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Elementary treatise on physics online

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MM. Bravais and Martins found, in 1844, that sound travelled with the
same velocity from the base, to the summit of the Faulhorn, as from the
summit to the base.

Mallet has investigated the velocity of the transmission of sound in
various rocks, and finds that it is as follows ;

,. . . 825 ft. in a second.

Contorted, stratified quartz and slate rock . 1088

Discontinuous granite 1306 ,,

Solid granite . . . . . . 1664





- 218] Velocity of Sound in Gases. 1 7 3



218. Formulae for calculating: the velocity of sound in gases.
For calculating the velocity of sound in gases Newton gave a rule equiva-
lent to the formula



in which v represents the velocity of the sound or the distance it travels
in a second, e the elasticity of the gas, and d its density.

This formula expresses that the velocity of the propagation of sound
in gases is directly as the square root of the elasticity of the gas, and in-
versely as the square root of its density. It follows that the velocity of
sound is the same under any pressure, for although the elasticity increases
with greater pressure, the density increases in the same ratio. At Quito,
where the mean pressure is only 21*8 inches, the velocity is the same as
at the sea level, provided the temperature is the same.

If g be the force of gravity, h the barometric height reduced to the
temperature zero, and o the density of mercury, also at zero, then for
a gas under the atmospheric pressure, and for zero, e -gh5 ; Newton's
formula accordingly becomes



/ gh*_
V a



Now if we suppose the temperature of a gas to increase from o to /,
its volume will increase from unity at zero to I +at at /, a being the
coefficient of expansion of the gas. But the density varies inversely as
the volume, therefore d becomes d - (\ + at}. Hence



The values of 77, obtained by this formula, are less than the experimen-
tal results. Laplace assigned as a reason for this discrepancy the heat
produced by pressure in the condensed waves ; and, by considerations
based on this idea, Poisson and Biot have found that Newton's formula

ought to be written v= / &i(iraf)' t c being the specific heat of the
\r d c

gas for a constant pressure, and c f its specific heat for a constant volume
(see Book VI.). When thus modified the results calculated by the for-
mula agree with the experimental results.

The physical reason for introducing the constant -^ into the equation

c

for the velocity of sound may be understood from the following con-
siderations. We have already seen that sound is propagated in air by a
series of alternate condensations and rarefactions of the layers. At each
condensation heat is evolved, and this heat increases the elasticity, and
thus the rapidity, with which each condensed layer acts on the next ; but,
in the rarefaction of each layer, the same amount of heat disappears as
was developed by the condensation, and its elasticity is diminished by the
cooling. The effect of this diminished elasticity of the cooled layer is



1/4 Acoustics. [218-

the same as if the elasticity of an adjacent wave had been increased, and
the rapidity with which this latter would expand upon the dilated wave
would be greater. Thus, while the average temperature of the air is
unaltered, both the heating which increases the elasticity and the chilling
which diminishes it concur in increasing velocity.

Knowing the velocity of sound, we can calculate approximately the
distance at which it is produced. Light travels with such velocity that
the flash or the smoke accompanying the report of a gun may be con-
sidered to be seen simultaneously with the explosion. Counting then
the number of seconds which elapse between seeing the flash and hear-
ing the sound, and multiplying this number by 1125, we get the distance
in feet at which the gun is discharged. In the same way the distance of
thunder may be estimated.

219. Velocity of sound in various gases. Approximately the same
results have been obtained for the velocity of sound in air, by another
method by which the velocity in other gases could be determined. As
the wave length A, is the distance which sound travels during the time of
one oscillation, that is n of a second, the velocity of sound or the distance
traversed in a second is i> = n\. Now the length of an open pipe is
half the wave length of the fundamental note of that pipe ; and that of
a closed pipe is a quarter of the wave length (259). Hence if we know
the number of vibrations of the note emitted by any particular pipe,
which can be easily ascertained by means of the syren, and we know the
length of this pipe, we can calculate v. Taking the temperature into
account, Wertheirrrfound 1086 feet for the velocity of sound at zero.

Further, since in different gases which have the same elasticity, but
differ in density, the velocity of sound varies inversely as the square
root of the density, knowing the velocity of sound in air, we may calculate
it for other gases ; thus, in hydrogen it will be



This number cannot be quite accurate, for the coefficient differs

somewhat in different gases. And when pipes were sounded with
different gases, and the number of vibrations of the notes multiplied
with twice the length of the pipe, numbers were obtained which differed
from those calculated by the above formula. When, however, the calcu-

lation was made, introducing for each gas the specia value of , the

c \
theoretical results agreed very well with the observed ones.

By the above method the following values have been obtained :

Carbonic acid ....... 856 ft. in a second.

Oxygen ........ 1040

Air ......... 1093

Carbonic oxide ...... 1106

Hydrogen ....... 4163



-221] Velocity of Sound in Liquids and in Solids. 175

220. Doppler's principle. When a sounding body approaches the
ear, the tone perceived is somewhat higher than the true one ; but if the
source of sound recedes from the ear, the tone perceived is lower. The
truth of this, which is known as Dopplefs principle, will be apparent from
the following considerations : When the source of sound and the ear
are at rest, the ear perceives n waves in a second ; but if the ear ap-
proaches the sound, or vice versa, it perceives more ; just as a ship
meets more waves when it ploughs through them than if it is at rest.
Conversely, the ear receives a smaller number when it recedes from the
source of sound. The effect in the first case is as if the sounding body
emitted more vibrations in a second than it really does, and in the
second case fewer. Hence in the first case the note appears higher ; in
the second case lower.

If the distance which the ear traverses in a second towards the source
of sound (supposed to be stationary) is s feet, and the wave length of the

particular tone is A feet, then there are waves in a second ; or also ,

A C

for A = , where c is the velocity of sound (216). Hence the ear receives

not only the n original waves, but also -^L in addition. Therefore the
number of vibrations which the ear actually perceives is

' = + _^L=(i+ -L)

c c

for an ear which approaches a tone ; and by similar reasoning it is

'



for an ear receding from a tone.

To test Doppler's theory Buys Ballot stationed trumpeters on the
Utrecht railway, and also upon locomotives, and had the height of the
approaching or receding tones compared with stationary ones by musicians.
He thus found both the principle and the formula fully confirmed.

221. Velocity of sound in liquids and in solids. The velocity of
sound in water was investigated in 1827 by Colladon and Sturm. They
moored two boats at a known distance in the lake of Geneva. The first
supported a bell immersed in water, and a bent lever provided at one end
with a hammer which struck the bell, and at the other with a lighted wick,
so arranged that it ignited some powder the moment the hammer struck
the bell. To the second boat was affixed an ear-trumpet, the bell of
which was in water, while the mouth was applied to the ear of the
observer, so that he could measure the time between the flash of light
and the arrival of sound by the water. By this method the velocity was
found to be 4708 feet in a second at the temperature 8-1, or four times
as great as in air.

The velocity of sound, which is different in different liquids, can be
calculated by a formula analogous to that given above (219) as applicable



176 Acoustics. [221-

to gases. In this way are obtained the number given in the following
table. As in the case of gases, the velocity varies with the temperature,
which is therefore appended in each case :

River water (Seine) . '. . I3C. = 4714 ft. in a second

, , . . . 30 5013

Artificial sea- water . . . 20 = 4761

Solution of common salt . .18 =5132 ,,

chloride of calcium . 23 - 6493

Absolute alcohol . . . . 23 = 3804

Turpentine . . . ., . 24 3976

Ether ...... = 3801

As a general rule, this elasticity of solids, as compared with the density,
is greater than that of liquids, and consequently the propagation of sound
is more rapid.

The difference is well seen in an experiment by M. Biot, who found
that when a bell was struck by a hammer, at one end of an iron tube
3120 feet long, two sounds were distinctly heard at the other end. The
first of these was transmitted by the tube itself with a velocity x; and the
second by the enclosed air with a known velocity a. The interval between
the sounds was 2*5 seconds. The value of x, obtained from the equation



shows that the velocity of sound in the tube is about 9 times as great as
that in air.

To this class of phenomena belongs the fact that if the ear is held
again s t a rock in which a blasting is being made at a distance, two
distinct reports are heard, one transmitted through the rock to the ear,
and the other transmitted through the air.

The velocity of sound in other solids has also been determined theo-
retically by Wertheim, by means of their coefficient of elasticity.

The following table gives the velocity, expressed in feet per second :

Lead . . ' . . 4030 Pine ..... 10900

Gold ..... 5717 Oak ..... 12622

Silver ..... 8553 Ash ..... I33'4

Copper .... 11666 - Elm . . . .. 13516

Steel wire .... 15470 Fir ..... 15218

Iron ..... 16822 Aspen .... 16677

The velocity in the direction of the fibres was greater than across them.

A direct method of determining the velocity of sound in solids, gases,
and vapours will be described further on.

222. Reflection of sound. So long as sonorous waves are not ob-
structed in their motion, they are propagated in the form of concentric
spheres ; but, when they meet with an obstacle, they follow the general
law of elastic bodies , that is, they return upon themselves, forming new
concentric waves, which seem to emanate from a second centre on the



-223]



Echoes and Resonances.



177



other side of the obstacle. This phenomenon constitutes the reflection
of sound.

Fig. 164 represents a series of incident waves reflected from an ob-
stacle, PQ. Taking, for example, the incident wave MCDN, emitted from




the centre A, the corresponding reflected wave is represented by the arc,
CKD, of a circle, whose centre a is as far beyond the obstacle PQ as A
is before it.

If any point, C, of the reflecting surface be joined to the sonorous
centre, and if the perpendicular CH be let fall on the surface of this body,
the angle ACH is called the angle of incidence, and the angle BCH.
formed by the prolongation of aC is the angle of reflection.

The reflection of sound is subject to the two following laws :

I. The angle of reflection is equal to the angle of incidence.

II. The incident sonorous ray and the reflected ray are in the same
plat'e perpendicular to the reflecting surface.

From these laws it follows that the wave which in the figure is pro-
pagated in the direction AC, takes the direction CB after reflection, so
that an observer placed at B hears, besides the sound proceeding from
the point A, a second sound, which appears to come from C.

The laws of the reflection of sound are the same as those for light and
radiant heat, and may be demonstrated by similar experiments. One of
the simplest of these is made with conjugate mirrors (see chapter on
Radiant Heat) ; if in the focus of one of these mirrors a watch is placed
the ear placed in the focus of the second mirror hears the ticking very
distinctly, even when the mirrors are at a distance of 12 or 13 yards.

223. Echoes and resonances. An echo is the repetition of a sound
in the air, caused by its reflection from some obstacle.

A very sharp quick sound can produce an echo when the reflecting
surface is 55 feet distant, but for articulate sounds at least double that
distance is necessary, for it may be easily shown that no one can pro-
nounce or hear distinctly more than five syllables in a second. Now, as
the velocity of sound at ordinary temperatures may be taken at 1125 feet
in a second, in a fifth of that time sound would travel 225 feet. If the

I 3



1 78 Acoustics. [223-

reflecting surface is 112*5 feet distant in going and returning, sound would
travel through 225 feet. The time which elapses between the articu-
lated and the reflected sound would, therefore, be a fifth of a second, the
two sounds would not interfere, and the reflected sound would be dis-
tinctly heard. A person speaking with a loud voice in front of a reflector,
at a distance of 112-5 f eet > can only distinguish the last reflected syllable :
such an echo is said to be monosyllabic. If the reflector were at a dis-
tance of two or three times 112-5 f eet > the echo would be dissyllabic,
trisyllabic, and so on.

When the distance of the reflecting surface is less than 112-5 ^ eet tne
direct and the reflected sound are confounded. They cannot be heard
separately but the sound is strengthened. This is what is called reso-
nance, and is often observed in large rooms. Bare walls are very reso-
'nant ; but tapestry and hangings, which are bad reflectors, deaden the
sound.

Multiple echoes are those which repeat the same sound several times :
this is the case when two opposite surfaces (for example, two parallel
walls) successively reflect sound. There are echoes which repeat the
same sound 20 or 30 times. Ah echo in the chateau of Simonetta, in
Italy, repeats a sound 30 times. At Woodstock there is one which
repeats from 17 to 20 syllables.

As the laws of reflection of sound are the same as those of light
and heat, curved surfaces produce acoustic foci like the luminous and
calorific foci produced by concave reflectors. If a person standing under
the arch of a bridge speaks with his face turned towards one of the piers,
the sound is reproduced near the other pier with such distinctness that
a conversation can be kept up in a low tone, which is not heard by any
one standing in the intermediate spaces.

There is a square room with an elliptical ceiling, on the ground floor ot
the Conservatoire des Arts et Metiers, in Paris, which presents this
phenomenon in a remarkable degree when persons stand in the two foci
of the ellipse.

It is not merely by solid surfaces, such as walls, rocks, ship's sails,
etc., that sound is reflected. It is also reflected by clouds, and it has even
been shown by direct experiment that a sound in passing from a gaseous
medium of one density into another is reflected at the surface as it would
be against a solid surface.

Whispering galleries are formed of smooth walls having a continuous
curved form. The mouth of the speaker is presented at one point, and
the ear of the hearer at another and distant point. In this case, the
sound is successively reflected from one point to the other until it reaches
the ear.

Different parts of the earth's surface are unequally heated by the sun,
owing to the shadows of trees, evaporation of water, and other causes, so
that in the atmosphere there are numerous ascending and descending
currents of air of different density. Whenever a sonorous wave passes
from a medium of one density into another it undergoes partial reflection,,
which, though not strong enough to form an echo, distinctly weakens



-224] Refraction of Sound. I / 9

the direct sound. This is doubtless the reason, as Humboldt remarks,
why sound travels further at night than at daytime ; even in the South
American forests, where the animals, which are silent by day, fill the
atmosphere in the night with thousands of confused sounds.

It has generally been considered that fog in the atmosphere is a great
deadener of sound, it being a mixture of air and globules of water, at
each of the innumerable surfaces of contact a portion of the vibration is
lost. The evidence as to the influence of this property is conflicting ; recent
researches of Tyndall show that a white fog, or snow,, or hail, are not im-
portant obstacles to the transmission of sound, but that aqueous vapour is.
Experiments made on a large scale, in order to ascertain ' the best
form of fog-signals, gave some remarkable results.

On some days which optically were quite clear, certain sounds could not
be heard at a distance far inferior to that at which they could be heard even
during a thick haze. Tyndall ascribes this result to the presence in the
atmosphere of aqueous vapour ; which forms in the air innumerable striae
that do not interfere with its optical clearness, but render it acoustically
turbid.

These conclusions first drawn from observations have been verified
by laboratory experiments. Tyndall has shown that a medium consisting
of alternate layers of a light and heavy gas deadens so.und, and also that
a medium consisting of alternate strata of heated and ordinary air exerts
a similar influence. The same is the case with an atmosphere containing
the vapours of volatile liquids. So long as the continuity of air is pre->
served, sound has great power of passing through the interstices of solids ;
thus it will pass through twelve folds of a dry silk handkerchief, but is
stopped by a single layer if it is wetted.

224. Refraction of sound. It will be found in the sequel that refrac-
tion is the change of direction which light and heat experience on passing
from one medium to another. Sondhauss has found that sonorous waves
are refracted like light and heat. He constructed gas lenses, by filling
spherical or lenticular collodion envelopes with carbonic acid. With
envelopes of paper or of goldbeater's skin the refraction of sound is not
perceptible.

Sondhauss cut equal segments out of a large collodion balloon, and
fastened them on the two sides of a sheet iron ring a foot in diameter, so
as to form a hollow biconvex lens about 4 inches thick in the centre.
This was filled with carbonic acid, and a watch was placed in the direc-
tion of the axis : the point was then sought, on the other side of the lens
at which the sound was most distinctly heard. It was found that when
the ear was removed from the axis, the sound was scarcely perceptible ;
but that at a certain point on the axial line it was very distinctly heard.
Consequently, the sonorous waves in passing from the lens had converged
towards the axis, their direction had been changed ; in other words, they
had been refracted.

The refraction of sound may be easily demonstrated by means of one
of the very thin india-rubber balloons used as children's toys, inflated by



Acoustics. [224-

carbonic acid. If the balloon be filled with hydrogen, no focus is detected ;
it acts like a convex lens, and the divergence of the rays is increased
instead of their being converged to the ear.

225. Speaking- trumpet. Ear trumpet. These instruments are
based both on the reflection of sound and on its conductibility in tubes.

The speaking trumpet, as its name implies, is used to render the voice
audible at great distances. It consists of a slightly conical tin or brass
tube (fig. 165), very much wider at one end (which is called the bell}, and




Fig. 165.

provided with a mouthpiece at the other. The larger the dimensions of
this instrument the greater is the distance at which the voice is heard.
Its action is usually ascribed to the successive reflections of sonorous
waves from the sides of the tube, by which the waves tend more and
more to pass in a direction parallel to the axis of the instrument. It has,
however, been objected to this explanation, that the sounds emitted by
the speaking trumpet are not stronger solely in the direction of the axis,
but in all directions, that the bell would not tend to produce parallelism
in the sonorous wave, whereas it certainly exerts considerable influence
in strengthening the sound. It must be said that no satisfactory explana-
tion has been gdven of the effect of the bell.

The ear trumpet is used by persons who are hard of hearing. It is
essentially an inverted speaking trumpet, and consists of a conical metallic
tube, one of whose extremities, terminating in a bell, receives the sound,
while the other end is introduced into the ear. This instrument is the
reverse of the speaking trumpet. The bell serves as a mouthpiece ; that
is, it receives the sound coming from the mouth of the person who
speaks. These sounds are transmitted by a series of reflections to the
interior of the trumpet, so that the waves which would become greatly
developed, are concentrated on the auditory apparatus, and produce a far
greater effect than divergent waves would have done.




Fig. 166.



Fig. 167.



226. Stethoscope. One of the most useful applications of acoustical
principles is the stethoscope. Figs. 166, 167 represent an improved form






-227] Measurement of the Number of Vibrations. 1 8 1

of this instrument devised by Konig. Two sheets of caoutchouc, c and a t
are fixed to the circular edge of a hollow metal hemisphere ; the edge is
provided with a stopcock, so that the plates can be inflated, and then
present the appearance of a double convex lens as represented in section
in fig. 1 66. To a tubulure on the hemisphere is fixed a caoutchouc tube
terminated by horn or ivory, o, which is placed in the ear (fig. 167).

When the membrane of the stethoscope is applied to the chest of a sick
person the beating of the heart and the sounds of respiration are trans-
mitted to the air in the chamber CA, and from thence to the ear by
means of the flexible tube. If several tubes are fixed to the instrument
as many observers may simultaneously auscultate the same patient.



CHAPTER II.

MEASUREMENT OF THE NUMBER OF VIBRATIONS.

227. Savart's apparatus. Savarfs tocthed wheel, so called from the
name of its inventor, is an apparatus by which the absolute number of
vibrations corresponding to a given note can be determined. It consists
of a solid oak frame in which there are two wheels, A and B (fig. 168);




Fig- '68ft

the larger wheel, A, is connected with fce^toothed wheel by means of a
strap and a multiplying wheel, thereby causing the toothed wheel to
revolve with great velocity ; a card, E, is fixed on the frame, and, in
revolving, the toothed wheel strikes against it, and causes it to vibrate.
The card being struck by each tooth, makes as many vibrations as there
are teeth. At the side of the apparatus there is an indicator, H, which
gives the number of revolutions of the wheel, and consequently the
number of vibrations in a given time.

When the wheel is moved slowly, the separate shocks against the
card are distinctly heard ; but if the velocity is gradually increased, the



182



Acoustics.



[227-



sound becomes higher and higher. Having obtained the sound whose
number of vibrations is to be determined, the revolution of the wheel is
continued with the same velocity for a certain number of seconds. The
number of turns of the toothed wheel B is then read off on the indicator,
and this multiplied by the number of teeth in the wheel gives the total
number of vibrations. Dividing this by the corresponding number of
seconds, the quotient gives the number of vibrations per second for the
given sound.

228. Syren. The syren is an apparatus which, like Savart's wheel, is
used to measure the number of vibrations of a body in a given time. The
name ' syren ' was given to it by its inventor, Cagniard Latour, because it
yields sounds under water.

It is made entirely of brass. Fig. 169 represents it fixed on the table
of a bellows, by which a continuous current of air can be sent through it.
Figs. 170 and 171 show the internal details. The lower part consists of




Fig. 169.



Fig. 171.



a cylindrical box, O, closed by a fixed plate, B. On this plate a vertical
rod, T, rests, to which is fixed a disc, A, moving with the rod. In the



Online LibraryAdolphe GanotElementary treatise on physics → online text (page 19 of 94)