Thus, if a common glass-jar be struck so as to yield an audible sound, the existence of a motion of this kind may be felt by the finger lightly applied to the edge of the glass; and, on increasing the pressure so as to destroy this motion, the sound forthwith ceases. Small pieces of cork put in the jar will be found to dance about during the continuance of the sound; water or spirits of wine poured into the glass will, under the same circumstances, exhibit a ruffled surface. The experiment is usually performed, in a more striking manner, with a bell-jar and a number of small light wooden balls suspended by silk strings to a fixed frame above the jar, so as to be just in contact with the widest part of the glass. On drawing a violin bow across the edge, the pendulums are thrown off to a considerable distance, and falling back are again repelled, &c.

It is also in many cases possible to follow with the eye the motions of the particles of the sounding body, as, for instance, in the case of a violin string or any string fixed at both ends, when the string will appear, by a law of optics, to occupy at once all the positions which it successively assumes during its vibratory motion.

3. It is, moreover, essential, in order that the ear may L be affected by a sounding body, that there be interposed between it and the ear one or more intermediate bodies (media), themselves capable of molecular vibration, which shall receive such motion from the source of sound, and transmit it to the external parts of the ear, and especially to the membrana tympani or drum of the ear. This statement is confirmed by the well-known effect of stopping the ear with soft cotton, or other substance possessing little elasticity.

The air around us forms the most important medium of communication of sound to our organs of hearing; in fact, were air devoid of this property, we should practically be without the sense of hearing. In illustration of the part thus assigned to the atmosphere in acoustics, an apparatus has been constructed, consisting of a glass receiver, in which is a bell and a hammer connected with clock-work, by which it can be made to strike the bell when required. The receiver is closed air-tight by a metal plate, through which passes, also air-tight, into the interior, a brass rod. By properly moving this rod with the hand, a detent is released, which checks the motion of the wheel-work, and the hammer strikes the bell continuously, till the detent is pushed into its original position. As long as the air in the receiver is of the usual atmospheric density, the sound is perfectly audible. But on rarefying the air by means of an air-pump (the clock-work apparatus having been separated from the plate of the pump by means of a padding of soft cotton), the sound grows gradually fainter, and at last becomes inaudible when the rarefaction of the air has reached a very low point. If, however, at this stage of the experiment, the metal rod be brought into contact with the bell, the sound will again be heard clearly, because now there is the necessary communication with the ear. On readmitting the air, the sound recovers its original intensity. This experiment was first performed by Hawksbee in 1705.

4. Inasmuch, then, as sound necessarily implies the existence in the sounding body, in the air, &c., and (we may add) in the ear itself, of vibratory motion of the particles of the various media concerned in the phenomenon, a general reference to the laws of such motion is essential to a right understanding of the principles of acoustics.

The most familiar instance of this kind of motion is afforded by the pendulum, a small heavy ball, for instance, attached to a fine string, which is fixed at its other end. There is but one position in which the ball will remain at rest, viz., when the string is vertical, there being then equilibrium between the two forces acting on the body, the tension of the string and the earth’s attractive force or gravity. Thus, in the adjoining fig., if C is the point of suspension, and CA the vertical through that point of length l, equal to the string, A is the equilibrium position of the particle.

Let now the ball be removed from A to P, the string being kept tight, so that P describes the arc AP of a circle of radius equal to l, and let the ball be there dropped. The tension of the string not being now directly opposite in direction to gravity (g ), motion will ensue, and the body will retrace the arc PA. Tn doing so, it will continually increase its velocity until it reaches the point A, where its velocity will be a maximum, and will consequently pass to the other side of A towards Q. But now gravity tends to draw it back towards A, and hence the motion becomes a retarded one; the velocity continually diminishes, and is ultimately destroyed at some point Q, which would be at a distance from A equal to that of P, but for the existence of friction, resistance of the air, &c., which make that distance less. From Q it will next move down with accelerated motion towards A, where it will have its greatest velocity in the direction from left to right, and whence it will pass onwards towards P, and so on. Thus the body will vibrate to and fro on either side of A, its amplitude of vibration or distance between its extreme positions gradually diminishing in consequence of the resistances before mentioned, and at last being sensibly reduced to nothing, the body 'then resuming its equilibrium-position A.

If the amplitude of vibration is restricted within inconsiderable limits, it is easy to prove that the motion takes place just as if the string were removed, the ball deprived altogether of weight and urged by a force directed to the point A, and proportional to the distance from that point. For then, if m be any position of the ball, the chord mA may be regarded as coincident with the tangent to the circie at m, and therefore as being perpendicular to Cm. Hence g, acting parallel to CA, being resolved along Cm and mA., the former component is counteracted by the tension of the string, and there remains as the only effective acceleration, the tangential component along mA, which, by the triangle of forces, is equal to or g/l. and is therefore proportional to Am.

On this supposition of indefinitely small vibrations, the pendulum is isochronous; that is, the time occupied in passing from one extreme position to the other is the same, for a given length l of the pendulum, whatever the extent of vibration.

We conclude from this that, whatever may be the nature of the forces by which a particle is urged, if the resultant of those forces is directed towards a fixed point, and is proportional to the distance from that point, the particle will oscillate to and fro about that point in times which are independent of the amplitudes of the vibrations, provided these are very small.

5. The particle, whose vibratory motion we have been considering, is a solitary particle acted on by external forces. But, in acoustics, we have to do with the motion of particles forming a connected system or medium, in which the forces to be considered arise from the mutual actions of the particles. These forces are in equilibrium with each other when the particles occupy certain relative positions. But, if any new or disturbing force act for a short time on any one or more of the particles, so as to cause a mutual approach or a mutual recession, on the removal of the disturbing force, the disturbed particles will, if the body be elastic, forthwith move towards their respective positions of equilibrium. Hence arises a vibratory motion to and fro of each about a given point, analogous to that of a pendulum, the velocity at that point being always a maximum, alternately in opposite directions. Thus, for example, if to one extremity of a pipe containing air were applied a piston, of section equal to that of the pipe, by pushing in the piston slightly and then removing it, we should cause particles of air, forming a thin section at the extremity of the pipe, to vibrate in directions parallel to its axis.

In order that a medium may be capable of molecular vibrations, it must, as we have mentioned, possess elasticity, that is, a tendency always to return to its original condition when slightly disturbed out of it.

6. We now proceed to show how the disturbance whereby certain particles of an elastic medium are displaced from their equilibrium-positions, is successively transmitted to the remaining particles of the medium, so as to cause these also to vibrate to and fro.

Let us consider a line of such particles y, x, a, b, &c.

y x a 1 a a 2 b c d e f g h i k l m n o p equidistant from each other, as above; and suppose one of them, say a, to be displaced, by any means, to a 1 . As we have seen, this particle will swing from a 1 to a 2 and back again, occupying a certain time T, to complete its double vibration. But it is obvious that, the distance between a and the next particle b to the right being diminished by the displacement of the former to a 1 , a tendency is generated in b to move towards a 1 , the mutual forces being no longer in equilibrium, but having a resultant in the direction ba 1 . The particle b will therefore also suffer displacement, and be compelled to swing to and fro about the point b. For similar reasons the particles c, d . . . will all likewise be thrown into vibration. Thus it is, then, that the disturbance propagates itself in the direction under consideration. There is evidently also, in the case supposed, a transmission from a to x, y, &c., i.e., in the opposite direction.

Confining our attention to propagation in the direction abc . . ., we have next to remark that each particle in that line will be affected by the disturbance always later than the particle immediately preceding it, so as to be found in the same stage of vibration a certain interval of time after the preceding particle.

7. Two particles which are in the same stage of vibration, that is, are equally displaced from their equilibriumpositions, and are moving in the same direction and with equal velocities, are said to be in the same phase. Hence we may express the preceding statement more briefly thus: Two particles of a disturbed medium at different distances from the centre of disturbance, are in the same phase at different times, the one whose distance from that centre is the greater being later than the other.

8. Let us in the meantime assume that, the intervals ab, bc, cd . . . . being equal, the intervals of time which elapse between the like phases of b and a, of c and b . . . . are also equal to each other, and let us consider what at any given instant are the appearances presented by the different particles in the row.

T being the time of a complete vibration of each particle, T

let — be the interval of time requisite for any phase of a to pass on to b. If then at a certain instant a is displaced to its greatest extent to the right, b will be somewhat short of, but moving towards, its corresponding position, c still further short, and so on. Proceeding in this way, we shall come at length to a particle p, for which the distance ap = p . ab, which therefore lags in its vibrations behind a T

by a time = p × T / p = T, and is consequently precisely in the same phase as a. And between these two particles a, p, we shall evidently have particles in all the possible phases of the vibratory motion. At h, which is at distance from a = ½ ap, the difference of phase, compared with a, will be ½T, that is, h will, at the given instant, be displaced to the greatest extent on the opposite side of its equilibrium-position from that in which a is displaced; in other words, h is in the exactly opposite phase to a.

9. In the case we have just been considering, the vibrations of the particles have been supposed to take place in a direction coincident with that in which the disturbance passes from one particle to another. The vibrations are then termed longitudinal.

But it need scarcely be observed that the vibrations may take place in any direction whatever, and may even be curvilinear. If they take place in directions at right angles to the line of progress of the disturbance, they are said to be transversal.

10. Now the reasoning employed in the preceding case will evidently admit of general application, and will, in particular, hold for transversal vibrations. Hence if we mark (as is done in fig. 2) the positions a 1 b 1 c 1 . . ., occupied by the various particles, when swinging transversely, at the instant at which a has its maximum displacement above its equilibrium-position, and trace a continuous line running through the points so found, that line will by its ordinates indicate to the eye the state of motion at the given instant.

Thus a and p are in the same phase, as are also b and q, c and r, &c. a and h are in opposite phases, as are also b and i, c and k, &c.

Distances a p, b q, &c., separating particles in the same phase, and each of which, as we have seen, is passed over by the disturbance in the time T of a complete vibration, include within them all the possible phases of the motion.

Beyond this distance, the curve repeats itself exactly, that is, the phases recur in the same order as before.

Now the figure so traced offers an obvious resemblance to the undulating surface of a lake or other body of water, after it has been disturbed by wind, exhibiting a wave with its trough A∕qB, and its crest B p 1 C. Hence have been introduced into Acoustics, as also into Optics, the terms wave and undulation. The distance ap, or bq . . . or A C, which separates two particles in same phase, or which includes both a wave-crest and a wave-trough, is termed the length of the wave, and is usually denoted by λ.

As the curve repeats itself at intervals each =λ, it follows that particles are in the same phase at any given moment, when the distances between them in the direction of transmission of the disturbance = λ, 2λ, 3λ .. . and generally = n λ , where n is any whole number.

Particles such as a and h, b and i, &c., which are at distances = ½λ , being in opposite phases, so will also be particles separated by distance, ½λ + λ = 3 /2λ,or, in general, by ½λ + m λ = (2 m + 1 ) λ / 2 , that is, by any odd multiple of λ / 2 .

11. A like construction to the one just adopted for the displacements of the particles at any given instant, may be also applied for exhibiting graphically their velocities at the same instant. Erect at the various points a, b, c, &c., perpendiculars to the line joining them, of lengths proportional to and in the direction of their velocities, and draw a line through the extreme points of these perpendiculars; this line will answer the purpose required. It is indicated by dots in the previous figure, and manifestly forms a wave of the same length as the wave of displacements, but the highest and lowest point's of the one wave correspond to the points in which the other wave crosses the line of equilibrium.

12. In order to a graphic representation of the displacements and velocities of particles vibrating longitudinally, it is convenient to draw the lines which represent those quantities, not in the actual direction in which the motion takes place and which coincides with the line a b c ... , but at right angles to it, ordinates drawn upwards indicating displacements or velocities to the right (i.e., in the direction of transmission of the disturbance), and ordinates drawn downwards indicating displacements or velocities in the opposite direction. When this is done, waves of displacement and velocity are figured identically with those for transversal vibrations, and are therefore subject to the same resulting laws.

13. But not only will the above waves enable us to see at a glance the circumstances of the vibratory motion at the instant of time for which it has been constructed, but also for any subsequent moment. Thus, if we desire to consider what is going on after an interval T / p , we have simply to conceive the whole wave (whether of displacement or velocity) to be moved to the right through a distance =ab. Then the state of motion in which a was before will have been transferred to b, that of b will have been transferred to c, and so on. At the end of another such interval, the state of the particles will in like manner be represented by the wave, if pushed onward through another equal space. In short, the whole circumstances may be pictured to the eye by two waves (of displacement and of velocity) advancing continuously in the line a b c ... with a velocity V which will take it over the distance ab in the time T / p ,V being therefore = ab / T = p.ab / T = AC / T or V = λ/ T .

This is termed the velocity of propagation of the wave, and, as we see, is equal to the length of the wave divided by the time of a complete vibration of each particle.

If, as is usually more convenient, we express T in terms of the number n of complete vibrations performed in a given time, say in the unit of time, we shall have 1 / T = n , and hence V = n λ

14. There is one very important distinction between the two cases of longitudinal and of transversal vibrations which now claims our attention, viz., that whereas vibrations of the latter kind, when propagated from particle to particle in an elastic medium, do not alter the relative distances of the particles, or, in other words, cause no change of density throughout the medium; longitudinal vibrations, on the other hand, by bringing the particles nearer to or further from one another than they are when undisturbed, are necessarily accompanied by alternate condensations and rarefactions.

Thus, in fig. 2, we see that at the instant to which that fig. refers, the displacements of the particles immediately adjoining a are equal and in the same direction; hence at that moment the density of the medium at a is equal to that of the undisturbed medium. The same applies to the points h, p, &c., in which the displacements are at their maxima and the velocities of vibration =0.

At any point, such as c, between a and A, the displacements of the two adjoining particles on either side are both...