horizontally in a direction perpendicular to the lines of its ridges and hollows. The actual motion of the water is, however, very different from its apparent motion, as may be ascertained by noticing the behaviour of a cork, or other body, floating on the surface of the sea, and therefore sharing its movement. Instead of steadily advancing, like the waves, the cork merely performs a heaving motion as the successive waves reach it, alternately riding over their crests and sinking into their troughs, as if anchored in the position it happens to occupy. Hence, while the waves travel steadily forward horizontally, the drops of water composing them are in a state of swaying to-and-fro motion, each separate drop rising and falling in a vertical straight line, but having no horizontal motion whatever1.

Thus, when we say that the waves advance horizontally, we mean, not that the masses of water of which they at any given instant consist, advance, but that these masses, by virtue of the separate vertical motions of their individual drops, successively arrange themselves in the same relative positions, so that the curved shapes of the surface, which we call waves, are transmitted without their materials sharing in the pro

1

This statement, though not strictly accurate, is sufficiently near the truth for our present purpose. See Weber's Wellenlehre.

Let ABCDEF represent a section of a part of the sea-surface at any given instant, and suppose that during, say, the next ensuing second of time, the separate drops in ABCDEF move vertically, either upwards or downwards as shown by the arrows, so that, at the end of that second, they all occupy positions along the dotted curved line A'B'C'D'EF". The two portions, ABCDE and B'C'D'EF', are exactly alike, and, therefore, the effect is just what it would have been had we pushed the curve ABCDE along horizontally until it came to occupy the position B'C'D'EF'.

In order further to illustrate this point, let us suppose that a hundred men are standing in a line and that the first ten are ordered to kneel down: a spectator who is too far off to distinguish individuals will merely see a broken line like that in the figure below.

(1)

Now, suppose that, after one second, the eleventh man is ordered to kneel and the first to stand; after two seconds the twelfth man to kneel and the second to stand; and so on. There will then continue to be a row of ten kneeling men, but, during each second, it will be shifted one place along the line. The distant observer will therefore see a depression steadily advancing along the line. The state of things presented to his eye after two, six, and nine seconds, respectively, is shown in Fig. 2.

Fig.2

There is here no horizontal motion on the part of the men composing the line, but their vertical motions give rise, in the way explained, to the horizontal transference of the depression along the line. The reader should observe that for no two consecutive seconds does the kneeling row consist of exactly the same men, while in such positions as

those shown in the figure, which are separated by more than ten seconds of time, the men who form it are totally different.

6. Let us now return to the sea-waves, and examine more closely the elements of which they consist.

Fig. 3 represents a vertical section of one complete wave.

The dotted line is that in which the horizontal plane, forming the surface of the sea when at rest, cuts the plane of the figure. The distance between the two extreme points of the wave, measured along this line, is called the length of the wave. C is the highest point of the crest DCB; E the lowest point of the trough AED. CF and GE are vertical straight lines through C and E; HCK and LEM are horizontal straight lines through the same two points. The vertical distance between the lines HK and LM is called the breadth or amplitude of the wave. Thus AB is the length of the wave, and, if we produce EG and CF to cut the lines HK and

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LM in N and P respectively, we have, for its amplitude, either of the equal lines EN, PC. Each of these is clearly equal to FC and GE together, that is to say, the amplitude of the wave is equal to the height of the crest above the levelline together with the depth of the trough below it. In addition to the length and amplitude of the wave, we have one more element, its form. The wave in the figure has its crest shorter than its trough and higher than its trough is deep. Moreover the part DC of the crest is steeper than the part CB, while, in the trough, the parts AE and EB are equally steep. Sea-waves have the most varied shapes according to the direction and force of the wind producing them. Hence, before we can lay down a wave in a figure, we must know the nature of the wave's curve, or, in other words, its form.

Since the crests of the waves are raised above the ordinary level of the sea, the troughs must necessarily be depressed below it, just as, in a ploughed field, the earth heaped up to form the ridges must be taken out of the furrows. Each crest being thus associated with a trough, it is convenient to regard one crest and one trough as forming together one complete wave. Thus each wave consists of a part raised above, and a part depressed below, the horizontal plane which would be the