will notice that each particle occupies in its own path a position one-eighth of a period of vibration behind that of the particle next to the left of it. (Compare p. 16.) (1), (2), (3)...show the positions of the particles after one-eighth, two-eighths, three-eighths... of a complete period of particle-vibration. By following any one of the vertical lines of spots it will be seen that in the instance selected for this figure, each particle moves more rapidly in the neighbourhood of its undisturbed position than it does near the extremities of its swing. In (8) the particles have returned to their original positions and the wave A is where B was in (0). The particles have completed one vibrational cycle and the wave has advanced by its own length. This result may be thus generalised: While an individual particle performs one complete vibration the wave advances one wave-length. The proposition proved above (p. 17) for water-waves is, therefore, also true of waves due to transverse vibrations, i.e. such as are executed perpendicularly to the direction of wave-propagation.

As waves thus produced are of leading importance in the theory of Sound, it is necessary to study them in some detail.

Let a particle originally at rest at O in the initial line (Fig. 7 bis) be cooperating in the transmission This wave is drawn in the figure in two

of a wave.

positions such that the two points of its curve the most distant from the initial line, A and A', are

Fig. 7 (bis)

A

situated in two straight lines OA and OA' drawn through O in opposite directions, each perpendicular to the initial line. It is evident that, at the moments when the wave is in these positions, the particle originally at O will be at A and A' respectively, and that these two points mark the limits of its vibration. Hence the line AA' is the extent of the particle's vibration. But by drawing parallels to the initial line through A and A' it will be seen, by reference to the definition in § 6, that AA' is also the amplitude of the wave. 'Extent of particle-vibration' and 'amplitude of corresponding wave' are, therefore, only different ways of expressing the same thing.

10. When a series of continuous equal waves are being transmitted, each particle, after completing one vibration, will instantly commence another precisely equal vibration, and go on doing so as long as the transmission of waves is maintained. This

kind of motion, in which the same movement is continuously repeated in successive equal intervals of time, is called 'periodic,' and the time which any one of the movements occupies is called its period.' Thus, to continuous equal waves correspond continuous periodic particle-vibrations.

We will next compare the periods of the vibrations which produce waves of different lengths advancing at the same speed.

In Fig. 8, waves of three different lengths are represented. One wave of (1) is as long as two of

(2), and as three of (3). Therefore, in virtue of the fact proved in § 9, a particle makes one complete

vibration in (1) while the long wave passes from A to B, two in (2) while the shorter waves there presented pass over the same distance, and three in the case of the shortest waves of (3). But the velocities of these waves being by our supposition equal, the times of describing the distance AB will be the same in (1), (2) and (3). Hence a particle in (2) vibrates twice as rapidly, and in (3) three times as rapidly, as in (1); or conversely, vibration in (1) is half as rapid as in (2), and one-third as rapid as in (3).

The rates of vibration in (1), (2) and (3) (by which we mean the numbers of vibrations performed in any given interval of time) are, therefore, proportional to the numbers 1, 2 and 3, which are themselves inversely proportional to the wave-lengths in the three cases respectively. We may express our result thus; The rate of particle-vibration is inversely proportional to the corresponding wave-length. Similar reasoning will apply equally well to any other case; the proposition, therefore, though deduced from the relations of particular waves, holds for waves in general.

The converse proposition admits of easy independent proof as follows. It has been shown (p. 21) that in one period of particle-vibration a wave traverses its own length. This length must therefore, if the velocity of the wave remain constant,

be proportional to the period, i.e. inversely proportional to the rate of vibration.

11. We have now connected the extent of the particle-vibration with the amplitude, and its rate with the length, of the corresponding wave. It remains to examine what feature of the vibratory movement corresponds to the third element, the form of the wave.

Fig.9.

A

Suppose that two boys start together to run a race from 0 to A, from A to B, and from B back to O, and that they reach the goal at the same moment. They may obviously do this in many different ways. For instance, they may keep abreast all through, or one may fall behind over the first half of the course and recover the lost ground in the second. Again, one may be in front over OAO, and the other over OBO, or each boy may pass, and be passed by, his competitor repeatedly during the race. We may regard the movement of each boy as constituting one complete vibration, and thus convince ourselves that a vibratory motion of given extent and period may be performed in an indefinitely numerous variety of modes. Let us now compare the positions of a particle at the expiration of successive equal intervals