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surface, being, in this manner, thrown successively into the same vibratory motion, give rise, by their consequent varieties of position at any assigned moment, to the transmission of the form which we call a wave.

Fig.7

When a series of continuous equal waves, such as those in Figure 7, are being transmitted, each oscillating drop, after completing one vibration, will instantly commence another precisely equal vibration, and go on doing so as long as the series of waves lasts. The 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 dropvibrations.

10. We will next compare the periods of the drop-vibrations corresponding to waves of different lengths advancing with equal velocities.

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 a drop makes

one complete vibration in (1) while the long wave passes from A to B, two in (2) while the shorter

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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 drop in (2) vibrates twice as rapidly, and a drop in (3) three times as rapidly, as a drop in (1); or conversely, a drop in (1) vibrates half as rapidly as a drop in (2), and one third as rapidly as a drop 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, propor

tional 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 drop-vibration is inversely proportional to the corresponding wave-length. The same reasoning will apply equally well to any other case; the proposition, therefore, though derived from particular relations of wave-lengths, is true universally.

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

Fig.9.

B

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 an oscillatory motion of given extent and period may be performed in an indefinitely numerous variety of modes. Let us now compare the positions of a drop at successive equal intervals of time, when cooperating in the transmission of waves of different forms.

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In each of the three cases in Fig. 10 the front of a wave-crest is shown in the positions it respectively occupies at the end of ten equal intervals of time (each one tenth of that occupied by the wave

in traversing a quarter-wave-length), the apex of the wave being successively at the equidistant points of the level line 1, 2, 3, 4, &c.

A drop whose place of rest is 0, will then assume the corresponding positions in the vertical line OA: thus the points where this line cuts the successive wave-fronts show the positions of the vibrating drop at equal intervals of time.

By comparing the three cases it will be seen that the mode of the drop's vibration is distinct in each. In (1), it moves fastest at O, and then slackens its pace up to 4. 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 waves in the figure have been purposely drawn of 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 for himself similar figures with other wave-forms, and so convince himself, more thoroughly, that every distinct form of wave has its own special mode of drop-vibration.

12. The converse of this proposition is also true, viz. that each distinct mode of drop-vibration gives rise to a special form of wave. We will show this by actually constructing the form of wavę

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