reproduced when each of these centres has advanced through a wave-length, so that the thirteenth row is merely a repetition of the first.

The velocities of the particles can be estimated by the comparison of successive rows. It is thus seen that the greatest forward velocity is at the centres of condensation, and the greatest backward velocity at the centres of rarefaction. Each particle has its greatest velocities, and greatest condensation and rarefaction, in passing through its mean position, and comes for an instant to rest in its positions of greatest displacement, which are also positions of mean density.

The distance between A and a remains invariable, being always a wave-length, and these two particles always agree in phase. Any other two particles represented in the diagram are always in different phases, and the phases of A and D, or B and E, or C and F, are always opposite; for example, when A is moving forwards with the maximum velocity, D is moving backwards with the same velocity.

The vibrations of the particles, in an undulation of this kind, are called longitudinal; and it is by such vibrations that sound is propagated through air. Fig. 605 illustrates the manner in which an undulation may be propagated by means of transverse vibrations, that is to say, by vibrations executed in a direction perpendicular to that in which the undulation advances. Thirteen particles A B C D EFGHIJKL a are represented in the positions which they occupy at successive times, whose interval is one-sixth of a period. At the instant first considered, D and J are the particles which are furthest displaced. At the end of the first interval, the wave has advanced two letters, so that F and L are now the furthest displaced. At the end of the next interval, the wave has advanced two letters further, and so on, the state of things at the end of the six intervals, or of one complete period, being the same as at the beginning, so that the seventh line is merely a repetition of the first. Some examples of this kind of wave-motion will be mentioned in later chapters.

871. Propagation in an Open Space.-When a sonorous disturbance occurs in the midst of an open body of air, the undulations to which it gives rise run out in all directions from the source. If the disturbance is symmetrical about a centre, the waves will be spherical; but this case is exceptional. A disturbance usually produces condensation on one side, at the same instant that it produces rarefaction on another. This is the case, for example, with a vibrating

PROPAGATION IN AN OPEN SPACE.

877

plate, since, when it is moving towards one side, it is moving away from the other. These inequalities which exist in the neighbourhood of the sonorous body, have, however, a tendency to become less marked, and ultimately to disappear, as the distance is increased. Fig. 606 represents a diametral section of a series of spherical waves. Their mode of propagation has some analogy to that of the circular

A B C D E F G H

J K L

a

A B C D E F G H I J

BCDEFGHIJ

C D E F G H

A

A B

з с

A B C D E

I J K L

F G H I J K L a

F G

HI
I J K L a

Fig. 605.-Transverse Vibration.

waves produced on water by dropping a stone into it; but the particles which form the waves of water rise and fall, whereas those which form sonorous waves merely advance and retreat, their lines of motion being always coincident with the directions along which the sound travels. In both cases it is important to remark that the undulation does not involve a movement of transference. Thus, when the surface of a liquid is traversed by waves, bodies floating on it rise and fall, but are not carried onward. This property is characteristic of undulations generally. An undulation may be defined as a system of movements in which the several particles move to and fro, or round and round, about definite points, in such a

manner as to produce the continued onward transmission of a condition, or series of conditions.

There is one important difference between the propagation of sound in a uniform tube and in an open space. In the former case, the layers of air corresponding to successive wave-lengths are of equal

mass, and their movements are precisely alike, except in so far as they are interfered with by friction. Hence sound is transmitted through tubes to great distances with but little loss of intensity, especially if the tubes are large.1

The same principle is illustrated by the ease with which a scratch

1 Regnault, in his experiments on the velocity of sound, found that in a conduit 108 of a metre in diameter, the report of a pistol charged with a gramme of powder ceased to be heard at the distance of 1150 metres. In a conduit of 3m, the distance was 3810". In the great conduit of the St. Michel sewer, of 1m 10, the sound was made by successive reflections to traverse a distance of 10,000 metres without becoming inaudible.-D.

DISSIPATION OF SONOROUS ENERGY.

879

on a log of wood is heard at the far end, the substance of the log acting like the body of air within a tube.

In an open space, each successive layer has to impart its own condition to a larger layer; hence there is a continual diminution of amplitude in the vibrations as the distance from the source increases. This involves a continual decrease of loudness. An undulation involves the onward transference of energy; and the amount of energy which traverses, in unit time, any closed surface described about the source, must be equal to the energy which the source emits in unit time. Hence, by the reasoning which we employed in the case of radiant heat (§ 465), it follows that the intensity of sonorous energy diminishes according to the law of inverse

squares.

The energy of a particle executing simple vibrations in obedience to forces of elasticity, varies as the square of the amplitude of its excursions; for, if the amplitude be doubled, the distance worked through, and the mean working force, are both doubled, and thus the work which the elastic forces do during the movement from either extreme position to the centre is quadrupled. This work is equal to the energy of the particle in any part of its course. At the extreme positions it is all in the shape of potential energy; in the middle position it is all in the shape of kinetic energy; and at intermediate points it is partly in one of these forms, and partly in the other.

It can be shown that exactly half the energy of a complete wave is kinetic, the other half being potential.

872. Dissipation of Sonorous Energy. The reasoning by which we have endeavoured to establish the law of inverse squares, assumes that onward propagation involves no loss of sonorous energy. This assumption is not rigorously true, inasmuch as vibration implies friction, and friction implies the generation of heat, at the expense of the energy which produces the vibrations. Sonorous energy must therefore diminish with distance somewhat more rapidly than according to the law of inverse squares. All sound, in becoming extinct, becomes converted into heat.

This conversion is greatly promoted by defect of homogeneity in the medium of propagation. In a fog, or a snow-storm, the liquid or solid particles present in the air produce innumerable reflections, in each of which a little sonorous energy is converted into heat.

873. Velocity of Sound in Air.—The propagation of sound through an elastic medium is not instantaneous, but occupies a very sensible time in traversing a moderate distance. For example, the flash of a gun at the distance of a few hundred yards is seen some time before the report is heard. The interval between the two impressions may be regarded as representing the time required for the propagation of the sound across the intervening distance, for the time occupied by the propagation of light across so small a distance is inappreciable.

It is by experiments of this kind that the velocity of sound in air has been most accurately determined. Among the best determinations may be mentioned that of Lacaille, and other members of a commission appointed by the French Academy in 1738; that of Arago, Bouvard, and other members of the Bureau de Longitudes in 1822; and that of Moll, Vanbeek, and Kuytenbrouwer in Holland, in the same year. All these determinations were obtained by firing cannon at two stations, several miles distant from each other, and noting, at each station, the interval between seeing the flash and hearing the sound of the guns fired at the other. If guns were fired only at one station, the determination would be vitiated by the effect of wind blowing either with or against the sound. The error from this cause is nearly eliminated by firing the guns alternately at the two stations, and still more completely by firing them simultaneously This last plan was adopted by the Dutch observers, the distance of the two stations in their case being about nine miles. Regnault has quite recently repeated the investigation, taking advantage of the important aid afforded by modern electrical methods for registering the times of observed phenomena. All the most careful determinations agree very closely among themselves, and show that the velocity of sound through air at 0° C. is about 332 metres, or 1090 feet per second.1 The velocity increases with the temperature, being proportional to the square root of the absolute temperature by air thermometer (§ 325). If t denote the ordinary Centigrade tempera

1 A recent determination by Mr. Stone at the Cape of Good Hope is worthy of note as being based on the comparison of observations made through the sense of hearing alone. It had previously been suggested that the two senses of sight and hearing, which are concerned in observing the flash and report of a cannon, might not be equally prompt in receiving impressions (Airy on Sound, p. 131). Mr. Stone accordingly placed two observers -one near a cannon, and the other at about three miles distance; each of whom on hearing the report, gave a signal through an electric telegraph. The result obtained was in precise agreement with that stated in the text.