surface of the sea were it not being traversed by

waves.

7. The length, amplitude, and form of a wave completely determine the wave, just as the length, breadth, and height of an oblong block of wood, i. e. its three dimensions, fix the size of the block. These three elements of a wave are mutually independent, that is to say, we may alter any one of them without altering the other two. This will be seen by a glance at the accompanying figures.

(2)

Fig. 4 (1)

А

(3)

(1) shows variation in length alone; (2) in am

plitude alone; (3) in form alone.

8. We will next study more closely the motion of an individual drop of water, in the surface of the sea, while a wave passes across it. Fig. 5 shows nine positions of the wave and moving drop at equal intervals of time, each one-eighth of the period during which the wave traverses a horizontal distance equal to its own wave-length. In (1), the front of the wave has just reached the drop previously at rest in the level-line represented by dots in the figure. In (2), the drop is a part of the way up the front of the crest; in (3), at the summit of the crest, and, therefore, at its greatest distance above the level-line. In (4), it is on the back of the crest, and, in (5), occupies its original position. It then crosses the level-line; is on the front of the trough in (6), and at its lowest point in (7), where it attains its greatest distance below the level-line. In (8), it is on the back of the trough, and, in (9), has once more returned to its starting-point in the level-line.

'We have here a vibratory or oscillatory movement, like that of the end, or 'bob,' of a clockpendulum, but executed in a vertical straight line. We call the distance between the two extreme positions of the bob, the extent of swing of the pendulum. The extent of the drop's oscillation will be seen, from (3) and (7), to be equal to the sum

of the height of the wave's crest above the levelline, and of the depth of the trough below it.

But this, as was shown in § 6, is equal to the amplitude of the wave. Hence 'extent of drop's vibration' and 'amplitude of corresponding wave' are only different ways of expressing the same thing.

Let the line A'OA be that in which the drop

Fig.6

0

under consideration vibrates, O being in the levelline, A and A' the limits of oscillation. The whole movement given in Fig. 5 will then be from 0 to A, from A through O to A', and from A' back again to 0. This is termed one complete vibration, and since, in the course of it, each portion of the drop's path is passed over twice, one complete vibration is equal to an upward swing from A' to A together with a downward swing from A to A. In the clock-pendulum we have, during each second, one complete oscillation, consisting of one swing from left to right and one from right to left.

Reference to Fig. 5 at once shows that, during the time occupied by the wave in traversing its own wave-length, the moving drop performs one complete vibration, or, to express the same fact in the reverse order, that while the drop makes one complete vibration, the wave advances through one wave-length. This is a most important principle, and should be thoroughly mastered and borne in mind by the student.

9. What has just been proved for a particular drop is, of course, equally true for any assigned drop in the surface passed over by a wave. All the drops, therefore, go through exactly the same vibrations in exactly equal times, but, since each drop can only start at the moment when the front of the wave reaches it [Fig. 5, (1)], they will in general occupy different positions in their paths at the same time. We may illustrate this by supposing a number of watches, which keep good time, to be set going successively in such a way that the first shall mark XII at twelve o'clock, the second at five minutes past twelve, the third at ten minutes past twelve, and so on. The hands of each watch will describe the same paths in equal times, but, at any assigned moment, will occupy different positions in those paths corresponding to the lateness of their several starts. The drops in the sea