmitted through the wall, will act upon its neighbor.
The quiescent pendulum, moved by a single tick, swings through an
extremely minute arc; but it returns to the limit of its swing just in
time to receive another impulse. By the continuance of this process,
the impulses so add themselves together as finally to set the clock
a-going. It is by this timing of impulses that a properly-pitched voice
can cause a glass to ring, and that the sound of an organ can break a
particular window-pane.


§ 4. _Sympathetic Vibration in Relation to the Human Ear_

If I dwell so fully upon this object, it is for the purpose of
rendering intelligible the manner in which sonorous motion is
communicated to the auditory nerve. In the organ of hearing, in man,
we have first of all the external orifice of the ear, closed at the
bottom by the circular tympanic membrane. Behind that membrane is the
drum of the ear, this cavity being separated from the space between it
and the brain by a bony partition, in which there are two orifices, the
one round and the other oval. These orifices are also closed by fine
membranes. Across the drum stretches a series of four little bones. The
first, called the _hammer_, is attached to the tympanic membrane; the
second, called the _anvil_, is connected by a joint with the hammer; a
third little round bone connects the anvil with the _stirrup-bone_, the
base of which is planted against the membrane of the oval orifice just
referred to. This oval membrane is almost covered by the stirrup-bone,
a narrow rim only of the membrane surrounding the bone being left
uncovered. Behind the bony partition, and between it and the brain,
we have the extraordinary organ called the _labyrinth_, filled with
water, over the lining membrane of which are distributed the terminal
fibres of the auditory nerve. When the tympanic membrane receives a
shock, it is transmitted through the series of bones above referred
to, being concentrated on the membrane against which the base of the
stirrup-bone is fixed. The membrane transfers the shock to the water of
the labyrinth, which, in its turn, transfers it to the nerves.

The transmission, however, is not direct. At a certain place within
the labyrinth exceedingly fine elastic bristles, terminating in sharp
points, grow up between the terminal nerve-fibres. These bristles,
discovered by Max Schultze, are eminently calculated to sympathize with
such vibrations of the water as correspond to their proper periods.
Thrown thus into vibration, the bristles stir the nerve-fibres which
lie between their roots. At another place in the labyrinth we have
little crystalline particles called _otolites_—the Hörsteine of the
Germans—imbedded among the nervous filaments, which, when they vibrate,
exert an intermittent pressure upon the adjacent nerve-fibres. The
otolites probably serve a different purpose from that of the bristles
of Schultze. They are fitted, by their weight, to accept and prolong
the vibrations of evanescent sounds, which might otherwise escape
attention, while the bristles of Schultze, because of their extreme
lightness, would instantly yield up an evanescent motion. They are, on
the other hand, eminently fitted for the transmission of continuous
vibrations.

Finally, there is in the labyrinth an organ, discovered by the
Marchese Corti, which is to all appearance a musical instrument, with
its chords so stretched as to accept vibrations of different periods,
and transmit them to the nerve-filaments which traverse the organ.
Within the ears of men, and without their knowledge or contrivance,
this lute of 3,000 strings[76] has existed for ages, accepting the
music of the outer world and rendering it fit for reception by the
brain. Each musical tremor which falls upon this organ selects from
the stretched fibres the one appropriate to its own pitch, and throws
it into unisonant vibration. And thus, no matter how complicated the
motion of the external air may be, these microscopic strings can
analyze it and reveal the constituents of which it is composed. Surely,
inability to feel the stupendous wonder of what is here revealed would
imply incompleteness of mind; and surely those who practically ignore,
or fear them, must be ignorant of the ennobling influence which such
discoveries may be made to exercise upon both the emotions and the
understanding of man.


§ 5. _Consonant Intervals in Relation to the Human Ear_

This view of the use of Corti’s fibres is theoretical; but it comes to
us commended by every appearance of truth. It will enable us to tie
together many things, whose relations it would be otherwise difficult
to discern. When a musical note is sounded its corresponding Corti’s
fibre resounds, being moved, as a string is moved by a second unisonant
string. And when two sounds coalesce to produce beats, the intermittent
motion is transferred to the proper fibre within the ear. But here it
is to be noted that, for the same fibre to be affected simultaneously
by two different sounds, it must not be far removed in pitch from
either of them. Call to mind our repetition of Melde’s experiments (in
Chapter III.). You then had frequent occasion to notice that, even
before perfect synchronism had been established between the string and
the tuning-fork to which it was attached, the string began to respond
to the fork. But you also noticed how rapidly the vibrating amplitude
of the string increased, as it came close to perfect synchronism
with the vibrating fork. On approaching unison the string would open
out, say to an amplitude of an inch; and then a slight tightening or
slackening, as the case might be, would bring it up to unison, and
cause it to open out suddenly to an amplitude of six inches.

So also in reference to the experiment made a moment ago with the
sonometer; you noticed that the unhorsing of the paper riders was
preceded by a fluttering of the bits of paper; showing that the
sympathetic response of the second string had begun, though feebly,
prior to perfect synchronism. Instead of two strings, conceive three
strings, all nearly of the same pitch, to be stretched upon the
sonometer; and suppose the vibrating period of the middle string to
lie midway between the periods of its two neighbors, being a little
higher than the one and a little lower than the other. Each of the side
strings, sounded singly, would cause the middle string to respond.
Sounding the two side strings together they would produce beats; the
corresponding intermittence would be propagated to the central string,
which would beat in synchronism with the beats of its neighbors. In
this way we make plain to our minds how a Corti’s fibre may, to some
extent, take up the vibrations of a note, nearly, but not exactly,
in unison with its own; and that when two notes close to the pitch
of the fibre act upon it together, their beats are responded to
by an intermittent motion on the part of the fibre. This power of
sympathetic vibration would fall rapidly on both sides of the perfect
unison, so that on increasing the interval between the two notes, a
time would soon arrive when the same fibre would refuse to be acted on
simultaneously by both. Here the condition of the organ, necessary for
the perception of audible beats, would cease.

In the middle region of the pianoforte, with the interval of a
semitone, the beats are sharp and distinct, falling indeed upon the
ear as a grating dissonance. Extending the interval to a whole tone,
the beats become more rapid, but less distinct. With the interval of a
minor third between the two notes, the beats in the middle region of
the scale cease to be sensible. But this smoothening of the sound is
not wholly due to the augmented rapidity of the beats. It is due in
part to the fact, for which the foregoing considerations have prepared
us, that the two notes here sounded are too far removed from that of
the intermediate Corti’s fibre to affect it powerfully. By ascending
to the higher regions of the scale we can produce, with a narrower
interval than the minor third, the same, or even a greater, number
of beats, which are sharply distinguishable because of the closeness
of their component notes. In the very highest regions of the scale,
however, the beats, when they become very rapid, cease to appeal as
roughness to the ear.

Hence both the rapidity of the beats, and the width of the interval,
enter into the question of consonance. Helmholtz judges that in the
middle and higher regions of the musical scale, when the beats reach 33
per second, the dissonance reaches its maximum. Both slower and quicker
beats have a less grating or dissonant effect. When the beats are very
slow, they may be of advantage to the music; and, when they reach 132
per second, their roughness is no longer discernible.

Thanks to Helmholtz, whose views I have here sought to express in the
briefest possible language, we are now in a condition to grapple with
the question of musical intervals, and to give the reason why some
are consonant and some dissonant to the ear. Circumstanced as we are
upon earth, all our feelings and emotions, from the lowest sensation
to the highest æsthetic consciousness, have a mechanical cause: though
it may be forever denied to us to take the step from cause to effect;
or to understand why the agitations of nervous matter can awaken the
delights which music imparts. Take, then, the case of a violin. The
fundamental tone of every string of this instrument is demonstrably
accompanied by a crowd of overtones; so that, when two violins are
sounded, we have not only to take into account the consonance or
dissonance of the fundamental tones, but also those of the higher tones
of both. Supposing two strings sounded whose fundamental tones, and
all of whose partial tones, coincide, we have then absolute unison;
and this we actually have when the ratio of vibration is 1:1. So
also when the ratio of vibration is accurately 1:2, each overtone of
the fundamental finds itself in absolute coincidence with either the
fundamental tone or some higher tone of the octave. There is no room
for beats or dissonance. When we examine the interval of a fifth, with
a ratio of 2:3, we find the coincidence of the partial tones of the
two so perfect as almost, though not wholly, to exclude every trace of
dissonance. Passing on to the other intervals, we find the coincidence
of the partial tones less perfect, as the numbers expressing the ratio
of the vibrations become more large. Thus, the dissonance of intervals
whose rates of vibration can only be expressed by large numbers, is
not to be ascribed to any mystic quality of the numbers themselves,
but to the fact that the fundamental tones which require such numbers
are inexorably accompanied by partial tones whose coalescence produces
beats, these producing the grating effect known as dissonance.


§ 6. _Graphic Representation of Consonance and Dissonance_

Helmholtz has attempted to represent this result graphically, and
from his work I copy, with some modification, the next two diagrams.
He assumes, as already stated, the maximum dissonance to correspond
to 33 beats per second; and he seeks to express different degrees of
dissonance by lines of different lengths. The horizontal line _c′ c″_,
Fig. 164, represents a range of the musical scale in which _c″_ is our
middle C, with 528 vibrations, and _c′_ the lower octave of _c″_. The
distance from any point of this line to the curve above it represents
the dissonance corresponding to that point. The pitch here is supposed
to ascend continuously, and not by jumps. Supposing, for example, two
performers on the violin to start with the same note _c′_, and that,
while one of them continues to sound that note, the other gradually
and continuously shortens his string, thus gradually raising its pitch
up to the octave _c″_. The effect upon the ear would be represented
by the irregular curved line in Fig. 164. Soon after the unison,
which is represented by contact at _c′_, is departed from, the curve
suddenly rises, showing the dissonance here to be the sharpest of all.
At _c′_, the curve approaches the straight line _c′ c″_, and this
point corresponds to the major third. At _f′_ the approach, is still
nearer, and this point corresponds to the fourth. At _g′_ the curve
almost touches the straight line, indicating that at this point, which
corresponds to the fifth, the dissonance almost vanishes. At _a′_ we
have the major sixth; while at _c″_, where the one note is an octave
above the other, the dissonance entirely vanishes. The _e s′_ and the
_a s′_, of this diagram are the German names of a third and a flat
sixth.

[Illustration: FIG. 164.]

Maintaining the same fundamental note _c′_, and passing through the
octave above _c″_, the various degrees of consonance and dissonance
are those shown in Fig. 165. That is to say, beginning with the octave
_c′-c″_, and gradually elevating the pitch of one of the strings till
it reaches _c″′_, the octave of _c″_, the curved line represents the
effect upon the ear. We see, from both these curves, that dissonance
is the general rule, and that only at certain definite points does the
dissonance vanish, or become so decidedly enfeebled as not to destroy
the harmony. These points correspond to the places where the numbers
expressing the ratio of the two rates of vibration are small whole
numbers. It must be remembered that these curves are constructed on the
supposition that the beats are the cause of the dissonance; and the
agreement between calculation and experience sufficiently demonstrates
the truth of the assumption.[77]

[Illustration: FIG. 165.]

You have thus accompanied me to the verge of the Physical portion of
the science of Acoustics, and through the æsthetic portion I have not
the knowledge of music necessary to lead you. I will only add that, in
comparing three or more sounds together, that is to say, in choosing
them for _chords_, we are guided by the principles just mentioned.
We choose sounds which are in harmony with the fundamental sound and
with each other. In choosing a series of sounds for combination two by
two, the simplicity alone of the ratios would lead us to fix on those
expressed by the numbers 1, 5/4, 4/3, 3/2, 5/3, 2; these being the
simplest ratios that we can have within an octave. But, when the notes
represented by these ratios are sounded in succession, it is found that
the intervals between 1 and 5/4, and between 5/3 and 2, are wider than
the others, and require the interpolation of a note in each case. The
notes chosen are such as form chords, not with the fundamental tone,
but with the note _3/2_ regarded as a fundamental tone. The ratios of
these two notes with the fundamental are 9/8 and 15/8. Interpolating
these, we have the eight notes of the natural or diatonic scale,
expressed by the following names and ratios:

  Names                C.    D.    E.    F.    G.    A.    B.    C′.
  Intervals            1st.  2d.   3d.   4th.  5th.  6th.  7th.  8th.
  Rates of vibration   1,    9/8,  5/4,  4/3,  3/2,  5/3,  15/8, 2.

Multiplying these ratios by 24, to avoid fractions, we obtain the
following series of whole numbers, which express the relative rates of
vibration of the notes of the diatonic scale:

  24, 27, 30, 32, 36, 40, 45, 48.

The meaning of the terms third, fourth, fifth, etc., which we have so
often applied to the musical intervals, is now apparent; the term has
reference to the position of the note in the scale.


§ 7. _Composition of Vibrations_

In our second lecture I referred to, and in part illustrated, a method
devised by M. Lissajous for studying musical vibrations. By means of a
beam of light reflected from a mirror attached to a tuning-fork, the
fork was made to write the story of its own motion. In our last lecture
the same method was employed to illustrate optically the phenomenon
of beats. I now propose to apply it to the study of the composition
of the vibrations which constitute the principal intervals of the
diatonic scale. We must, however, prepare ourselves for the thorough
comprehension of this subject by a brief preliminary examination of the
vibrations of a common pendulum.

Such a pendulum hangs before you. It consists of a wire carefully
fastened to a plate of iron at the roof of the house, and bearing a
copper ball weighing 10 lbs. I draw the pendulum aside and let it go;
it oscillates to and fro almost in the same plane.

I say “almost,” because it is practically impossible to suspend a
pendulum without some little departure from perfect symmetry around
its point of attachment. In consequence of this, the weight deviates
sooner or later from a straight line, and describes an oval more or
less elongated. Some years ago this circumstance presented a serious
difficulty to those who wished to repeat M. Foucault’s celebrated
experiment, demonstrating the rotation of the earth.

Nevertheless, in the case now before us, the pendulum is so carefully
suspended that its deviation from a straight line is not at first
perceptible. Let us suppose the amplitude of its oscillation to be
represented by the dotted line _a b_, Fig. 166. The point _d_, midway
between _a_ and _b_, is the pendulum’s point of rest. When drawn aside
from this point to _b_, and let go, it will return to _d_, and in
virtue of its momentum will pass on to _a_. There it comes momentarily
to rest, and returns through _d_ to _b_. And thus it will continue to
oscillate until its motion is expended.

The pendulum having first reached the limit of its swing at _b_, let us
suppose a push in a direction perpendicular to _a b_ imparted to it;
that is to say, in the direction _b c_. Supposing the time required by
the pendulum to swing from _b_ to _a_ to be one second,[78] then the
time required to swing from _b_ to _d_ will be half a second. Suppose,
further, the force applied at _b_ to be such as would carry the bob,
if free to move in that direction alone, to _c_ in half a second, and
that the distance _b c_ is equal to _b d_, the question then occurs,
where will the bob really find itself at the end of half a second? It
is perfectly manifest that both forces are satisfied by the pendulum
reaching the point _e_, exactly opposite the centre _d_, in half a
second. To reach this point, it can be shown that it must describe the
circular arc _b e_, and it will pursue its way along the continuation
of the same arc, to _a_, and then pass round to _b_. Thus, by the
rectangular impulse the rectilinear oscillation is converted into a
rotation, the pendulum describing a circle, as shown in Fig. 167.

[Illustration: FIG. 166.]

[Illustration: FIG. 167.]

If the force applied at _b_ be sufficient to urge the weight in half
a second through a greater distance than _b c_, the pendulum will
describe an ellipse, with the lines _a b_ for its smaller axis; if,
on the contrary, the force applied at _b_ urge the pendulum in half a
second through a distance less than _b c_, the weight will describe an
ellipse, with the line _a b_ for its greater axis.

[Illustration: Fig. 167.]

Let us now inquire what occurs when the rectangular impulse is applied
at the moment the ball is passing through its position of rest at _d_.

Supposing the pendulum to be moving from _a_ to _b_, Fig. 168, and that
at _d_ a shock is imparted to it sufficient of itself to carry it in
half a second to _c_; it is here manifest that the resultant motion
will be along the straight line _d g_ lying between _b d_ and _d c_.
The pendulum will return along this line to _d_, and pass on to _h_. In
this case, therefore, the pendulum will describe a straight line, _g
h_, oblique to its original direction of oscillation.

Supposing the direction of motion at the moment the push is applied to
be from _b_ to _a_, instead of from _a_ to _b_, it is manifest that the
resultant here will also be a straight line oblique to the primitive
direction of oscillation; but its obliquity will be that shown in Fig.
169.

[Illustration: FIG. 168.]

[Illustration: FIG. 169.]

When the impulse is imparted to the pendulum neither at the centre nor
at the limit of its swing, but at some point between both, we obtain
neither a circle nor a straight line, but something between both. We
have, in fact, a more or less elongated ellipse with its axis oblique
to _a b_, the original direction of vibration. If, for example, the
impulse be imparted at _d′_, Fig. 170, while the pendulum is moving
toward _b_, the position of the ellipse will be that shown in Fig. 170;
but if the push at _d′_ be given when the motion is toward _a_, then
the position of the ellipse will be that represented in Fig. 171.

[Illustration: FIG. 170.]

[Illustration: FIG. 171.]

[Illustration: FIG. 172.]

By the method of M. Lissajous we can combine the rectangular vibrations
of two tuning-forks, a subject which I now wish to illustrate before
you. In front of an electric lamp, L, Fig. 172, is placed a large
tuning-fork, T′, fixed in a stand horizontally, and provided with a
mirror, on which a narrow beam of light, L T′, is permitted to fall.
The beam is thrown back, by reflection. In the path of the reflected
beam is placed a second upright tuning-fork, T, also furnished with
a mirror. By the horizontal fork, when it vibrates, the beam is
tilted laterally; by the vertical fork, vertically. At the present
moment both forks are motionless, the beam of light being reflected
from the mirror of the horizontal to that of the vertical fork, and
from the latter to the screen, on which it prints a brilliant disk. I
now agitate the upright fork, leaving the other motionless. The disk
is drawn out into a fine luminous band, 3 feet long. On sounding the
second fork, the straight band is instantly transformed into a white
ring _o p_, Fig. 172, 36 inches in diameter. What have we done here?
Exactly what we did in our first experiment with the pendulum. We have
caused a beam of light to vibrate simultaneously in two directions, and
have accidentally hit upon the phase when one fork has just reached
the limit of its swing and come momentarily to rest, while the beam is
receiving the maximum impulse from the other fork.

That the _circle_ was obtained is, as stated, a mere accident; but it
was a fortunate accident, as it enables us to see the exact similarity
between the motion of the beam and that of the pendulum. I stop both
forks, and, agitating them afresh, obtain an ellipse with its axis
oblique. After a few trials we obtain the straight line, indicating
that both the forks then pass simultaneously through their positions of
equilibrium. In this way, by combining the vibrations of the two forks,
we reproduce all the figures obtained with the pendulum.

When the vibrations of the two forks are, in all respects, absolutely
alike, whatever the figure may be which is first traced upon the
screen, it remains unchanged in form, diminishing only in size as
the motion is expended. But the slightest difference in the rates of
vibration destroys this fixity of the image. I endeavored before the
lecture to reader the unison between these two forks as perfect as
possible, and hence you have observed very little alteration in the
shape of the figure. But by moving a small weight along the prong of
either fork, or by attaching to either of them a bit of wax, the unison
is impaired. The figure then obtained by the combination of both passes
slowly from a straight line into an oblique ellipse, thence into a
circle; after which it narrows again to an ellipse with an opposed
obliquity, it then passes again into a straight line, the direction of
which is at right angles to the first direction. Finally, it passes, in
the reverse order, through the same series of figures to the straight
line with which we began. The interval between two successive identical
figures is the time in which one of the forks succeeds in executing one
complete vibration more than the other. Loading the fork still more
heavily, we have more rapid changes; the straight line, ellipse, and
circle being passed through in quick succession. At times the luminous
curve exhibits a stereoscopic depth, which renders it difficult to
believe that we are not looking at a solid ring of white-hot metal.

[Illustration: FIG. 173.]

By causing the mirror of the fork, T, to rotate through a small arc,
the steady circle first obtained is drawn out into a luminous scroll
stretching right across the screen, Fig. 173. The same experiment
made with the changing figure, obtained by throwing the forks out of
unison, gives us a scroll of irregular amplitude, Fig. 174.[79]

[Illustration: FIG. 174.]

[Illustration: FIG. 175.]

We have next to combine the vibrations of two forks, one of which
oscillates with twice the rapidity of the other; in other words, to
determine the figure corresponding to the combination of a note and
its octave. To prepare ourselves for the mechanics of the problem,
we must resort once more to our pendulum; for it also can be caused
to oscillate in one direction twice as rapidly as in another. By a
complicated mechanical arrangement this might be done in a very perfect
manner, but at present simplicity is preferable to completeness. The
wire of our pendulum is therefore permitted to descend from its point
of suspension, A, Fig. 175, midway between two horizontal glass rods,
_a b_, _a′ b′_, supported firmly at their ends, and about an inch
asunder. The rods cross the wire at a height of 7 feet above the bob of
the pendulum. The whole length of the pendulum being 28 feet, the glass
rods intercept one-fourth of this length. On drawing the pendulum aside
in the direction of the rods, _a b_, _a′ b′_, and letting it go, it
oscillates freely between them. I bring it to rest and draw it aside
in a direction perpendicular to the last; a length of 7 feet only can
now oscillate, and by the laws of oscillation a pendulum 7 feet long
vibrates with twice the rapidity of a pendulum 28 feet long.

I wish to show you the figure described by the combination of these two
rates of vibration. Attached to the copper ball, _p_, is a camel’s-hair
pencil, intended to rub lightly upon a glass plate placed on black
paper and over which is strewed white sand. Allowing the pendulum to
oscillate as a whole, the sand is rubbed away along a straight line
which represents the amplitude of the vibration. Let _a b_, Fig. 176,
represent this line, which, as before, we will assume to be described
in one second. When the pendulum is at the limit, _b_, of its swing,
let a rectangular impulse be imparted to it sufficient to carry it to
_c_ in one-fourth of a second. If this were the only impulse acting
on the pendulum, the bob would reach _c_ and return to _b_ in half a
second. But under the actual circumstances it is also urged toward _d_,
which point, through the vibration of the whole pendulum, it ought also
to reach in half a second. Both vibrations, therefore, require that the
bob shall reach _d_ at the same moment; and to do this it will have to
describe the curve _b c′ d_. Again, in the time required by the long
pendulum to pass from _d_ to _a_, the short pendulum will pass _to and
fro_ over the half of its excursion; both vibrations must therefore
reach _a_ at the same moment, and to accomplish this the pendulum
describes the lower curve between _d_ and _a_. It is manifest that
these two curves will repeat themselves at the opposite sides of _a b_,
the combination of both vibrations producing finally a figure of 8,
which you now see fairly drawn upon the sand before you.

The same figure is obtained if the rectangular impulse be imparted when
the pendulum is passing its position of rest, _d_.

[Illustration: FIG. 176.]

[Illustration: FIG. 177.]

[Illustration: FIG. 178.]

I have here supposed the time occupied by the pendulum in describing
the line _a b_ to be one second. Let us suppose three-fourths of the
second exhausted, and the pendulum at _d′_, Fig. 177, in its excursion
toward _b_; let the rectangular impulse then be imparted to it,
sufficient to carry it to _c_ in one-fourth of a second. Now the long
pendulum requires that it should move from _d′_ to _b_ in one-fourth of
a second; both impulses are therefore satisfied by the pendulum taking
up the position _c′_ at the end of a quarter of a second. To reach
this position it must describe the curve _d′ c′_. It will manifestly
return along the same curve, and at the end of another quarter of a
second find itself again at _d′_. From _d′_ to _d_ the long pendulum
requires a quarter of a second. But at the end of this time the short
pendulum must be at the lower limit of its swing: both requirements are
satisfied by the pendulum being at _e_. We thus obtain one arm, _c′
e_, of a curve, which repeats itself to the left of _e_; so that the
entire curve, due to the combination of the two vibrations, is that
represented in Fig. 165. This figure is a parabola, whereas the figure
of 8 before obtained is a lemniscata.

We have here supposed that, at the moment when the rectangular impulse
was applied, the motion of the pendulum was _toward_ _b_: if it were
toward _a_ we should obtain the inverted parabola, as shown in Fig. 178.

Supposing, finally, the impulse to be applied, not when the pendulum
is passing through its position of equilibrium, nor when it is passing
a point corresponding to three-fourths or one-fourth of the time of
its excursion, but at some other point in the line, _a b_, between its
end and centre. Under these circumstances we should have neither the
parabola nor the perfectly symmetrical figure of 8, but a distorted 8.

And now we are prepared to witness with profit the combined vibration
of our two tuning-forks, one of which sounds the octave of the other.
Permitting the vertical fork, T, Fig. 172, to remain undisturbed
in front of the lamp, we can oppose to it a horizontal fork, which
vibrates with twice the rapidity. The first passage of the bow across
the two forks reveals the exact similarity of this combination, and
that of our pendulum. A very perfect figure of 8 is described upon
the screen. Before the lecture the vibrations of these two forks were
fixed as nearly as possible to the ratio of 1:2, and the steadiness
of the figure indicates the perfection of the tuning. Stopping both
forks, and again agitating them, we have the distorted 8 upon the
screen. A few trials enable me to bring out the parabola. In all these
cases the figure remains fixed upon the screen. But if a morsel of
wax be attached to one of the forks, the figure is steady no longer,
but passes from the perfect 8 into the distorted one, thence into
the parabola, from which it afterward opens out to an 8 once more. By
augmenting the discord, we can render those changes as rapid as we
please.

When the 8 is steady on the screen, a rotation of the mirror of the
fork, T, produces the scroll shown in Fig. 179.

[Illustration: FIG. 179.]

Our next combination will be that of two forks vibrating in the ratio
of 2:3. Observe the admirable steadiness of the figure produced by
the compounding of these two rates of vibration. On attaching a
four-penny-piece with wax to one of the forks the steadiness ceases,
and we have an apparent rocking to and fro of the luminous figure.
Passing on to intervals of 3:4, 4:5, and 5:6, the figures become more
intricate as we proceed. The last combination, 5:6, is so entangled
that to see the figure plainly a very narrow band of light must be
employed. The distance existing between the forks and the screen also
helps us to unravel the complication.

[Illustration: FIG. 180.]

And here it is worth noting that, when the figure is fully developed,
the loops along the vertical and horizontal edges express the ratio of
the combined vibrations. In the octave, for example, we have two loops
in one direction, and one in another; in the fifth, two loops in one
direction, and three in another. When the combination is as 1:3, the
luminous loops are also as 1:3. The changes which some of these figures
undergo, when the tuning is not perfect, are extremely remarkable. In
the case of 1:3, for example, it is difficult at times not to believe
that you are looking at a solid link of white-hot metal. The figure
exhibits a depth, apparently incompatible with its being traced upon a
plane surface.

[Illustration: FIG. 181.]

Fig. 180 (page 445) is a diagram of these beautiful figures, including
combinations from 1:1 to 5:6. In each case, the characteristic phases
of the vibration are shown; and through all of these each figure passes
when the interval between the two forks is not pure. I also add here,
Fig. 181, two phases of the combination 8:9.

[Illustration: FIG. 182. 1:2.]

[Illustration: FIG. 183. 2:3.]

To these illustrations of rectangular vibrations I add two others,
Figs. 182 and 183, from a very beautiful series obtained by Mr. Herbert
Airy with a compound pendulum. The experiments are described in
“Nature” for August 17 and September 7, 1871. As their loops indicate,
the figures are those of an octave and a twelfth.

[Illustration: FIG. 184. 2:3.]

[Illustration: FIG. 185. 3:4.]

But the most instructive apparatus for the compounding of rectangular
vibrations is that of Mr. Tisley. Figs. 184 and 185 are copies of
figures obtained by him through the joint action of two distinct
pendulums; the rates of vibration corresponding to these particular
figures being 2:3 and 3:4 respectively. The pen which traces the
figures is moved simultaneously by two rods attached to the pendulums
above their places of suspension. These two rods lie in the two planes
of vibration, being at right angles to the pendulums, and to each
other. At their place of intersection is the pen. By means of a ball
and socket, of a special kind, the rods are enabled to move with a
minimum of friction in all directions, while the rates of vibration are
altered, in a moment, by the shifting of movable weights. The figures
are drawn either with ink on paper, or, when projection on a screen is
desired, by a sharp point on smoked glass. When the pendulums, having
gone through the entire figure, return to their starting-point, they
have lost a little in amplitude. The second excursion will, therefore,
be smaller than the first, and the third smaller than the second. Hence
the series of fine lines, inclosing gradually-diminishing areas, shown
in these exquisite figures.[80] Mr. Tisley’s apparatus reflects the
highest credit upon its able constructor.

[Illustration: FIG. 186.]

Sir Charles Wheatstone devised, many years ago, a small and very
efficient apparatus for the compounding of rectangular vibrations.
A drawing, Fig. 186, and a description of this beautiful little
instrument, for both of which I am indebted to its eminent inventor,
may find a place here: _a_ is a steel rod polished at its upper end so
as to reflect a point of light; this rod moves in a ball-and-socket
joint at _b_, so that it may assume any position. Its lower end is
connected with two arms _c_ and _d_, placed at right angles to each
other, the other ends of which are respectively attached to the
circumferences of the two circular disks _e_ and _f_. The axis of
the disk _e_ carries at its opposite end another large disk _g_,
which gives motion to the small disk _h_, placed on the axis which
carries the disk _f_; and, according as this small disk _h_ is placed
nearer to or further from the centre of the disk _g_, it communicates
a different relative motion to the disk _f_. The nut and screw _i_
enable the disk _h_ to be placed in any position between the centre,
and circumference of the larger disk _g_; and by means of the fork
_j_ the disk _f_ is caused to revolve, whatever may be the position
of the disk _h_. By this arrangement, while the wheel _k_ is turned
regularly, the rod _a_ is moved backward and forward by the disk _e_
in one direction, and by the disk _f_, with any relative oscillatory
motion, in the rectangular direction. The end of the rod is thus made
to describe and to exhibit optically all the beautiful acoustical
figures produced by the composition of vibrations of different periods
in directions rectangular to each other. A lever _l_, bearing against
the nut _i_, indicates, on a scale _m_, the numerical ratio of the two
vibrations.[81]

I close these remarks on the combination of rectangular vibrations with
a brief reference to an apparatus constructed by Mr. A. E. Donkin, of
Exeter College, Oxford, and described in the “Proceedings of the Royal
Society,” vol. xxii., p. 196. In its construction great mechanical
knowledge is associated with consummate skill. I saw the apparatus as
a wooden model, before it quitted the hands of its inventor, and was
charmed with its performance. It is now constructed by Messrs. Tisley
and Spiller.


SUMMARY OF CHAPTER IX

By the division of a string Pythagoras determined the consonant
intervals in music, proving that, the simpler the ratio of the
two parts into which the string was divided, the more perfect is
the harmony of the sounds emitted by the two parts of the string.
Subsequent investigators showed that the strings act thus because of
the relation of their lengths to their rates of vibration.

With the double siren this law of consonance is readily illustrated.
Here the most perfect harmony is the unison, where the vibrations are
in the ratio of 1:1. Next comes the octave, where the vibrations are
in the ratio of 1:2. Afterward follow in succession the fifth, with a
ratio of 2:3; the fourth, with a ratio of 3:4; the major third, with a
ratio of 4:5; and the minor third, with a ratio of 5:6. The interval
of a tone, represented by the ratio 8:9, is dissonant, while that of a
semitone, with a ratio of 15:16, is a harsh and grating dissonance.

The musical interval is independent of the absolute number of the
vibrations of the two notes, depending only on the _ratio_ of the two
rates of vibration.

The Pythagoreans referred the pleasing effect of the consonant
intervals to number and harmony, and connected them with “the music of
the spheres.” Euler explained the consonant intervals by reference to
the constitution of the mind, which, he affirmed, took pleasure in
simple calculations. The mind was fond of order, but of such order as
involved no weariness in its contemplation. This pleasure was afforded
by the simpler ratios in the case of music.

The researches of Helmholtz prove the rapid succession of beats to be
the real cause of dissonance in music.

By means of two singing-flames, the pitch of one of them being
changeable by the telescopic lengthening of its tube, beats of any
degree of slowness or rapidity may be produced. Commencing with beats
slow enough to be counted, and gradually increasing their rapidity, we
reach, without breach of continuity, downright dissonance.

But, to grasp this theory in all its completeness, we must refer to the
constitution of the human ear. We have first the tympanic membrane,
which is the anterior boundary of the drum of the ear. Across the drum
stretches a series of little bones, called respectively the _hammer_,
the _anvil_, and the _stirrup-bone_; the latter abutting against a
second membrane, which forms part of the posterior boundary of the
drum. Beyond this membrane is the labyrinth filled with water, and
having its lining membrane covered with the filaments of the auditory
nerve.

Every shock received by the tympanic membrane is transmitted through
the series of bones to the opposite membrane; thence to the water of
the labyrinth, and thence to the auditory nerve.

The transmission is not direct. The vibrations are in the first place
taken up by certain bodies, which can swing sympathetically with
them. These bodies are of three kinds: the otolites, which are little
crystalline particles; the bristles of Max Schultze; and the fibres of
Corti’s organ. This latter is to all intents and purposes a stringed
instrument, of extraordinary complexity and perfection, placed within
the ear.

As regards our present subject, the strings of Corti’s organ probably
play an especially important part. That one string should respond, in
some measure, to another, it is not necessary that the unison should
be perfect; a certain degree of response occurs in the immediate
neighborhood of unison.

Hence each of two strings, not far removed from each other in
pitch, can cause a third string, of intermediate pitch, to respond
sympathetically. And if the two strings be sounded together, the beats
which they produce are propagated to the intermediate string.

So, as regards Corti’s organ, when single sounds of various pitches, or
rather when vibrations of various rapidities, fall upon its strings,
the vibrations are responded to by the particular string whose period
coincides with theirs. And when two sounds, close to each other in
pitch, produce beats, the intermediate Corti’s fibre is acted on by
both, and responds to the beats.

In the middle and upper portions of the musical scale the beats are
most grating and harsh when they succeed each other at the rate of 33
per second. When they occur at the rate of 132 per second, they cease
to be sensible.

The perfect consonance of certain musical intervals is due to the
absence of beats. The imperfect consonance of other intervals is due
to their existence. And here the overtones play a part of the utmost
importance. For, though the primaries may sound together without any
perceptible roughness, the overtones may be so related to each other as
to produce harsh and grating beats. A strict analysis of the subject
proves that intervals which require large numbers to express them are
invariably accompanied by overtones which produce beats; while in
intervals expressed by small numbers the beats are practically absent.

The graphic representation of the consonances and dissonances of
the musical scale, by Helmholtz, furnishes a striking proof of this
explanation.

The optical illustration of the musical intervals has been effected in
a very beautiful manner by Lissajous. Corresponding to each interval is
a definite figure, produced by the combination of its vibrations.

The compounding of vibrations has, of late years, been beautifully
illustrated by apparatus constructed by Sir C. Wheatstone, Mr. Herbert
Airy, and Mr. A. E. Donkin; and by the beautiful pendulum apparatus of
Mr. Tisley, of the firm of Tisley and Spiller.


The pressure which, on a former occasion, prevented me from adding a
“summary” to this chapter, was also the cause of hastiness, and partial
inaccuracy, in its sketch of the theory of Helmholtz. That the sketch
needed emendation I have long known, but I did not think it worth while
to anticipate the correction here made; as the chapter, imperfect as
it was, had been published, without comment, in Germany, by Helmholtz
himself.




APPENDICES




APPENDIX I

ON THE INFLUENCE OF MUSICAL SOUNDS ON THE FLAME OF A JET OF COAL-GAS.
BY JOHN LE CONTE, M.D.[82]


A short time after reading Prof. John Tyndall’s excellent article
“On the Sounds produced by the Combustion of Gases in Tubes,”[83] I
happened to be one of a party of eight persons assembled after tea
for the purpose of enjoying a private musical entertainment. Three
instruments were employed in the performance of several of the grand
trios of Beethoven, namely, the piano, violin, and violoncello. Two
“_fish-tail_” gas-burners projected from the brick wall near the piano.
Both of them burned with remarkable steadiness, the windows being
closed and the air of the room being very calm. Nevertheless, it was
evident that _one_ of them was under a pressure nearly sufficient to
make it _flare_.

Soon after the music commenced, I observed that the flame of the
last-mentioned burner exhibited pulsations in height which were
_exactly synchronous_ with the audible beats. This phenomenon was very
striking to every one in the room, and especially so when the strong
notes of the violoncello came in. It was exceedingly interesting
to observe how perfectly even the _trills_ of this instrument were
reflected on the sheet of flame. _A deaf man might have seen
the harmony_. As the evening advanced, and the diminished consumption
of gas in the city _increased the pressure_, the phenomenon became
more conspicuous. The _jumping_ of the flame gradually increased,
became somewhat irregular, and finally it began to flare continuously,
emitting the characteristic sound indicating the escape of a greater
amount of gas than could be properly consumed. I then ascertained
by experiment that the phenomenon _did not_ take place unless the
discharge of gas was so regulated that the flame approximated to the
condition of _flaring_. I likewise determined by experiment that the
effects _were not_ produced by jarring or shaking the floor and walls
of the room by means of repeated concussions. Hence it is obvious that
the pulsations of the flame _were not_ owing to _indirect_ vibrations
propagated through the medium of the walls of the room to the burning
apparatus, but must have been produced by the _direct_ influence of the
aërial sonorous pulses on the burning jet.

In the experiments of M. Schaffgotsch and Prof. J. Tyndall, it is
evident that “the shaking of the singing-flame within the glass
tube,” produced by the voice or the siren, was a phenomenon perfectly
analogous to what took place under my observation _without the
intervention of a tube_. In my case the discharge of gas was so
regulated that there was a tendency in the flame to flare, or to emit a
“_singing-sound_.” Under these circumstances, strong aërial pulsations
occurring at _regular intervals_ were sufficient to develop synchronous
fluctuations in the height of the flame. It is probable that the
effects would be more striking when the tones of the musical instrument
are _nearly_ in unison with the sounds which would be produced by the
flame under the slight increase in the rapidity of discharge of gas
required to manifest the phenomenon of flaring. This point might be
submitted to an experimental test.

As in Prof. Tyndall’s experiments on the jet of gas burning within
a tube, clapping of the hands, shouting, etc., were ineffectual in
converting the “silent” into the “singing-flame,” so, in the case under
consideration, _irregular_ sounds did not produce any perceptible
influence. It seems to be necessary that the impulses should
_accumulate_, in order to exercise an appreciable effect.

With regard to the mode in which the sounds are produced by the
combustion of gases in tubes, it is universally admitted that the
explanation given by Prof. Faraday in 1818 is essentially correct.
It is well known that he referred these sounds to the successive
explosions produced by the periodic combination of the atmospheric
oxygen with the issuing jet of gas. While reading Prof. J. Plateau’s
admirable researches (third series) on the “Theory of the Modifications
experienced by Jets of Liquid issuing from Circular Orifices when
exposed to the Influence of Vibratory Motions,”[84] the idea flashed
across my mind that the phenomenon which had fallen under my
observation was nothing more than a _particular case_ of the effects
of sounds on _all kinds of fluid jets_. Subsequent reflection has only
served to fortify this first impression.

The beautiful investigations of Felix Savart, on the influence of
sounds on jets of water, afford results presenting so many points of
analogy with their effects on the jet of burning gas, that it may be
well to inquire whether both of them may be referred to a common cause.
In order to place this in a striking light, I shall subjoin some of the
results of Savart’s experiments. Vertically-descending jets of water
receive the following modifications under the influence of vibrations:

1. The continuous portions become shortened; the vein resolves itself
into separate drops nearer the orifice than when _not_ under the
influence of vibrations.

2. Each of the masses, as they detach themselves from the extremity of
the continuous part, becomes flattened alternately in a vertical and
horizontal direction, presenting to the eye, under the influence of
their translatory motion, regularly-disposed series of maxima and
minima of thickness, or ventral segments and nodes.

3. The foregoing modifications become much more developed and regular
when a note, in unison with that which would be produced by the shock
of the discontinuous part of the jet against a stretched membrane, is
sounded in its neighborhood. The continuous part becomes considerably
shortened, and the ventral segments are enlarged.

4. When the note of the instrument is _almost_ in unison, the
continuous part of the jet is alternately lengthened and shortened
and the beats which coincide with these variations in length _can be
recognized by the ear_.

5. Other tones act with less energy on the jet, and some produce no
sensible effect.

When a jet is made to ascend _obliquely_, so that the discontinuous
part appears scattered into a kind of _sheaf_ in the same vertical
plane, M. Savart found:

_a._ That, under the influence of vibrations of a determinate period,
this sheaf may form itself into _two_ distinct jets, each possessing
regularly-disposed ventral segments and nodes; sometimes with a
different node the sheaf becomes replaced by _three_ jets.

_b._ The note which produces the greatest shortening of the continuous
part always reduces the whole to a _single_ jet, presenting a perfectly
regular system of ventral segments and nodes.

In the last memoir of M. Savart—a posthumous one, presented to the
Academy of Sciences of Paris, by M. Arago, in 1853[85]—several
remarkable acoustic phenomena are noticed in relation to the musical
tones produced by the efflux of liquids through short tubes. When
certain precautions and conditions are observed (which are minutely
detailed by this able experimentalist), the discharge of the liquid
gives rise to a succession of musical tones of great intensity and of a
peculiar quality, somewhat analogous to that of the human voice. That
these notes were not produced by the descending drops of the liquid
vein was proved by permitting it to discharge itself into a vessel of
water, while the orifice was below the surface of the latter. In this
case the jet of liquid must have been _continuous_, but nevertheless
the notes were produced. These unexpected results have been entirely
confirmed by the more recent experiments of Prof. Tyndall.[86]

According to the researches of M. Plateau, all the phenomena of
the influence of vibrations on jets of liquid are referable to the
conflict between the vibrations and the _forces of figure_ (“_forces
figuratrices_”). If the physical fact is admitted—and it seems to be
indisputable—that a liquid cylinder attains a _limit of stability_ when
the proportion between its length and its diameter is in the ratio
of twenty-two to seven, it is almost a _physical necessity_ that the
jet should assume the constitution indicated by the observations of
Savart. It likewise seems highly probable that a liquid jet, while
in a transition stage to discontinuous drops, should be exceedingly
sensitive to the influence of all kinds of vibrations. It must be
confessed, however, that Plateau’s beautiful and coherent theory does
not appear to embrace Savart’s last experiment, in which the musical
tones were produced by a jet of water issuing under the surface of the
same liquid. It is rather difficult to imagine what agency the “forces
of figure” could have, under such circumstances, in the production of
the phenomenon. This curious experiment tends to corroborate Savart’s
original idea, that the vibrations which produce the sounds must
take place in the glass reservoir itself, and that the cause must be
inherent in the phenomenon of the flow.

To apply the principles of Plateau’s theory to gaseous jets, we are
compelled to abandon the idea of the _non-existence of molecular
cohesion in gases_. But is there not abundant evidence to show that
cohesion _does exist_ among the particles of gaseous masses? Does not
the deviation from rigorous accuracy, both in the law of Mariotte and
Gay-Lussac—especially in the case of condensable gases, as shown by the
admirable experiments of M. Regnault—clearly prove that the hypothesis
of the non-existence of cohesion in aëriform bodies is fallacious? Do
not the expanding rings which ascend when a bubble of phosphuretted
hydrogen takes fire in the air indicate the existence of some cohesive
force in the gaseous product of combustion (aqueous vapor), whose
outlines are marked by the opaque phosphoric acid? In short, does not
the very _form_ of the flame of a “fish-tail” burner demonstrate that
cohesion _must exist_ among the particles of the issuing gas? It is
well known that in this burner the single jet which issues is formed by
the union of _two oblique jets_ immediately before the gas is emitted.
The result is a perpendicular _sheet of flame_. How is such a result
produced by the mutual action of two jets, unless the force of cohesion
is brought into play? Is it not obvious that such a fanlike flame must
be produced by the same causes as those varied and beautiful forms of
aqueous sheets, developed by the mutual action of jets of water, so
strikingly exhibited in the experiments of Savart and of Magnus?

If it be granted that gases possess molecular cohesion, it seems to be
physically certain that jets of gas must be subject to the same laws as
those of liquid. Vibratory movements excited in the neighborhood ought,
therefore, to produce modifications in them analogous to those recorded
by M. Savart in relation to jets of water. Flame or incandescent gas
presents gaseous matter in a _visible_ form, admirably adapted for
experimental investigation; and, _when produced by a jet_, should be
amenable to the principles of Plateau’s theory. According to this view,
the pulsations or _beats_ which I observed in the gas-flame when under
the influence of musical sounds, are produced by the conflict between
the aërial vibrations and the “forces of figure” (as Plateau calls
them) giving origin to periodical fluctuations of intensity, depending
on the sonorous pulses.

If this view is correct, will it not be necessary for us to modify
our ideas in relation to the agency of tubes in developing musical
sounds by means of burning jets of gas? Must we not look upon all
burning jets—as in the case of water-jets—as _musically inclined_; and
that the use of tubes merely places them in a condition favorable for
developing the tones? It is well known that burning jets frequently
emit a _singing-sound_ when they are perfectly _free_. Are these
sounds produced by successive explosions analogous to those which take
place in glass tubes? It is very certain that, under the influence
of molecular forces, any cause which tends to elongate the flame,
without affecting the velocity of discharge, must tend to render it
discontinuous, and thus bring about that mixture of gas and air which
is essential to the production of the explosions. The influence of
tubes, as well as of aërial vibrations, in establishing this condition
of things, is sufficiently obvious. Was not the “beaded line” with its
succession of “luminous stars,” which Prof. Tyndall observed when a
flame of olefiant gas, burning in a tube, was examined by means of a
moving mirror, an indication that the flame became _discontinuous_,
precisely as the continuous part of a jet of water becomes _shortened_,
and resolved into isolated drops, under the influence of sonorous
pulsations? But I forbear enlarging on this very interesting subject,
inasmuch as the accomplished physicist last named has promised
to examine it at a future period. In the hands of so sagacious a
philosopher, we may anticipate a most searching investigation of the
phenomena in all their relations. In the meantime I wish to call the
attention of men of science to the view presented in this article, in
so far as it groups together several classes of phenomena under one
head, and may be considered a partial generalization.—From Silliman’s
“American Journal” for January, 1858.




APPENDIX II

ON ACOUSTIC REVERSIBILITY[87]


On the 21st and 22d of June, 1822, a commission, appointed by the
Bureau des Longitudes of France, executed a celebrated series of
experiments on the velocity of sound. Two stations had been chosen,
the one at Villejuif, the other at Montlhéry, both lying south of
Paris, and 11·6 miles distant from each other. Prony, Mathieu, and
Arago were the observers at Villejuif, while Humboldt, Bouvard, and
Gay-Lussac were at Montlhéry. Guns, charged sometimes with two pounds
and sometimes with three pounds of powder, were fired at both stations,
and the velocity was deduced from the interval between the appearance
of the flash and the arrival of the sound.

On this memorable occasion an observation was made which, as far as
I know, has remained a scientific enigma to the present hour. It was
noticed that while every report of the cannon fired at Montlhéry was
heard with the greatest distinctness at Villejuif, by far the greater
number of the reports from Villejuif failed to reach Montlhéry. Had
wind existed, and had it blown from Montlhéry to Villejuif, it would
have been recognized as the cause of the observed difference; but the
air at the time was calm, the slight motion of translation actually
existing being from Villejuif toward Montlhéry, or against the
direction in which the sound was best heard.

So marked was the difference in transmissive power between the two
directions, that on June 22d, while every shot fired at Montlhéry was
heard _à merveille_ at Villejuif, but one shot out of twelve fired at
Villejuif was heard, and that feebly, at the other station.

With the caution which characterized him on other occasions, and
which has been referred to admiringly by Faraday,[88] Arago made no
attempt to explain this anomaly. His words are: “Quant aux différences
si remarquables d’intensité que le bruit du canon a toujours
présentées suivant qu’il se propageait du nord au sud entre Villejuif
et Montlhéry, ou du sud au nord entre cette seconde station et la
première, nous ne chercherons pas aujourd’hui à l’expliquer, parce que
nous ne pourrions offrir au lecteur que des conjectures denuées de
preuves.”[89]

I have tried, after much perplexity of thought, to bring this subject
within the range of experiment, and have now to submit the following
solution of the enigma: The first step was to ascertain whether the
sensitive flame, referred to in my recent paper in the “Philosophical
Transactions,” could be safely employed in experiments on the mutual
reversibility of a source of sound and an object on which the sound
impinges. Now, the sensitive flame usually employed by me measures from
eighteen to twenty-four inches in height, while the reed employed as a
source of sound is less than a square quarter of an inch in area. If,
therefore, the whole flame, or the pipe which fed it, were sensitive to
sonorous vibrations, strict experiments on reversibility with the reed
and flame might be difficult, if not impossible. Hence my desire to
learn whether the seat of sensitiveness was so localized in the flame
as to render the contemplated interchange of flame and reed permissible.

The flame being placed behind a cardboard screen, the shank of a funnel
passed through a hole in the cardboard was directed upon the middle
of the flame. The sound-waves issuing from the vibrating reed, placed
within the funnel, produced no sensible effect upon the flame. Shifting
the funnel so as to direct its shank upon the root of the flame, the
action was violent.

To augment the precision of the experiment, the funnel was connected
with a glass tube three feet long and half an inch in diameter, the
object being to weaken, by distance, the effect of the waves diffracted
round the edge of the funnel, and to permit those only which passed
through the glass tube to act upon the flame.

Presenting the end of the tube to the orifice of the burner (_b_, Fig.
1), or the orifice to the end of the tube, the flame was violently
agitated by the sounding-reed, R. On shifting the tube, or the burner,
so as to concentrate the sound on a portion of the flame about half an
inch above the orifice, the action was _nil_. Concentrating the sound
upon the burner itself, about half an inch below its orifice, there was
no action.

[Illustration: FIG. 1.]

These experiments demonstrate the localization of “the seat of
sensitiveness,” and they prove the flame to be an appropriate
instrument for the contemplated experiments on reversibility.

The experiments then proceeded thus: The sensitive flame being placed
close behind a screen of cardboard 18 inches high by 12 inches wide, a
vibrating reed, standing at the same height as the root of the flame,
was placed at a distance of 6 feet on the other side of the screen. The
sound of the reed, in this position, produced a strong agitation of the
flame.

The whole upper half of the flame was here visible from the reed; hence
the necessity of the foregoing experiments to prove the action of the
sound on the upper portion of the flame to be _nil_, and that the
waves had really to bend round the edge of the screen, so as to reach
the seat of sensitiveness in the neighborhood of the burner.

The positions of the flame and reed were reversed, the latter being now
close behind the screen, and the former at a distance of 6 feet from
it. The sonorous vibrations were without sensible action upon the flame.

The experiment was repeated and varied in many ways. Screens of various
sizes were employed; and, instead of reversing the positions of the
flame and reed, the screen itself was moved, so as to bring in some
experiments the flame, and in other experiments the reed, close behind
it. Care was also taken that no reflected sound from the walls or
ceiling of the laboratory, or from the body of the experimenter, should
have anything to do with the effect. In all cases it was shown that the
sound was effective when the reed was at a distance from the screen,
and the flame close behind it; while the action was insensible when
these positions were reversed.

[Illustration: FIG. 2.]

Thus, let _s e_, Fig. 2, be a vertical section of the screen. When the
reed was at A and the flame at B there was no action; when the reed was
at B and the flame at A the action was decided. It may be added that
the vibrations communicated to the screen itself, and from it to the
air beyond it, were without effect; for when the reed, which at B was
effectual, was shifted to C, where its action on the screen was greatly
augmented, it ceased to have any action on the flame at A.

We are now, I think, prepared to consider the failure of reversibility
in the larger experiments of 1822. Happily an incidental observation
of great significance comes here to our aid. It was observed and
recorded at the time that, while the reports of the guns at Villejuif
were without echoes, a roll of echoes lasting from 20 to 25 seconds
accompanied every shot at Montlhéry, being heard by the observers
there. Arago, the writer of the report, referred these echoes to
reflection from the clouds, an explanation which I think we are now
entitled to regard as problematical. The report says that “tous
les coups tirés à Montlhéry y étaient _accompagnés_ d’un roulement
semblable à celui du tonnerre.” I have italicized a very significant
word—a word which fairly applies to our experiments on gun-sounds
at the South Foreland, where there was no sensible interval between
explosion and echo, but which could hardly apply to echoes coming
from the clouds. For supposing the clouds to be only a mile distant,
the sound and its echo would have been separated by an interval of
nearly ten seconds. But there is no mention of any interval; and, had
such existed, surely the word “followed,” instead of “accompanied,”
would have been the one employed. The echoes, moreover, appear to
have been _continuous_, while the clouds observed seem to have been
_separate_. “Ces phénomènes,” says Arago, “n’ont jamais eu lieu qu’au
moment de l’apparition de quelques nuages.” But from separate clouds a
continuous roll of echoes could hardly come. When to this is added the
experimental fact that clouds far denser than any ever formed in the
atmosphere are demonstrably incapable of sensibly reflecting sound,
while cloudless air, which Arago pronounced echoless, has been proved
capable of powerfully reflecting it, I think we have strong reason to
question the hypothesis of the illustrious French philosopher.[90]

And, considering the hundreds of shots fired at the South Foreland,
with the attention especially directed to the aërial echoes, when no
single case occurred in which echoes of measurable duration did not
accompany the report of the gun, I think Arago’s statement, that at
Villejuif no echoes were heard when the sky was clear, must simply
mean that they vanished with great rapidity. Unless the attention
was specially directed to the point, a slight prolongation of the
cannon-sound might well escape observation; and it would be all the
more likely to do so if the echoes were so loud and prompt as to form
apparently part and parcel of the direct sound.

I should be very loth to transgress here the limits of fair criticism,
or to throw doubt, without good reason, on the recorded observations
of illustrious men. Still, taking into account what has been just
stated, and remembering that the minds of Arago and his colleagues
were occupied by a totally different problem (that the echoes were an
incident rather than an object of observation), I think we may justly
consider the sound which he called “instantaneous” as one whose aërial
echoes did not differentiate themselves from the direct sound by any
noticeable fall of intensity, and which rapidly died into silence.

Turning now to the observations at Montlhéry, we are struck by the
extraordinary duration of the echoes heard at that station. At the
South Foreland the charge habitually fired was equal to the largest of
those employed by the French philosophers; but on no occasion did the
gun-sounds produce echoes approaching to 20 or 25 seconds’ duration.
The time rarely reached half this amount. Even the siren-echoes,
which were more remarkable and more long-continued than those of the
gun, never reached the duration of the Montlhéry echoes. The nearest
approach to it was on October 17, 1873, when the siren-echoes required
15 seconds to subside into silence.

On this same day, moreover (and this is a point of marked
significance), the transmitted sound reached its maximum range, the
gun-sounds being heard at the Quenocs buoy, 16-1/2 nautical miles from
the South Foreland. I have stated in another place that the duration
of the air-echoes indicates “the atmospheric depths” from which they
came. An optical analogy may help us here. Let light fall upon chalk,
the light is wholly scattered by the superficial particles; let the
chalk be powdered and mixed with water, light reaches the observer from
a far greater depth of the turbid liquid. The solid chalk typifies the
action of exceedingly dense acoustic clouds; the chalk and water that
of clouds of more moderate density. In the one case we have echoes of
short, in the other echoes of long, duration. These considerations
prepare us for the inference that Montlhéry, on the occasion referred
to, must have been surrounded by a highly-diacoustic atmosphere; while
the shortness of the echoes at Villejuif shows that the atmosphere
surrounding that station must have been, in a high degree, acoustically
opaque.

Have we any clew to the cause of the opacity? I think we have.
Villejuif is close to Paris, and over it, with the observed light
wind, was slowly wafted the air from the city. Thousands of chimneys
to windward of Villejuif were discharging their heated currents; so
that an exceeding non-homogeneous atmosphere must have surrounded that
station.[91] At no great height in the atmosphere the equilibrium of
temperature would be established. This non-homogeneous air surrounding
Villejuif is experimentally typified by our screen, with the source
of sound close behind it, the upper edge of the screen representing
the place where equilibrium of temperature was established in the
atmosphere above the station. In virtue of its proximity to the screen,
the echoes from our sounding-reed would, in the case here supposed,
so blend with the direct sound as to be practically indistinguishable
from it, as the echoes at Villejuif followed the direct sound so hotly,
and vanished so rapidly, that they escaped observation. And as our
sensitive flame, at a distance, failed to be affected by the sounding
body placed close behind the cardboard screen, so, I take it, did the
observers at Montlhéry fail to hear the sounds of the Villejuif gun.

Something further may be done toward the experimental elucidation of
this subject. The facility with which sounds pass through textile
fabrics has been already illustrated,[92] a layer of cambric or calico,
or even of thick flannel or baize, being found competent to intercept
but a small fraction of the sound from a vibrating reed. Such a layer
of calico may be taken to represent a layer of air, differentiated
from its neighbors by temperature or moisture; while a succession of
such sheets of calico may be taken to represent successive layers of
non-homogeneous air.

[Illustration: FIG. 3.]

Two tin tubes (M N and O P, Fig. 3) with open ends were placed so
as to form an acute angle with each other. At the end of one was
the vibrating reed _r_; opposite the end of the other, and in the
prolongation of P O, the sensitive flame _f_, a second sensitive flame
(_f′_) being placed in the continuation of the axis of M N. On sounding
the reed, the direct sound through M N agitated the flame _f′_.
Introducing the square of calico _a b_ at the proper angle, a slight
decrease of the action on _f′_ was noticed, and the feeble echo from
_a b_ produced a barely perceptible agitation of the flame f. Adding
another square, _c d_, the sound transmitted by _a b_ impinged on _c
d_; it was partially echoed, returned through _a b_, passed along P O,
and still further agitated the flame _f_. Adding a third square, _e
f_, the reflected sound was still further augmented, every accession
to the echo being accompanied by a corresponding withdrawal of the
vibrations from _f′_, and a consequent stilling of that flame.

With thinner calico or cambric it would require a greater number of
layers to intercept the entire sound; hence with such cambric we
should have echoes returned from a greater distance, and therefore
of greater duration. Eight layers of the calico employed in these
experiments, stretched on a wire frame and placed close together as a
kind of pad, may be taken to represent a dense acoustic cloud. Such a
pad, placed at the proper angle beyond N, cuts off the sound, which
in its absence reaches _f′_, to such an extent that the flame _f′_,
when not too sensitive, is thereby stilled, while _f_ is far more
powerfully agitated than by the reflection from a single layer. With
the source of sound close at hand, the echoes from such a pad would be
of insensible duration. Thus close at hand do I suppose the acoustic
clouds surrounding Villejuif to have been, a similar shortness of echo
being the consequence.

A further step is here taken in the illustration of the analogy between
light and sound. Our pad acts chiefly by internal reflection. The sound
from the reed is a composite one, made up of partial sounds differing
in pitch. If these sounds be ejected from the pad in their pristine
proportions, the pad is acoustically _white_; if they return with their
proportions altered, the pad is acoustically _colored_.

[Illustration: FIG. 4.]

In these experiments my assistant, Mr. Cottrell, has rendered me
material assistance.[93]


NOTE, _June 3d_.—I annex here a sketch of an apparatus[94] devised by
my assistant, Mr. Cottrell, and constructed by Tisley and Spiller, for
the demonstration of the law of reflection of sound. It consists of two
tubes (A F, B R) with a source of sound at the end R of one of them,
and a sensitive flame at the end F of the other. The axes of the tube
converge upon the mirror, M, and they are capable of being placed so as
to inclose any required angle. The angles of incidence and reflection
are read off on the graduated semicircle. The mirror M is also movable
round a vertical axis.




FOOTNOTES:

[1] It will be borne in mind that the Washington Appendix was published
nearly a year after my Report to the Trinity House.

[2] That is to say, homogeneous air with an opposing wind is frequently
more favorable to sound than non-homogeneous air with a favoring wind.
We had the same experience at the South Foreland.—J. T.

[3] Had this observation been published, it could only have given
me pleasure to refer to it in my recent writings. It is a striking
confirmation of my observations on the Mer de Glace in 1859.

[4] Had I been aware of its existence I might have used the language of
General Duane to express my views on the point here adverted to. See
Chap. VII., pp. 340-341.

[5] This does not seem more surprising than the passage of light, or
radiant heat, through rock salt.

[6] Also “Proceedings of the Royal Society,” vol. xxiii., p. 159, and
“Proceedings of the Royal Institution,” vol. vii., p. 344.

[7] See page 372 of this volume.

[8] The rapidity with which an impression is transmitted through
the nerves, as first determined by Helmholtz, and confirmed by Du
Bois-Reymond, is 93 feet a second.

[9] And long previously by Robert Boyle.

[10] A very effective instrument, presented to the Royal Institution by
Mr. Warren De La Rue.

[11] By directing the beam of an electric lamp on glass bulbs filled
with a mixture of equal volumes of chlorine and hydrogen, I have caused
the bulbs to explode in vacuo and in air. The difference, though not so
striking as I at first expected, was perfectly distinct.

[12] It may be that the gas fails to throw the vocal chords into
sufficiently strong vibration. The _laryngoscope_ might decide this
question.

[13] Poisson, “Mécanique,” vol. ii., p. 707.

[14] To converge the pulse upon the flame, the tube was caused to end
in a cone.

[15] It is recorded that a bell placed on an eminence in Heligoland
failed, on account of its distance, to be heard in the town. A
parabolic reflector placed behind the bell, so as to reflect the
sound-waves in the direction of the long, sloping street, caused
the strokes of the bell to be distinctly heard at all times. This
observation needs verification.

[16] “Encyclopædia Metropolitana,” art. “Sound.”

[17] Placing himself close to the upper part of the wall of the London
Colosseum, a circular building one hundred and thirty feet in diameter,
Mr. Wheatstone found a word pronounced to be repeated a great many
times. A single exclamation appeared like a peal of laughter, while the
tearing of a piece of paper was like the patter of hail.

[18] “Poggendorff’s Annalen,” vol. lxxxv., p. 378; “Philosophical
Magazine,” vol. v., p. 73.

[19] Thin India-rubber balloons also form excellent sound lenses.

[20] For the sake of simplicity, the wave is shown broken at _o′_ and
its two halves straight. The surface of the wave, however, is really a
curve, with its concavity turned in the direction of its propagation.

[21] See “Heat as a Mode of Motion,” chap. iii.

[22] In fact, the prompt abstraction of the motion of heat from the
condensation, and its prompt communication to the rarefaction by the
contiguous luminiferous ether, would prevent the former from ever
rising so high, or the latter from ever falling so low, in temperature
as it would do if the power of radiation was absent.

[23] “Heat a Mode of Motion,” chap. x.

[24] According to Burmeister, through the injection and ejection of air
into and from the cavity of the chest.

[25] On July 27, 1681, “Mr. Hooke showed an experiment of making
musical and other sounds by the help of teeth of brass wheels; which
teeth were made of equal bigness for musical sounds, but of unequal for
vocal sounds.”—Birch’s “History of the Royal Society,” p. 96, published
in 1757.

The following extract is taken from the “Life of Hooke,” which precedes
his “Posthumous Works,” published in 1705, by Richard Waller, Secretary
of the Royal Society: “In July the same year he (Dr. Hooke) showed a
way of making musical and other sounds by the striking of the teeth
of several brass wheels, proportionally cut as to their numbers, and
turned very fast round, in which it was observable that the equal or
proportional strokes of the teeth, that is, 2 to 1, 4 to 3, etc., made
the musical notes, but the unequal strokes of the teeth more answered
the sound of the voice in speaking.”

[26] Galileo, finding the number of notches on his metal to be great
when the pitch of the note was high, inferred that the pitch depended
on the rapidity of the impulses.

[27] When a rough tide rolls in upon a pebble beach, as at Blackgang
Chine or Freshwater Gate in the Isle of Wight the rounded stones are
carried up the slope by the impetus of the water and when the wave
retreats the pebbles are dragged down. Innumerable collisions thus
ensue of irregular intensity and recurrence. The union of these shocks
impresses us as a kind of scream. Hence the line in Tennyson’s “Maud”

  “Now to the scream of a maddened beach dragged down by the wave.”

The height of the note depends in some measure upon the size of the
pebble, varying from a kind of roar—heard when the stones are large—to
a scream; from a scream to a noise resembling that of frying bacon; and
from this, when the pebbles are so small as to approach the state of
gravel, to a mere hiss. The roar of the breaking wave itself is mainly
due to the explosion of bladders of air.

[28] The error of Savart consists, according to Helmholtz, in having
adopted an arrangement in which overtones (described in Chapter III.)
were mistaken for the fundamental one.

[29] “The deepest tone of orchestra instruments is the E of the
double-bass, with 41-1/4 vibrations. The new pianos and organs go
generally as far as C^{1}, with 33 vibrations; new grand pianos may
reach A^{11}, with 27-1/2 vibrations. In large organs a lower octave is
introduced, reaching to C^{11}, with 16-1/2 vibrations. But the musical
character of all these tones under E is imperfect, because they are
near the limit where the power of the ear to unite the vibrations to
a tone ceases. In height the pianoforte reaches to a^{iv}, with 3,520
vibrations, or sometimes to c^{v}, with 4,224 vibrations. The highest
note of the orchestra is probably the d^{v} of the piccolo flute,
with 4,752 vibrations.”—Helmholtz, “Tonempfindungen,” p. 30. In this
notation we start from C, with 66 vibrations, calling the first lower
octave C^{1}, and the second C^{11}; and calling the first highest
octave c, the second c^{1}, the third c^{11}, the fourth c^{12}, etc.
In England the deepest tone, Mr. Macfarren informs me, is not E, but A,
a fourth above it.

[30] It is hardly necessary to remark that the quickest vibrations and
shortest waves correspond to the extreme violet, while the slowest
vibrations and longest waves correspond to the extreme red, of the
spectrum.

[31] Experiments on this subject were first made by M. Buys Ballot
on the Dutch railway, and subsequently by Mr. Scott Russell in this
country. Doppler’s idea is now applied to determine, from changes of
wave-length, motions in the sun and fixed stars.

[32] An ordinary musical box may be substituted for the piano in this
experiment.

[33] To show the influence of a large vibrating surface in
communicating sonorous motion to the air, Mr. Kilburn incloses a
musical box within cases of thick felt. Through the cases a wooden rod,
which rests upon the box, issues. When the box plays a tune, it is
unheard as long as the rod only emerges; but when a thin disk of wood
is fixed on the rod, the music becomes immediately audible.

[34] Chladni remarks (“Akustik,” p. 55) that it is usual to ascribe to
Sauveur the discovery, in 1701, of the nodes of vibration corresponding
to the higher tones of strings; but that Noble and Pigott had made the
discovery in Oxford in 1676, and that Sauveur declined the honor of the
discovery when he found that others had made the observation before him.

[35] The first experiment really made in the lecture was with a bar
of steel 62 inches long, 1-1/2 inch wide, and 1/2 an inch thick, bent
into the shape of a tuning-fork, with its prongs 2 inches apart, and
supported on a heavy stand. The cord attached to it was 9 feet long
and a quarter of an inch thick. The prongs were thrown into vibration
by striking them briskly with two pieces of lead covered with pads and
held one in each hand. The prongs vibrated transversely to the cord.
The vibrations produced by a single stroke were sufficient to carry the
cord through several of its subdivisions and back to a single ventral
segment. That is to say, by striking the prongs and causing the cord
to vibrate as a whole, it could, by relaxing the tension, be caused
to divide into two, three, or four vibrating segments; and then, by
increasing the tension, to pass back through four, three, and two
divisions, to one, _without renewing the agitation of the prongs_. The
cord was of such a character that, instead of oscillating to and fro
in the same plane, each of its points described a circle. The ventral
segments, therefore, instead of being flat surfaces were surfaces of
revolution, and were equally well seen from all parts of the room. The
tuning-forks employed in the subsequent illustrations were prepared for
me by that excellent acoustic mechanician, König, of Paris, being such
as are usually employed in the projection of Lissajou’s experiments.

[36] A string steeped in a solution of the sulphate of quinine, and
illuminated by the violet rays of the electric lamp, exhibits brilliant
fluorescence. When the fork to which it is attached vibrates, the
string divides itself into a series of spindles, and separated from
each other by more intensely luminous nodes, emitting a light of the
most delicate greenish-blue.

[37] The subject of musical intervals will be treated in a subsequent
lecture.

[38] “This quality of sound, sometimes called its register, color, or
timbre.”—Thomas Young, “Essay on Music.”

[39] “Lehre von den Tonempfindungen,” p. 135.

[40] The action of such a string is substantially the same as that of
the siren. The string renders intermittent the current of air. Its
action also resembles that of a _reed_. See Lecture V.

[41] Chladni also observed this compounding of vibrations, and executed
a series of experiments, which, in their developed form, are those of
the kaleidophone. The composition of vibrations will be studied at some
length in a subsequent lecture.

[42] I copy this figure from Sir C. Wheatstone’s memoir; the nodes,
however, ought to be nearer the ends, and the free terminal portions
of the dotted lines ought not to be bent upward or downward. The nodal
lines in the next two figures are also drawn too far from the edge of
the plates.

[43] Under the shoulder of the Wetterhorn I found in 1867 a pool of
clear water into which a driblet fell from a brow of overhanging
limestone rock. The rebounding water-drops, when they fell back, rolled
in myriads over the surface. Almost any fountain, the spray of which
falls into a basin, will exhibit the same effect.

[44] This experiment succeeds almost equally well with a glass tube.

[45] This experiment is more easily executed with hydrogen than with
coal-gas.

[46] Only an extremely small fraction of the fork’s motion is, however,
converted into sound. The remainder is expended in overcoming the
internal friction of its own particles. In other words, nearly the
whole of the motion is converted into heat.

[47] The clear illustrations of organ-pipes and reeds introduced here,
and at page 226, have been substantially copied from the excellent work
of Helmholtz. Pipes opening with hinges, so as to show their inner
parts, were shown in the lecture.

[48] I owe it to this eminent artist to direct attention to his
experiments communicated to the Royal Society in May, 1855, and
recorded in the “Philosophical Magazine” for 1855, vol. x., page 218.

[49] The velocity in glass varies with the quality; the result of each
experiment has therefore reference only to the particular kind of glass
employed in the experiment.

[50] This experiment was first made with a hydrogen-flame by Sir C.
Wheatstone.

[51] A gas-jet, for example, can be ignited five inches above the tip
of a visible gas-flame, where platinum-leaf shows no redness.

[52] “Philosophical Magazine,” March, 1858, p. 235. In the Appendix
Prof. Le Conte’s interesting paper is given _in extenso_. Some years
subsequently Mr. (now Professor) Barrett, while preparing some
experiments for my lectures, observed the action of a musical sound
upon a flame, and by the selection of suitable burners he afterward
succeeded in rendering the flame extremely sensitive. Le Conte, of
whose discovery I informed Mr. Barrett, was my own starting-point.

[53] A gas-bag properly weighted also answers for these experiments.

[54] In the actions described in the case of the blow-pipe and
candle-flames, it was the jet of air issuing from the blow-pipe, and
not the flame itself, that was directly acted on by the external
vibrations.

[55] Numerous modifications of these experiments are possible. Other
inflammable gases than coal-gas may be employed. Mixtures of gases
have also been found to yield beautiful and striking results. An
infinitesimal amount of mechanical impurity has been found to exert a
powerful influence.

[56] Referring to these effects, Helmholtz says: “Die erstaunliche
Empfindlichkeit eines mit Rauch imprägnirten cylindrischen Luftstrahls
gegen Schall ist von Herrn Tyndall beschrieben worden; ich habe
dieselbe bestätigt gefunden. Es ist dies offenbar eine Eigenschaft
der Trennungsflächen die für das Anblasen der Pfeifen von grösster
Wichtigkeit ist.”—“Discontinuirliche Luftbewegung,” Monatsbericht,
April, 1868.

[57] When these two tuning-forks were placed _in contact_ with a
vessel from which a liquid vein issued, the visible action on the vein
continued long after the forks had ceased to be heard.

[58] The experiments on sounding flames have been recently considerably
extended by my assistant, Mr. Cottrell. By causing flame to rub against
flame, various musical sounds can be obtained—some resembling those
of a trumpet, others those of a lark. By the friction of unignited
gas-jets, similar though less intense effects are produced. When the
two flames of a fish-tail burner are permitted to impinge upon a plate
of platinum, as in Scholl’s “perfectors,” the sounds are trumpet-like,
and very loud. Two ignited gas-jets may be caused to flatten out like
Savart’s water-jets. Or they may be caused to roll themselves into two
hollow horns, forming a most instructive example of the _Wirbelflächen_
of Helmholtz. The carbon-particles liberated in the flame rise through
the horns in continuous red-hot or white-hot spirals, which are
extinguished at a height of some inches from their place of generation.

[59] “Essay on Sound,” par. 21.

[60] “Report of the British Association for 1863,” page 105.

[61] A very sagacious remark, as observation proves.

[62] Powerful electric lights have since been established and found
ineffectual.

[63] This is also Sir John Herschel’s way of regarding the subject.
“Essay on Sound,” par. 38.

[64] In all cases nautical miles are meant.

[65] Sir John Herschel gives the following account of Arago’s
observation: “The rolling of thunder has been attributed to echoes
among the clouds; and, if it is considered that a cloud is a collection
of particles of water, however minute, in a liquid state, and therefore
each individually capable of reflecting sound, there is no reason why
very large sounds should not be reverberated confusedly (like bright
lights) from a cloud. And that such is the case has been ascertained
by direct observation on the sound of cannon. Messrs. Arago, Matthieu,
and Prony, in their experiments on the velocity of sound, observed
that under a perfectly clear sky the explosions of their guns were
always single and sharp; whereas, when the sky was overcast, and
even when a cloud came in sight over any considerable part of the
horizon, they were frequently accompanied by a long-continued roll like
thunder.”—“Essay on Sound,” par. 38. The distant clouds would imply
a long interval between sound and echo, but nothing of the kind is
reported.

[66] A friend informs me that he has followed a pack of hounds on a
clear calm day without hearing a single yelp from the dogs; while on
calm foggy days from the same distance the musical uproar of the pack
was loudly audible.

[67] The horn here was temporarily suspended, but doubtless would have
been well heard.

[68] Experiments so important as those of De la Roche ought not to be
left without verification. I have made arrangements with a view to this
object.

[69] The Elder Brethren have already had plans of a new signal-gun laid
before them by the constructors of the War Department.

[70] Described in Chapter V., p. 229.

[71] The figure is but a meagre representation of the fact. The band of
light was two inches wide, the depth of the sinuosities varying from
three feet to zero.

[72] In his admirable experiments on tuning, Scheibler found in the
beats a test of differences of temperature of exceeding delicacy.

[73] Sir John Herschel and Sir C. Wheatstone, I believe, made this
experiment independently.

[74] A subject to be dealt with in Chapter IX.

[75] Nor indeed any of those tones whose rates of vibration are _even_
multiples of the rate of the fundamental.

[76] According to Kolliker, this is the number of fibres in Corti’s
organ.

[77] The comparison employed by Mr. Sedley Taylor appeals with graphic
truth to a mountaineer. Considering, the above curve to represent
a mountain-chain, he calls the discords _peaks_, and the concords
_passes_.

[78] This supposition is of course made for the sake of simplicity, the
real period of oscillation of a pendulum 28 feet long being between two
and three seconds.

[79] This figure corresponds to the interval 15:16. For it and some
other figures, I am indebted to that excellent mechanician, M. König,
of Paris.

[80] For some beautiful figures of this description I am indebted to
Prof. Lyman, of Yale College.

[81] Mr. Sang, of Edinburgh, was, I believe, the first to treat this
subject analytically.

[82] This able paper was the starting-point of the experiments on
sensitive flames, recorded in Chapters VI. and VII.; the researches of
Thomas Young and Savart being the starting-point of the experiments on
smoke-jets and water-jets.—J. T.

[83] “Philosophical Magazine,” section 4, vol. xiii., p. 413, 1857.

[84] “Philosophical Magazine,” section 4, vol. xiv., p. 1, _et seq._,
July, 1857.

[85] “Comptes Rendus” for August, 1853. Also “Philosophical Magazine,”
section 4, vol. vii., p. 186, 1854.

[86] “Philosophical Magazine,” section 4, vol. viii., p. 74, 1854.

[87] “Proceedings of the Royal Institution,” January 15, 1875.

[88] “Researches in Chemistry and Physics,” p. 484.

[89] “Connaissance des Temps,” 1825, p. 370.

[90] See Chapter VII., Part II.

[91] The effect of the air of London is sometimes strikingly evident.

[92] “Philosophical Transactions,” 1874, Part I., p. 208, and Chapter
VII. of this volume.

[93] Since this was written I have sent the sound through fifteen
layers of calico, and echoed it back through the same layers, in
strength sufficient to agitate the flame. Thirty layers were here
crossed by the sound. The sound was subsequently found able to
penetrate two hundred layers of cotton net; a single layer of wetted
calico being competent to stop it.

[94] The cut reached me too late for introduction at the proper place.


INDEX


  A

  Acoustic clouds, echoes from, 325
  —— reversibility, 461-469
  —— transparency, great change of, 323

  Air, process of the propagation of sound through the, 33
  —— propagation of sound through air of varying density, 41
  —— elasticity and density of air, 54
  —— influence of temperature on the velocity of sound, 55
  —— thermal changes produced by the sonorous wave, 60
  —— ratio of specific heats at constant pressure and at constant volume
       deduced from velocities of sound, 62
  —— mechanical equivalent of heat deduced from this ratio, 64
  —— inference that atmospheric air possesses no sensible power
       to radiate heat, 66
  —— velocity of sound in, 69
  —— musical sounds produced by puffs of air, 89
  —— other modes of throwing the air into a state of periodic motion, 91
  —— reflection from heated air, 338

  Albans, St., echo in the Abbey Church of, 50

  Amplitude of the vibration of a sound-wave, 42

  Arago, his report on the velocity of sound, 328

  Atmosphere, reflection from atmospheric air, 335

  Atmosphere, its effect on sound, 365

  Auditory nerve, office of the, 32
  —— manner in which sonorous motion is communicated to the, 33


  B

  Bars, heated, musical sounds produced by, 87
  —— examination of vibrating bars by polarized light, 209

  Beats, theory of, 385
  —— action of, on flame, 387
  —— optical illustration of, 390
  —— various illustrations of, 397
  —— dissonance due to beats, 399, 428

  Bell, experiments on a, placed _in vacuo_, 36-37

  Bells, analysis of vibrations of, 190, 198
  —— fluctuations of, 351-354

  Bourse, at Paris, echoes of the gallery of the, 49

  Burners, fish-tail and bat’s-wing, experiments with, 277


  C

  Carbonic acid, velocity of sound in, 65
  —— reflection from, 335
  —— oxide, velocity of sound in, 69

  Chladni, his tonometer, 168
  —— his experiments on the modes of vibration possible to rods free
       at both ends, 174
  —— his analysis of the vibrations of a tuning-fork, 176

  Chladni, his device for rendering the vibrations visible, 178
  —— illustrations of his experiments, 180

  Chords, musical, 432

  Clang, definition of, 153

  Claque-bois, formation of the, 175, 197

  Clarinet, tones of the, 237

  Clouds, sounds reflected from the, 49

  Corti’s fibres, in the mechanism of the ear, 426

  Cottrell, Mr., his experiment of an echo from flame, 339


  D

  Derham, Dr., on fog-signals, 306

  Diatonic scale, 263

  Difference-tones, 404

  Diffraction of sound, 76 _note_, 78

  Disks, analysis of vibrations of, 187, 198

  Dissonance, cause of, 428
  —— graphic representation of, 430

  Doppler, his theory of the colored stars, 113


  E

  Ear, limits of the range of hearing of the, 106, 118
  —— causes of artificial deafness, 108, 119
  —— mechanism of the ear, 424
  —— consonant intervals in relation to, 426

  Echoes, 48
  —— instances of, 48-50
  —— aërial, production of, 328-329
  —— from flame, 339-340
  —— reputed cloud echoes, 328

  Eolian harp, formation of the, 159-160

  Erith, effects of the explosion of 1864 on the village and church
    of, 53

  Eustachian tube, the, 198
  —— mode of equalizing the air on each side of the tympanic membrane,
      109, 119


  F

  Falsetto voice, causes of the, 239

  Faraday, Mr., his experiment on sonorous ripples, 195

  Fiddle, formation of the, 123
  —— sound-board of the, 123
  —— the iron fiddle, 169, 197
  —— the straw-fiddle, 175, 197

  Flames, sounding, 261, 302
  —— rhythmic character of friction, 260, 301
  —— influence of the tube surrounding the flame, 263, 302
  —— singing-flames, 264, 302
  —— effect of unisonant notes on singing-flames, 275
  —— action of sound on naked flames, 275, 302-304
  —— influence of pitch, 283
  —— extraordinary delicacy of flames as acoustic reagents, 285
  —— the vowel-flame, 286
  —— discovery of a new sensitive flame by Philip Barry, 288
  —— echo from, 339
  —— action of beats on flame, 387

  Flute, tones of the, 237

  Fog, its want of power to obstruct sound, 348
  —— observations in London, 348
  —— fog-signals in, 355
  —— artificial, experiments on, 357

  Fog-signals, researches on the acoustic transparency of the atmosphere
    in relation to the question of, 305
  —— station at South Foreland, 309
  —— instruments and observations, 309
  —— variations of range, 315-316
  —— contradictory results, 317
  —— solution of contradictions, 317-323
  —— extraordinary case of acoustic opacity, 318
  —— in fogs, 355
  —— minimum range of, 371
  —— its position, 370
  —— disadvantages of the gun, 368

  Foreland, South, fog-signal station at, 309
  —— fog at, 354


  G

  Gaines’s Farm, account of the battle of, 324

  Gases, velocity of sound in, 69

  Gun, range of, for fog-signals, 312-313
  —— inferiority to the siren, 369
  —— its disadvantages as a signal, 368


  H

  Hail, doubt as to its power to obstruct sound, 342

  Harmonic tones of strings, 152-154

  Harmony, 410
  —— notions of the Pythagoreans, 411
  —— Euler’s theory, 419
  —— conditions of harmony, 411
  —— influence of overtones on harmony, 429
  —— graphic representations of consonance and dissonance, 431

  Harmonica, the glass, 176

  Hawksbee, his experiment on sounding bodies placed _in vacuo_, 36

  Hearing, mechanism of, 424

  Heat, thermal changes in the air produced by the sonorous wave, 60
  —— ratio of specific heats at constant pressure and at constant volume
       deduced from velocities of sound, 64

  —— mechanical equivalent of heatdeduced from this ratio, 66
  —— inference that atmospheric air possesses no sensible power
       to radiate heat, 68
  —— musical sounds produced by heated bars, 87

  Helmholtz, his theory of resultant tones, 405, 406
  —— —— consonance, 414, 420

  Herschel, Sir John, his article on “Sound” quoted, 50
  —— his account of Arago’s observation on velocity of sound, 328

  Hooke, Dr. Robert, his anticipation of the stethoscope, 75
  —— his production of musical sounds by the teeth of a rotating wheel, 85

  Horn, as an instrument for fog-signalling, 310-311

  Hydrogen, action of, upon the voice, 40
  —— deadening of sound by, 38
  —— velocity of sound in, 55, 69


  I

  Inflection of sound, 53
  —— case of the Erith explosion, 53

  Interference of sonorous waves, 381-382, 407
  —— extinction of sound by sound, 383, 408
  —— theory of beats, 385, 408

  Intervals, optical illustration of, 440


  J

  Joule’s equivalent, 67

  Jungfrau, echoes of the, 49


  K

  Kaleidophone, Wheatstone’s, formation of, 170, 196

  Kundt, M., his experiments, 344



  L

  Laplace, his correction of Newton’s formula for the velocity of sound,
    58-59

  Le Conte, Professor, his observation upon sensitive naked flames,
    274-275
  —— on the influence of musical sounds on the flame of a jet
        of coal-gas, 454-460

  Lenses, refraction of sound by, 51

  Light, analogy between sound and, 45-50
  —— analogy of, 320

  Liquids, velocity of sound in, 69
  —— transmission of musical sounds through, 113
  —— constitution of liquid veins, 291
  —— action of sound on liquid veins, 294, 303-304
  —— Plateau’s theory of the resolution of a liquid vein into drops, 295
  —— delicacy of liquid veins, 300

  Lissajous, M., his method of giving optical expression to the
    vibrations of a tuning-fork, 93
  —— illustration of beats of two tuning-forks, 390


  M

  Mayer, his formula of the equivalent of heat, 66

  Melde, M., his experiments with vibrating strings, 141, 427
  —— and with sonorous ripples, 194

  Metals, velocity of sound transmitted through, 72
  —— determination of velocity in, 73

  Molecular structure, influence of, on the velocity of sound, 73, 212

  Monochord or sonometer, the, 121

  Motion, conveyed to the brain by the nerves, 31
  —— sonorous motion. See SOUND.

  Mouth, resonance of the, 241-242

  Music, physical difference between noise and, 82, 117
  —— a musical tone produced by periodic, noise by unperiodic,
       impulses, 83, 117
  —— production of musical sounds by taps, 84, 117
  —— —— by puffs of air, 89, 117
  —— pitch and intensity of musical sounds, 90, 92, 117
  —— description of the siren, 97
  —— definition of an octave, 105
  —— description of the double siren, 110
  —— transmission of musical sounds through liquids and solids, 113
  —— musical chords, 432-433
  —— the diatonic scale, 432-433
  —— See also HARMONY.

  Musical-box, formation of the, 169, 197


  N

  Nerves of the human body, motion conveyed by the, to the brain, 31
  —— rapidity of impressions conveyed by, 31 _note_

  Newton, Sir Isaac, his calculation of the velocity of sound, 58

  Nodes, 131-132
  —— the nodes not points of absolute rest, 135
  —— nodes of a tuning-fork, 175, 177
  —— rendered visible, 177, 180
  —— a node the origin of vibration, 251

  Noise, physical difference between music and, 82, 117


  O

  Octave, definition of an, 105

  Organ-pipes, 219, 256


  Organ-pipes, vibrations of stopped pipes, 221, 256
  —— —— Pandean pipes, 224
  —— —— open pipes, 224, 256, 260
  —— state of the air in sounding-pipes, 227, 257
  —— reeds and reed-pipes, 234

  Otolites of the ear, 425

  Overtones, definition of, 153
  —— relation of the point plucked to the, 155
  —— corresponding to the vibrations of a rod fixed at both ends, 165
  —— of a tuning-fork, 177
  —— rendered visible, 177, 179
  —— of rods vibrating longitudinally, 207
  —— of the siren, 415
  —— influence of overtones on harmony, 429


  P

  Pandean pipes, the, 224

  Piano-wires, clang of, 158
  —— curves described by vibrating, 160

  Pipes. See ORGAN-PIPES

  Pitch of musical sounds, 90
  —— illustration of the dependence of pitch on rapidity of vibration, 100
  —— relation of velocity to pitch, 211-212
  —— velocity deduced from pitch, 233

  Plateau, his theory of the resolution of a liquid vein into drops, 295

  Pythagoreans, notions of the, regarding musical consonance, 410


  R

  Rain, reputed power of obstructing sound, 341-342
  —— artificial, passage of sound through, 345

  Reeds and reed-pipes, 234

  Reeds, the clarinet and flute, 237

  Reflection of sound, 45
  —— from gases, 332
  —— aërial, proved experimentally, 258

  Refraction of sound, 51

  Resonance, 213
  —— of the air, 213-214, 256
  —— of coal-gas, 216
  —— of the mouth, 242

  Resonators, 213

  Resultant tones, discovery of, 399
  —— conditions of their production, 400
  —— experimental illustrations, 401
  —— theories of Young and Helmholtz, 404, 406

  Reversibility, acoustic, 461-469

  Robinson, Dr., his summary of existing knowledge of fog-signals, 307
  —— Professor, his production of musical sounds by puffs of air, 89

  Rod, vibrations of a, fixed at both ends; its subdivisions and
    corresponding overtones, 165, 197
  —— vibrations of a rod fixed at one end, 166, 197
  —— —— of rods free at both ends, 173, 197


  S

  Savart’s experiments on the influence of sounds on jets of water, 457

  Schultze’s bristles in the mechanism of hearing, 425

  Sea-water, velocity of sound in, 70

  Sensitive flames, 274

  Smoke-jets, action of musical sounds on, 290

  Snow, its reputed power to obstruct sound, 344

  Solids, velocity of sound transmitted through, 69, 72
  —— musical sounds transmitted through, 115-116, 122
  —— determination of velocity in, 211

  Sonometer, or monochord, the, 121

  Sorge, his discovery of resultant tones, 399

  Sound, production and propagation of, 32, 77
  —— experiments on sounding bodies placed _in vacuo_, 36, 77
  —— deadened by hydrogen, 38
  —— action of hydrogen upon the voice, 40
  —— propagation of sound through air of varying density, 40
  —— amplitude of the vibration of a sound-wave, 42, 77
  —— the action of sound compared with that of light and radiant heat, 45
  —— reflection of, 45, 77
  —— echoes, 48-50, 78
  —— sounds reflected from the clouds, 49-50
  —— refraction of sound, 51, 77
  —— diffraction of sound, 53, 78
  —— influence of density and elasticity on velocity, 54, 78
  —— influence of temperature on velocity of sound, 55, 78
  —— determination of velocity, 57, 78
  —— Newton’s calculation, 58, 80
  —— Laplace’s correction of Newton’s formula, 59, 80
  —— thermal changes produced by the sonorous wave, 60, 80
  —— velocity of sound in different gases, 69, 81
  —— —— in liquids and solids, 70-73, 81
  —— influence of molecular structure on the velocity of sound, 73, 81
  —— velocity of sound transmitted through wood, 74, 81
  —— diffraction of, 76 _note_, 78
  —— physical distinction between noise and music, 82
  —— musical sounds periodic, noise unperiodic, impulses, 83
  —— —— produced by taps, 84
  —— —— by puffs of air, 89
  —— pitch and intensity o£ musical sounds, 90
  —— vibrations of a tuning-fork, 91
  —— M. Lissajous’s method of giving optical expression to the vibrations
       of a tuning-fork, 93
  —— definition of the wave-length, 96
  —— description of the siren, 97
  —— determination of the rapidity of vibration, 101
  —— and of the length of the corresponding sonorous wave, 102
  —— various definitions of vibration and of sound-wave, 103
  —— limits of range of hearing of the ear: highest and deepest tones, 106
  —— double siren, 110
  —— transmission of musical sounds through liquids and solids, 113-117
  —— vibrations of strings, 120
  —— the sonometer, or monochord, 121
  —— influence of sound-boards, 123
  —— laws of vibrating strings, 125
  —— direct and reflected pulses, 129
  —— stationary and progressive waves, 130
  —— nodes and ventral segments, 130, 133
  —— application of the results to the vibration of musical strings, 138
  —— M. Melde’s experiments, 141, 427
  —— longitudinal and transverse impulses, 144
  —— laws of vibration thus demonstrated, 148, 162
  —— harmonic tones of strings, 152, 163-164
  —— definitions of timbre, or quality, of overtones and clang, 153, 164
  —— relation of the point of string plucked to overtones, 155-156
  —— vibrations of a rod fixed at both ends; its subdivisions
       and corresponding overtones, 165
  —— —— of a rod fixed at one end, 166
  —— Chladni’s tonometer, 168
  —— Wheatstone’s kaleidophone, 170, 196
  —— vibrations of rods free at both ends, 173, 197
  —— nodes and overtones of a tuning-fork, 175-178, 197
  —— —— rendered visible, 177-179, 197-198
  —— vibrations of square plates, 184, 198
  —— —— of disks and bells, 187-190, 198-199
  —— sonorous ripples in water, 193
  —— Faraday’s and Melde’s experiments on sonorous ripples, 194-195
  —— longitudinal vibrations of a wire, 200
  —— relative velocities of sound in brass and iron, 203, 206
  —— examination of vibrating bars by polarized light, 209
  —— determination of velocity in solids, 211
  —— relation of velocity to pitch, 212
  —— resonance, 213, 253, 256
  —— —— of the air, 213, 256
  —— —— of coal-gas, 216, 256
  —— description of vowel-sounds, 240
  —— Kundt’s experiments on sound-figures within tubes, 244-251, 259
  —— new methods of determining velocity of sound, 247-251, 259
  —— causes that obstruct the propagation of, 306
  —— action of fog upon sound, 307
  —— contradictory results of fog-signalling, 317
  —— solution of contradictions of fog-signalling, 317-318
  —— extraordinary case of acoustic opacity, 318
  —— great change of acoustic transparency, 323
  —— noise of battle unheard, 324
  —— echoes from invisible acoustic clouds, 325, 375
  —— report of Arago on the velocity of, 328
  —— aërial echoes of, 330
  —— demonstration of reflection from gases, 332
  —— reflection from vapors, 336
  —— —— heated air, 337
  —— echo from flame, 340
  —— investigations of the transmission of sound through
       the atmosphere, 341
  —— action of hail and rain, 341
  —— action of snow, 344
  —— passage through tissues, 345
  —— —— artificial showers, 346
  —— action of fog, 347
  —— fluctuations of bells, 351-354
  —— action of wind, 361
  —— atmospheric selection, 365
  —— law of vibratory motions in water and air, 377, 407
  —— superposition of vibrations, 381
  —— interference and coincidence of sonorous waves, 382-383, 407
  —— extinction of sound by sound, 384, 407
  —— theory of beats, 385, 408
  —— action of beats on flame, 387
  —— optical illustration of beats, 390, 408
  —— various illustrations of beats, 397
  —— resultant tones, 399, 409
  —— —— conditions of their production, 400
  —— —— experimental illustrations, 401
  —— —— theories of Young and Helmholtz, 405-406
  —— difference-tones and summation-tones, 405
  —— combination of musical sounds, 410
  —— sympathetic vibrations, 421
  —— mode in which sonorous motion is communicated to the auditory
       nerve, 426

  Sound-boards, influence of, 123-124

  Sound-figures within tubes, M. Kundt’s experiments with, 244-251

  Stars, Doppler’s theory of the colored, 113

  Steam-siren, description of, 309
  —— conclusive opinion as to its power for a fog-signal, 370

  Stethoscope, Dr. Hook’s anticipations of the, 74

  Stokes, Professor, his explanation of the action of sound-boards, 124
  —— his explanation of the effect of wind on sound, 363

  Straw-fiddle, formation of the, 175, 197

  Strings, vibration of, 120
  —— laws of vibrating strings, 125
  —— combination of direct and reflected pulses, 129
  —— stationary and progressive waves, 130
  —— nodes and ventral segments, 130-133
  —— experiments of M. Melde, 141, 427
  —— longitudinal and transverse impulses, 144
  —— laws of vibration thus demonstrated, 148, 162
  —— harmonic tones of strings, 152, 163-164
  —— timbre, or quality, and overtones and clang, 153, 164
  —— Dr. Young’s experiments on the curves described by vibrating
       piano-wires, 160
  —— longitudinal vibrations of a wire, 200
  —— —— with one end fixed, 204
  —— —— with both ends free, 206

  Summation-tones, 405

  Siren, description of the, 97
  —— sounds, description of the, 97
  —— its determination of the rate of vibration, 101
  —— the double siren, 110, 411-412
  —— the echoes of the, 330


  T

  Tartini’s tones, 399. See RESULTANT TONES.

  Timbre, or quality of sound, definition of, 153

  Tisley, Mr., his apparatus for the compounding of rectangular
    vibrations, 447

  Toepler, M., his experiment on the rate of vibration of the flame, 268

  Tonometer, Chladni’s, 168

  Trumpets, range of, for fog-signals, 313

  Tuning-fork, vibrations of a, 93
  —— M. Lissajous’s method of giving optical expression
       to the vibrations, 93
  —— strings set in motion by tuning-forks, 142
  —— vibrations of the tuning-forks as analyzed by Chladni, 176
  —— nodes and overtones of a, 171, 197
  —— interference of waves of the, 395


  V

  Vapors, reflection from, 336

  Velocity of sound, influence of density and elasticity on, 54
  —— influence of temperature on, 55
  —— determination of, 57
  —— Newton’s calculation, 58
  —— velocity of sound in different gases, 69
  —— and transmitted through various liquids and solids, 70-73
  —— relative velocities of sound in brass and iron, 203, 206
  —— relation of velocity to pitch, 212
  —— velocity deduced from pitch, 233

  Ventral segments, 132

  Vertical jets, action of sound on, 297, 304

  Vibrations of a tuning-fork, 93
  —— method of giving optical expression to the vibrations
       of a tuning-fork, 93
  —— illustration of the dependence of pitch on rapidity
       of vibration, 101
  —— the rate of vibration determined by the siren, 101
  —— determination of the length of the sound-wave, 102, 118
  —— various definitions of vibrations, 103, 118
  —— vibrations of strings, 120
  —— laws of vibrating strings, 125
  —— direct and reflected pulses illustrated, 129
  —— application of the result to the vibration of musical strings, 138
  —— M. Melde’s experiments on the vibration of strings, 141, 427
  —— longitudinal and transverse impulses, 144
  —— Vibrations of a red-hot wire, 147
  —— laws of vibration thus demonstrated, 148, 162
  —— new mode of determining the law of vibration, 148, 150
  —— harmonic tones of strings, 152, 163
  —— vibrations of a rod fixed at both ends; its subdivisions
       and corresponding overtones, 165
  —— vibrations of a rod fixed at one end, 166
  —— Chladni’s tonometer, 168
  —— Wheatstone’s kaleidophone, 170
  —— vibrations of rods free at both ends, 173
  —— nodes and overtones rendered visible, 177-179
  —— vibrations of square plates, 184
  —— —— of disks and bells, 187-190
  —— longitudinal vibrations of a wire, 200, 255
  —— —— with one end fixed, 204
  —— —— with both ends free, 206
  —— divisions and overtones of rods, vibrating longitudinally, 207
  —— examination of vibrating bars by polarized light, 209
  —— vibrations of stopped pipes, 221
  —— —— of open pipes, 224
  —— a node the origin of vibration, 251
  —— law of vibratory motions in water and air, 377
  —— superposition of vibrations, 381
  —— theory of beats, 385
  —— sympathetic vibrations, 421
  —— M. Lissajous’s method of studying musical vibrations, 433
  —— apparatus for the compounding of rectangular vibrations, 447

  Violin, formation of the, 123
  —— sound-board of the, 123
  —— the iron fiddle, 169, 197

  Voice, human, action of hydrogen upon the, 40
  —— sonorous waves of the, 104
  —— description of the organ of voice, 238
  —— causes of the roughness of the voice in colds, 239
  —— causes of the squeaking falsetto voice, 239
  —— Müller’s imitation of the action of the vocal chords, 240
  —— formation of the vowel-sounds, 240-241
  —— synthesis of vowel-sounds, 242-243

  Vowel-flame, the, 286

  Vowel-sounds, formation of the, 240
  —— synthesis of, 242-243


  W

  Water-Waves, stationary, phenomena of, 136

  Water, velocity of sound in, 70
  —— transmission of musical sounds through, 113
  —— effects of musical sounds on jets of water, 291-292
  —— delicacy of liquid veins, 294
  —— theory of the resolution of a liquid vein into drops, 295, 304
  —— law of vibratory motions in water, 377

  Wave-length, definition of, 96
  —— determination of the length of the sonorous wave, 102

  Wave-length, definition of sonorous wave, 104

  Wave-motion, illustration, 128-133
  —— stationary waves, 133
  —— law of, 377

  Waves of the sea, causes of the roar of the breaking, 88 _note_

  Weber, Messrs., their researches on wave-motion, 133

  Wetterhorn, echoes of the, 49

  Wheatstone, Sir Charles, his kaleidophone, 170
  —— his apparatus for the compounding of rectangular vibrations, 448

  Whistles, range of, for fog-signals, 313

  Wind, effect on sound, 361

  Wires. See STRINGS

  Wood, velocity of sound transmitted through, 74
  —— musical sounds transmitted through, 115
  —— the claque-bois, 175
  —— determination of velocity in wood, 211

  Woodstock Park, echoes in, 56


  Y

  Young, Dr. Thomas, his proof of the relation of the point of a string
    plucked to the overtones, 155
  —— on the curves described by vibrating piano-wires, 160-161
  —— his theory of resultant tones, 404