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36. Propagation of Waves. — We have now to consider briefly, in general principle, the mode or physical process by which variations of velocity produced in the particles of an aeriform medium, including those periodic increments and decrements of velocity termed "waves," are propagated to a distance by the medium.

We may, as an illustrative example, take the case of a tuning- fork vibrating in air. Then it is to be noted that before the prong is put in motion the air molecules are already in motion (having a speed of about 1600 feet per second), rebounding from the prong in all directions and exerting an equal pressure.

Then, after the fork has been put in vibration, if we regard one forward movement or semivibration of the prong, then the rebounding air molecules being struck by the prong, receive an increment of velocity (small compared with their normal velocity). This increment of velocity is instantly transferred to the neighbouring air molecules in the collisions (mutual exchange of motion) continually occurring; this transference of motion from molecule to molecule taking place in accordance with the simple principles of impact in the case of equal masses. The air molecules next the prong, therefore, lose their increment of velocity by transferring it to the molecules in advance, the molecules returning to the prong with their normal velocities to receive a fresh increment, which they again transfer, &c.; and thus during the advance of the prong a succession of small increments of velocity is imparted to the air molecules in the form of a pulse or wave, which is transmitted through the air by exchange of motion with a velocity dependent on the normal velocity of the air molecules. Since the air molecules forming this pulse or half of the wave possess, during the interval occupied by the passage of the pulse, a velocity slightly in excess of that of the molecules around them, this portion of the wave is somewhat pushed forward, and a "condensation" is the result of the increment of velocity.

We have up to this point regarded one forward movement or semivibration of the prong : the latter gradually comes to rest before commencing its backward movement, and thus the first

part of the wave is succeeded by a portion of air whose molecules ave received no perceptible increment of velocity. On the prong commencing its backward movement to finish one complete vibration, the impinging air molecules now striking against its receding surface lose a portion of their normal velocity by transference to the prong, the decrement of velocity thus sustained by the air molecules being transmitted in precisely the same manner by

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exchange of motion from molecule to molecule through the surrounding air; and since the air molecules forming this portion of the wave possess during the passage of the wave a velocity slightly less than that of the molecules in the advanced part of the wave, the mass of air is accordingly shifted backwards in relation to the advanced part of the wave, a " rarefaction " being the result.

37. The principle involved admits of further illustration from a somewhat different point of view. Since the condition for the equilibrium of pressure of an aeriform medium requires that the particles should not be moving in one special direction in preference to another, but in every possible direction, it follows, there- fore, that if any imaginary straight line be taken in an aeriform medium, such as the ether for example, then since the particles are moving in every possible direction, and the medium is in equilibrium of pressure along this line (as in every direction), there must therefore be at any given Instant, when a large number of particles are taken, on an average as many particles whose direction of motion is towards one extremity of the line, as there are particles whose direction of motion is towards the opposite extremity, the resolved components of the motions in the direction of the line being taken when the motions are oblique; for evidently, in order that equilibrium of pressure may exist, or in order that the particles in their mutual interchange of motion may balance each other's effects, as many particles must be moving in any one given direction as in the opposite.

The principle involved in the movement admits, therefore, of being illustrated in a very simple manner; and as regards method of illustration we are to a certain extent indebted to a paper by Waterston, 'On the Theory of Sound;'* this paper treating specially of the mode of propagation of waves by the air molecules, whose normal motion is considered to take place in accordance with Fig. 1. the theory of Joule and Clausius. Let

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us suppose a line or row of spheres A| 1, 2, 3, &c. (Fig. 1), of which as many at any given instant are moving towards one extremity of the line as towards the opposite; or we may suppose all those spheres designated with the odd numbers to be moving simultaneously in one direction, while those designated with the even numbers are moving simultaneously in the opposite direction. The motion of the spheres may be supposed to take place without resistance, and so to go on continuously; the row of spheres being further supposed as placed between two plane surfaces A and B, from which the end ones rebound, the whole line of spheres being thus in equilibrium and exerting by their impacts a pressure tending to separate the controlling surfaces A and B. Each sphere, therefore, performs a simple reciprocating

• « Philosophical Magazine,' Jan., 1859, Supp. to vol. xvi.

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movement within the space bounded by the dotted lines in the diagram, the spheres continually coming into collision or rebound- ing from each other; one half their number, represented by the odd numbers (i. e. one half the total mass of matter forming the row of spheres), moving simultaneously in one direction, daring the time that the other half, marked with the even numbers, moves in the reverse direction, the whole line of spheres being thus in perfect equilibrium, and not tending bodily as a whole to move in any particular direction, but simply tending to open out or expand, and to separate the controlling surfaces A and B.

This it may be observed is the only mode of motion possible to the spheres by which equilibrium can be maintained; in fact, this constitutes in principle the only mode of motion possible to matter by which it can be in rapid motion, and yet, regarded as a whole, can maintain a fixed position; for an oscillatory movement is the only means by which a mass can be in motion and yet not deviate from a given spot; and the movement of an equal quan- tity of the matter in any two opposite directions is the only means by which equilibrium can be maintained, the two moving portions thus precisely balancing each other's effects; this being the case with particles of an aeriform medium, which move in such a way that at any given instant along any imaginary straight line in the

^; i • I lique and irregular along such an > ! «i» j 4b imaginary line; but since the; ! 2,s ! *? particles maintain an equilibrium I, f % t Djr their collisions when a suffi- V | *•" ' "I* | *1~ m cien * number are taken into ao- 1 i 2 J a \ 4 | count, the principle involved in

the movement cannot therefore be at all affected bv making the motions regular; and the mode of illustration, as shown in the diagram, will therefore serve to give a perfectly just idea of the principle involved, and to show the mode in which waves are propagated.

38. We will suppose now that a forward and backward motion is communicated to the plane A, in the form of a vibratory movement; also the line of spheres may be supposed to be extended indefinitely from the plane A, the movement of vibration of the .. plane being also supposed slow, compared with the normal velocity

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of the spheres. In that case the sphere 1 would strike against the plane A a number of times during one forward movement or semivibration of the plane. On the commencement of the first forward motion of the plane, the plane moving towards the sphere 1, the latter receives a small increment of velocity, which it transfers at the collision to sphere 2, the two spheres simply exchanging velocities. The sphere 1, therefore, returns to the plane with its original normal velocity unchanged, and receives a second similar increment of velocity from the plane, which it again transfers, &c. The sphere 2 at once transfers to sphere 3 the increment of velocity received from sphere 1, the sphere 2 returning with its normal velocity to repeat the process. In this way, during one forward movement or semivibration of the plane A, a series of small, successive increments of velocity are propagated in the form of a pulse, by exchange of motion along the line of spheres, the velocity of transmission of the pulse being that of the spheres themselves, or the velocity of transmission is equal to the normal velocity of the spheres. The length of this pulse or semiwave evidently must depend on the time taken by the plane to complete a semivibration, or wave length is proportional to vibrating period. The wave length will also evidently depend on the normal velocity of the spheres.

These points are practically illustrated in the case of gases, or aeriform media generally, the different rates of transmission of waves (waves of sound, for example) by various gases being proportional to the different normal speeds of the component molecules of the gases. The molecules of hydrogen have a normal speed about four times greater than the molecules of air, and waves (such as waves of sound) are transmitted by hydrogen at a speed about four times greater than in the case of air; also a wave due to a given vibrating period, when generated in hydrogen, is about four times longer than when generated in air.

39. Returning to our illustrative case. The pulse due to the forward motion of the plane A being made up of a succession of spheres, moving (at the time of passage of the pulse) with a velocity slightly in excess of the normal velocity of the spheres forming the advanced portion of the line, this portion of the wave, or the spheres forming it, would therefore be slightly shifted forwards, producing a "condensation."

The plane comes to rest gradually before changing the direction of its motion; and the opposite movement is also commenced gradually, the motion of the plane being therefore almost nothing for a short interval, so that the first portion of the wave is separated from the second by a succession of spheres whose normal velocities have not been appreciably changed, an effect corresponding to the normal mass of air which separates the two portions of a wave of sound.

By the backward movement of the plane at the next semi-

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vibration, a series of small decrements of velocity, forming the second half of the wave, is transmitted in the same manner along the line of spheres; and the velocity of the spheres at the instant when this portion of the wave passes them, being slightly less than the normal velocity of the adjacent spheres, this half of the wave, or the spheres forming it, are therefore shifted backwards in relation to the advanced portion of the wave, an opening out of the line of spheres, or a u rarefaction/' being the result It is clear that in the actual fact in the case of an aeriform medium, since the motions of the particles take place also obliquely to the line of propagation of the wave, the rate of propagation cannot therefore equal the normal speed of the particles, but must be to a certain degree slower. This, however, does not affect in the least the principle involved, and, accordingly, the above mode of illustration will serve to convey a perfectly just idea of the mode of the normal motion of the component particles of an aeriform medium, and the physical means by which through this mode of motion "waves," or any changes of velocity experienced by the particles, are propagated to a distance along the medium.