V in diffe

rent gases.

Experiments for determining V in air.

of sound.

=

is

1099 982

or

545 491

= 1.110 ft.

√y? that, at a

21. If the value of y were the same for different gases, it is obvious from formula V a given γ temperature, the velocities of sound in those gases would be to each other inversely as the square roots of their densities. Regnault has found that this is so for common air, carbonic acid, nitrous oxide, hydrogen and ammoniacal gas (though less so as regards the two last).

22. The experimental determination of the velocity of sound in air has been carried out by ascertaining accurately the time intervening between the flash and report of a gun as observed at a given distance, and dividing the distance by the time. A discussion of the many experiments conducted on this principle in various countries and at various periods, by Van Der Kolk (Lond. and Edin. Phil. Mag., July 1865), assigns to the velocity of sound in dry air at 32° Fahr., 1091 ft. 8 in. per second, with a probable error of ± 3.7 ft.; and still more recently (in 1871) Mr Stone, the Astronomer Royal at the Cape of Good Hope, has found 1090-6 as the result of careful experiments by himself there. The coincidence of these numbers with that we have already obtained theoretically sufficiently establishes the general accuracy of the theory.

V depends 23. Still it cannot be overlooked that the formula for on intensity V is founded on assumptions which, though approximately, are not strictly correct. Thus, the air is not a perfect gas, nor is the variation of elastic force, caused by the passage through it of a wave of disturbance always very small in comparison with the elastic force of the undisturbed air. Earnshaw (1858) first drew attention to these points, and came to the conclusion that the velocity of sound increases with its loudness, that is, with the violence of the disturbance. In confirmation of this statement, he appeals to a singular fact, viz., that, during experiments made by. Captain Parry, in the North Polar Regions, for determining the velocity of sound, it was invariably found that the report of the discharge of cannon was heard, at a distance of 2 miles, perceptibly earlier than the sound of the word fire, which, of course, preceded the discharge.

V depends on the pitch of sound.

As, in the course of propagation in unlimited air, there is a gradual decay in the intensity of sound, it would fol low that the velocity must also gradually decrease as the sound proceeds onwards. This curious inference has been verified experimentally by Regnault, who found the velocity of sound to have decreased by 2.2 ft. per second in passing from a distance of 4000 to one of 7500 feet.

24. Among other interesting results, derived by the accurate methods adopted by Regnault, but which want of space forbids us to describe, may be mentioned the dependence of the velocity of sound on its pitch, lower notes being, cat. par., transmitted at a more rapid rate than higher ones. Thus, the fundamental note of a trumpet travels faster than its harmonies.

D

(IV.)

(which is the length in feet of a column that would be increased 1 foot by the weight of 1 cubic foot) by l, V = √gl (V.)

which shows that the velocity is that due to a fall through

1 η

=

Ex. 1. For water,

=

(VI.

and

20,000 very nearly; H=34 ft. V in water. and hence V = 4680 feet.

This number coincides very closely with the value obtained, whether by direct experiment, as by Colladon and Sturm on the Lake of Geneva in 1826, who found 4708, or by indirect means which assign to the velocity in the water of the River Seine at 59° Fahr. a velocity of 4714 ft. (Wertheim).

Ex. 2. For iron. Let the weight necessary to double V in iron. the length of an iron bar be 4260 millions of lbs. on the square foot. Then a length will be extended to +1 by 4260 millions lbs. a force of on the sq. ft. This, therefore, 1 by our definition of 7, must be the weight of a cubic foot of the iron. Assuming the density of iron to be 7.8, and 62-32 lbs. as the weight of a cubic foot of water, we get 7.8 x 62.32 or 486 lbs. as the weight of an equal bulk of 4260 millions iron. Hence = 486 and 7:

26. Biot, about 50 years ago, availed himself of the Experigreat difference in the velocity of the propagation of sound mental dethrough metals and through air, to determine the ratio of terminathe one velocity to the other. A bell placed near one ex- tion of V tremity of a train of iron pipes forming a joint length of upwards of 3000 feet, being struck at the same instant as the same extremity of the pipe, a person placed at the other extremity heard first the sound of the blow on the pipe, conveyed through the iron, and then, after an interval

sound.

heard by the same ear at the distances OA and OB are to each other as OB2 to OA2.

Influence of 30. In order to verify the above law when the atmodiminished sphere forms the intervening medium, it would be necessary density of to test it at a considerable elevation above the earth's the air on intensity of surface, the ear and the source of sound being separated by air of constant density. As the density of the air diminishes, we should then find that the loudness of the sound at a given distance would decrease, as is the case in the air-pump experiment previously described. This arises from the decrease of the quantity of matter impinging on the ear, and the consequent diminution of its vis-viva. The decay of sound due to this cause is observable in the rarefied air of high mountainous regions. De Saussure, the celebrated Alpine traveller, mentions that the report of a pistol at a great elevation appeared no louder than would a small cracker at a lower level.

Laws of

But it is to be remarked that, according to Poisson, when air-strata of different densities are interposed between the source of sound and the ear placed at a given distance, the intensity depends only on the density of the air at the source itself; whence it follows that sounds proceeding from the surface of the earth may be heard at equal distances as distinctly by a person in a floating balloon as by one situated on the surface itself; whereas any noise originating in the balloon would be heard at the surface as faintly as if the ear were placed in the rarefied air on a level with the balloon. This was exemplified during a balloon ascent by Glaisher and Coxwell, who, when at an elevation of 20,000 feet, heard with great distinctness the whistle of a locomotive passing beneath them.

PART III.

Reflexion and Refraction of Sound.

31. When a wave of sound travelling through one refraction. medium meets a second medium of a different kind, the vibrations of its own particles are communicated to the particles of the new medium, so that a wave is excited in the latter, and is propagated through it with a velocity dependent on the density and elasticity of the second medium, and therefore differing in general from the previous velocity. The direction, too, in which the new wave travels is different from the previous one. This change of direction is termed refraction, and takes place according to the same laws as does the refraction of light, viz., (1.) The new direction or refracted ray lies always in the plane of incidence, or plane which contains the incident ray (i.e., the direction of the wave in the first medium), and the normal to the surface separating the two media, at the point in which the incident ray meets it; (2.) The sine of the angle between the normal and the incident ray bears to the sine of the angle between the normal and the refracted ray, a ratio which is constant for the same pair of media. For a theoretical demonstration of these laws, we must refer to the art. OPTICS, where it will be shown that the ratio involved in the second law is always equal to the ratio of the velocity of the wave in the first medium to the velocity in the second; in other words, the sines of the angles in question are directly proportional to the velocities. 32. Hence sonorous rays, in passing from one medium into another, are bent in towards the normal, or the reverse, according as the cording to velocity of propagation in the former relative exceeds or falls short of that in the latter. values of Thus, for instance, sound is refracted towards the perpendicular when passing into air from water, or into carbonic acid gas from air; the converse is the case when the passage takes place the opposite way.

Refraction is to or

from the normal ac

the velocities.

Limiting angle and

total reflexion.

M

N

Fig. 6.

33. It further follows, as in the analogous case of light,

that there is a certain angle termed the limiting angle, whose sine is found by dividing the less by the greater velocity, such that all rays of sound meeting the surface separating two different bodies will not pass onward, but suffer total reflexion back into the first body, if the velocity in that body is less than that in the other body, and if the angle of incidence exceeds the limiting angle.

The velocities in air and water being respectively 1090 and 4700 feet, the limiting angle for these media may be easily shown to be slightly above 15°. Hence, rays of sound proceeding from a distant source, and therefore nearly parallel to each other, and to PO (fig. 6), the angle POM being greater than 1510, will not pass into the water at all, but suffer total reflexion. Under such circumstances, the report of a gun, however powerful, would be inaudible by an ear placed in the water.

lenses.

34. As light is concentrated into a focus by a convex Acoustic glass lens (for which the velocity of light is less than for the air), so sound ought to be made to converge by passing through a convex lens formed of carbonic acid gas. On the other hand, to produce convergence with water or hydrogen gas, in both which the velocity of sound exceeds its rate in air, the lens ought to be concave. These results have been confirmed experimentally by Sondhaus and Hajech, who also succeeded in verifying the law of the equality of the index of refraction to the ratio of the velocities of sound.

35. When a wave of sound falls on a surface separating Laws of two media, in addition to the refracted wave transmitted reflexion. into the new medium, which we have just been considering, there is also a fresh wave formed in the new medium, and travelling in it in a different direction, but, of course, with the same velocity. This reflected wave is subject to the same laws as regulate the reflexion of light, viz., (1.) the coincidence of the planes of incidence and of reflexion, and (2.) the equality of the angles of incidence and reflexion, that is, of the angles made by the incident and reflected rays with the normal.

36. As in an ellipse (fig. 7), the normal PG at any point Reflexion bisects the angle SPH (S, H by a sphe being the foci), rays of sound roid. diverging from S, and falling on the spheroidal surface formed by 4 the revolution of the ellipse about the longest diameter AB, will be reflected to H. Also, since SP

B

G H

Fig. 7.

Echoes.

38. As a luminous object may give a succession of images when placed between two or more reflecting surfaces, so also in like circumstances may a sound suffer repetition.

To these principles are easily traceable all the peculiarities of echoes. A wall or steep cliff may thus send back, somewhat reduced in intensity, a shout, the report of a pistol, &c. The time

Fig. 9.

S

which elapses between the sound and its echo may be easily deduced from the known velocity of sound in air, if the distance of the wall be given. Thus, for a distance of 37 yards, the interval will be found by dividing the double of that or 74 yards by 370 yards, the velocity of sound at 50° Fahr., to amount to of a second. Hence, if we assume that the rate at which syllables can be distinctly uttered is five per second, the wall must be at a distance exceeding 37 yards to allow of the echo of a word of one syllable reaching the ear after the word has been uttered, 74 yards for a word of two syllables, and so on.

If the reflecting surface consists of one or more walls, cliffs, &c., forming together a near approach in shape to that of a prolate spheroid or of a double parabolic surface, then two points may be found, at one of which if a source of sound be placed, there will be produced, by convergence, a distinct echo at the other. As examples of this may be mentioned the whispering gallery in St. Paul's, London, and the still more remarkable case of the Cathedral of Girgenti in Sicily mentioned by Sir John Herschel.

Sound con. 39. On similar principles of repeated reflexion may be veyed over explained the well-known fact that sounds may be conwater, &c. veyed to great distances with remarkably slight loss of intensity, on a level piece of ground or smooth sheet of water or ice, and still more so in pipes, chimneys, tunnels, &c. Thus, in one of Captain Parry's Polar expeditions, a conversation was on one occasion carried on, at a distance of 1 mile, between two individuals separated by a frozen sheet of water. M. Biot heard distinctly from one end of the train of pipes of a mile long, previously referred to, a low whisper proceeding from the opposite end.

Thunder.

Practical illustrations are afforded by the system of communication by means of tubing now so extensively adopted in public and private buildings, and by the speak ing trumpet and the ear trumpet.

The ex

40. The prolonged roll of thunder, with its manifold varieties, is partly to be ascribed to reflexion by mountains, clouds, &c.; but is mainly accounted for on a different acoustic principle, viz., the comparatively low rate of transmission of sound through air, as was first shown by Dr Hooke at the close of the 17th century. planation will be more easily understood by adverting to the case of a volley fired by a long line of troops. A person situated at a point in that line produced, will first it is evident hear the report of the nearest musket, followed by that of the one following, and so down to the last one in the line, which will close the prolonged roll thus reaching his ear; and as each single report will appear to him less intense according as it proceeds from a greater distance, the roll of musketry thus heard will be one of gradually decreasing loudress. But if he were to place. himself at a relatively great distance right opposite to the centre of the line, the separate reports from each of the two wings would reach him nearly at the same moment, and hence the sound of the volley would now approach more nearly to that of a single loud crash. If the line of

soldiers formed an arc of a circle having its centre in his position, then the distances gone over by the separate reports being equal, they would reach his ear at the same absolute instant of time, and with exactly equal intensities; and the effect produced would be strictly the same as that of a single explosion, equal in violence to the sum of all the separate discharges, occurring at the same distance. It is easy to see that, by varying the form of the line of troops and the position of the observer, the sonorous effect will be diversified to any extent desired. If then we keep in view the great diversity of form exhibited by lightning-flashes, which may be regarded as being lines, at the points of which are generated explosions at the same instant of time, and the variety of distance and relative position at which the observer may be placed, we shall feel no difficulty in accounting for all those acoustic pheno mena of thunder to which Hooke's theory is applicable.

PART IV.

The Principles of Musical Harmony.

41. A few words on the subject of musical harmony must be introduced here for the immediate purposes of this article, further details being reserved for the special article on that subject.

Sounds in general exhibit three different qualities, so far as their effect on the ear is concerned, viz., loudness, pitch, and timbre.

extent of

Loudness depends, cat. par., on the violence with which Loudness the vibrating portions of the ear are excited; and there- depends on fore on the extent or amplitude of the vibrations of the vibration. body whence the sound proceeds. Hence, after a bell has been struck, its effect on the ear gradually diminishes as its vibration becomes less and less extensive. By the theory of vibrations, loudness or intensity is measured by the vis-viva of the vibrating particles, and is consequently proportional to the square of their maximum velocity or to the square of their maximum displacement. Helmholtz, however, in his remarkable work on the perception of tone, observes that notes differing in pitch differ also in loudness, where their vis viva is the same, the higher note always exhibiting the greater intensity.

vibration.

42. Difference of pitch is that which finds expression in Pitch dethe common terms applied to notes: Acute, shrill, high, pends on sharp, grave, deep, low, flat. We will point out presently in rapidity of what manner it is established that this quality of sound depends on the rapidity of vibration of the particles of air in contact with the external parts of the ear. The pitch of a note is higher in proportion to the number of vibrations of the air corresponding to it, in a given time, such as one V second. If n denote this number, then, by § 13, n = and hence, V being constant, the pitch is higher the less the length λ of the wave.

43. Timbre, or, as it is termed by German authors, Timbre. klang-farbe, rendered by Tyndall into clang-colour or clangtint, but for which we would substitute the expression acoustic colour, denotes that peculiarity of impression produced on the ear by sounds otherwise, in pitch, loudness, &c., alike, whereby they are recognisable as different from each other. Thus human voices are readily interdistinguishable; so are notes of the same pitch and intensity, produced by different instruments. The question whence arises this distinction must be deferred for the present.

Laws of musical

periodic vibrations of the air-particles acting on the ear, and therefore also of the body whence they proceed, each particle passing through the same phase at stated intervals of time. On the other hand, the motion to which noise is due is irregular and flitting, alternately fast and slow, and creating in the mind a bewildering and confusing effect of a more or less unpleasant character. Noise may also be produced by combining in an arbitrary manner several musical notes, as when one leans with the fore-arm against the keys of a piano. In fact, the composition of regular periodic motions, thus effected, is equivalent to an irregular motion.

45. We now proceed to state the laws of musical harmony, and to describe certain instruments by means of harmony. which they admit of being experimentally established. The chief of these laws are as follow:

Ratios of (1.) The notes employed in music always correspond vibrations. to certain definite and invariable ratios between the numbers of vibrations performed in a given time by the air when conveying these notes to the ear, and these ratios are of a very simple kind, being restricted to the various permutations of the first four prime numbers 1, 2, 3, 5, and their powers.

Unison.

Octave.

Twelfth and Fifth

Major third.

n

=

By interposing an additional note between each pair of Chromatie notes whose interval is a major or a minor tone, the result- scale. ing series of notes may be made to exhibit a nearer apthem, which will be very nearly semi-tones. proach to equality in the intervals successively separating This sequence

of twelve notes forms the chromatic scale. The note inter

== n

3 is termed the twelfth, and if we reduce the higher note of the pair by an 8, i.e., divide its number of vibrations by 2, we obtain the interval

22

=

3

n 2'

(5.) The interval "1 5 has no particular name at tached to it, but if we lower the higher note by two gror divide n, by 4, we get the interval or the

5

n

4'

is termed the major sixth.

2 x 3

21 2× 2

=

=

n 9

3

is termed the fourth.

which, being =

garded as formed by taking in the first place a note onefifth higher than the key-note or fundamental, i.e., higher than the latter by the interval thence ascending by

3

2'

another fifth, which gives us X and lowering this by 9 an octave, which results in which is called the second. " 8 15 3 5 (10.) The interval or 8 major third of the seventh.

of the fifth

be regarded as the may ,and is called the interval

46. If the key-note or fundamental be denoted by C, and the notes, whose intervals above C are those just enumerated, by D, E, F, G, A, B, C, we form what is

Savart's toothed wheel apparatus consists of a brass wheel, whose edge is divided into a number of equal pro- toothed jecting teeth distributed uniformly over the circumference, wheel ap and which is capable of rapid rotation about an axis per- paratus. pendicular to its plane and passing through its centre, by means of a series of multiplying wheels, the last of which is turned round by the hand. The toothed wheel being

set in motion, the edge of a card or of a funnel-shaped piece of common note paper is held against the teeth, when a note will be heard arising from the rapidly succeeding displacements of the air in its vicinity. The pitch of this note will, agreeably to the theory, rise as the rate of rotation increases, and becomes steady when that rotation is maintained uniform. It may thus be brought into unison with any sound of which it may be required to determine the corresponding number of vibrations per second, as for instance the note A3, three 8ves higher than the A which is indicated musically by a small circle placed between the second and third lines of the G clef, which A is the note of the tuning-fork usually employed for regulating concert-pitch. As may be given by a piano. Now, suppose that the note produced with Savart's apparatus is in unison with A3, when the experimenter turns round the first wheel at the rate of 60 turns per minute or one per second, and that the circumferences of the various multiplying wheels are such that the rate of revolution of the toothed wheel is thereby increased 44 times, then the latter wheel will perform 44 revolutions in a second, and hence, if the number of its teeth be 80, the number of taps imparted to the card every second will amount to 44 × 80 or 3520. This, therefore, is the number of vibrations corresponding to the note A3. If we divide this by 23 or 8, we obtain 440 as the number of vibrations answering to the note A. This, however, tacitly assumes that the bands by which motion is transmitted from wheel to wheel do not slip during the experiment. If, as is always more or less the case, slipping occurs, a different mode for determining the rate at which the toothed wheel revolves, such as is employed in the syren of De la Tour (vide below), must be adopted.