Waves and ripples in water, air, and æther

by Sir J. A. Fleming

Fig. 12.

The difference may be illustrated as follows: Consider a row of glass or steel balls suspended by threads so hung as to be quite close to each other ([see Fig. 12]). Withdraw the first ball, and let it fall against the second one. The result is that the last ball of the row flies off with a jerk. In this case the whole energy imparted to the first ball is transmitted along the row of balls. The first ball, on falling against the second one, exerts on it a pressure which slightly squeezes both out of shape. This pressure is just sufficient to bring the first ball to rest. The second ball, in turn, expands after the blow and squeezes the third, and so on. Hence, in virtue of Newton’s Third Law of Motion, that “action and reaction are equal and opposite,” it follows that the pressure produced by the blow of the first ball is handed on from ball to ball, and finally causes the last ball to fly off.

In this case, owing to the rigid connection between the elastic balls, each one hands on to its neighbour the whole of the energy it receives. Supposing, however, that we separate the balls slightly, and give the first ball a transverse, or side-to-side swing. Then, owing to the fact that there is no connection between the balls, the energy imparted to the first ball would not be handed on at all, and no wave would be propagated.

Between these two extremes of the whole energy transferred and a solitary wave produced, and no energy transferred and no wave produced, we have a condition in which an initial disturbance of one ball gives rise to a wave-train and part of the energy is transferred.

For if we interconnect the balls by loose elastic threads, and then give, as before, a transverse or sideways impulse to the first ball, this will pull the second one and set it swinging, but it will be pulled back itself, and will be to some extent deprived of its motion. The same sharing or division of energy will take place between the second and third, and third and fourth balls, and so on. Hence the initial solitary vibration of the first ball draws out into a wave-train, and the originally imparted energy is spread out over a number of balls, and not concentrated in one of them. Accordingly, as time goes on, the wave-train is ever extending in length and the oscillatory motion of each ball is dying away, and the original energy gets spread over a wider and wider area or number of balls, but is propagated with less speed than the wave-velocity for that medium.

There need be no difficulty in distinguishing between the notion of a wave-velocity and a wave-train velocity, if we remember that the wave travels a distance equal to a wave-length in the time taken by one oscillation. Hence the wave-velocity is measured by taking the quotient of the wave-length by the time of one complete vibration.

If, for example, the wave-length of a water wave is 4 inches, and we observe that twelve waves pass any given point in 3 seconds, we can at once infer that the wave-velocity is 16 inches per second. The transference of energy may, however, take place so that the whole group of waves moves forward much more slowly. They move forward because the waves are dying out in the rear of the group and being created in the front, and the rate of movement of the group is, in the case of deep-water waves, equal to half that of the single-wave velocity.

A very rough illustration of this difference between a group velocity and an individual velocity may be given by supposing a barge to be slowly towed along a river. Let a group of boys run along the barge, dive over the bows, and reappear at the stern and climb in again. Then the velocity of the group of boys on the barge is the same as the speed of the barge, but the speed of each individual boy in space is equal to the speed of the barge added to the speed of each boy relatively to the barge. If the barge is being towed at 3 miles an hour, and the boys run along the boat also at 3 miles an hour, then the velocity of the group of boys is only half that of the individual boy, because the former is 3 miles an hour and the latter is 6 miles an hour.

Before leaving the subject of sea waves there are two or three interesting matters which must be considered. In the first place, the breaking of a wave on the shore or on shallow water calls for an explanation. If we watch a sea wave rolling in towards the beach, we shall notice that, as it nears the shore, it gets steeper on the shore side, and gradually curls over until it falls and breaks into spray. The reason is because, as the wave gets into the shallow water, the top part of the wave advances more rapidly than the bottom portion. It has already been explained that the path of the water-particle is a circle, with its plane vertical and perpendicular to the wave-front or line.

Accordingly, if the wave is moving in shallow water, the friction of the water against the bottom retards the backward movement at the lowest position of the water, but no such obstacle exists to the forward movement of the water at its highest position. An additional reason for the deformation of the wave on a gently sloping shore may be found in the fact that the front part of the wave is then in shallower water, and hence moves more slowly than the rearward portion in deeper water. From both causes, however, the wave continually gets steeper and steeper on its landward side until it curls over and tumbles down like a house which leans too much on one side. The act of curling over in a breaking wave is a beautiful thing to watch, and one which attracts the eye of every artist who paints seascapes and storm waves, or of any lover of Nature who lingers by the shore.