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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

of time, when cooperating in the transmission of waves of different forms.

In each of the three cases in Fig. 10 the front of a wave is shown in the positions it respectively occupies at the end of ten equal intervals of time during which its intersection with the level-line moves from O through the equidistant points 1, 2, 3, 4, &c. of the initial line.

A particle whose place of rest is O will necessarily assume corresponding positions in the vertical line OA: thus the points where this line cuts the successive wave-fronts show the positions

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of the vibrating particle at equal intervals of time.

On comparing the three cases it will be seen that the mode of the particle's vibration is distinct in each. In (1), it moves fastest at O, and then slackens its pace up to A. In (2), it starts more slowly than in (1), attains its greatest speed near the middle of OA, and again slackens on approaching A. In (3), the pace steadily increases from 0 to A. The different wave-fronts shown in the figure have been purposely constructed with the same amplitude and length, in order that only such variations as were due to differences of form might come into consideration. The reader should construct similar figures with other forms, and so convince himself more thoroughly that to every distinct form of wave there corresponds a special mode of particle-vibration.

12. Conversely each distinct mode of particlevibration gives rise to a special form of wave. We will show this by actually constructing the form of wave produced by a given mode of particle-vibration when the mode in which a particle moves is given.

Suppose that each particle makes one complete vibration per second about its position of original rest in the initial line and that the law of vibration is roughly indicated in Fig. 11.

AB is the path described by a particle; O its

position when in the initial line; 1, 2, 3, 4, 5, 6...12, 13...16 its positions after 16 equal intervals of time

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each one-sixteenth of a second: 16 coincides with O, as the particle has returned to its starting-point.

Next, select a series of particles originally at rest in equidistant positions along the initial line, and so situated that each commences a vibration identical with that above laid down in Fig. 11 one-sixteenth of a second after the particle to the left of it has started on an equal vibration. Fig. 12 shows the rest-positions of the series of particles

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in the initial line, and their contemporaneous positions

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