Waves and ripples in water, air, and æther

by Sir J. A. Fleming

Fig. 15.

It may be well, as a little digression, to point out how the law connecting height fallen through and velocity acquired by the falling body may be experimentally illustrated for teaching purposes.

The apparatus is shown in [Fig. 15]. It consists of a long board placed in a horizontal position and held with the face vertical. This board is about 16 feet long. Attached to this board is a grooved railway, part of which is on a slope and part is horizontal. A smooth iron ball, A, about 2 inches in diameter, can run down this railway, and is stopped by a movable buffer or bell, B, which can be clamped at various positions on the horizontal rail. At the bottom of the inclined plane is a light lever, T, which is touched by the ball on reaching the bottom of the hill. The trigger releases a pendulum, P, which is held engaged on one side, and, when released, it takes one swing and strikes a bell, G. The pendulum occupies half a second in making its swing. An experiment is then performed in the following manner: The iron ball is placed at a distance, say, of 1 foot up the hill and released. It rolls down, detaches the pendulum at the moment it arrives at the bottom of the hill, and then expends its momentum in running along the flat part of the railway. The buffer must be so placed by trial that the iron ball hits it at the instant when the pendulum strikes the bell. The distance which the buffer has to be placed from the bottom of the hill is a measure of the velocity acquired by the iron ball in falling down the set distance along the hill. The experiment is then repeated with the iron ball placed respectively four times and nine times higher up the hill, and it will be found that the distances which the ball runs along the flat part in one half-second are in the ratio of 1, 2, and 3, when the heights fallen through down the hill are in the ratio of 1, 4, and 9.

The inference we make from this experiment is that the velocity acquired by a body in falling through any distance is proportional to the square root of the height. The same law holds good, no matter how steep the hill, and therefore it holds good when the body, such as a stone or ball, falls freely through the air.

The experiment with the ball rolling down a slope is an instructive one to make, because it brings clearly before the mind what is meant by saying, in scientific language, that one thing “varies as the square root” of another. We meet with so many instances of this mode of variation in the study of physics, that the reader, especially the young reader, should not be content until the idea conveyed by these words has become quite clear to him or her.

Thus, for instance, the time of vibration of a simple clock pendulum “varies as the square root of the length;” the velocity of a canal wave “varies as the square root of the depth of the canal;” and the velocity or speed acquired by a falling ball “varies as the square root of the distance fallen through.” These phrases mean that if we have pendulums whose lengths are in the ratio of 1 to 4 to 9, then the respective times of their vibration are in the ratio of 1 to 2 to 3. Also a similar relation connects the canal-depth and wave-velocity, or the ball-velocity and height of fall.

Returning again to canal waves, it should be pointed out that the real path of a particle of water in the canal, when long waves are passing along it, is a very flat oval curve called an ellipse. In the extreme cases, when the canal is very wide and deep, this ellipse will become nearly a circle; and, on the other hand, when narrow and shallow, it will be nearly a straight line. Hence, if long waves are created in a canal which is shallow compared with the length of the wave, the water-particles simply oscillate to and fro in a horizontal line. There is, however, one important fact connected with wave-propagation in a canal, which has a great bearing on the mode of formation of what is called a “bore.”

As a wave travels along a canal, it can be shown, both experimentally and theoretically, that the crest of the wave travels faster than the hollow, and as a consequence the wave tends to become steeper on its front side, and its shape then resembles a saw-tooth.

A very well known and striking natural phenomenon is the so-called “bore” in certain tidal rivers or estuaries. It is well seen on the Severn in certain states of the tide and wind. The tidal wave returning along the Severn channel, which narrows rapidly as it leaves the coast, becomes converted into a “canal wave,” and travels with great rapidity up the channel. The front side of this great wave takes an almost vertical position, resembling an advancing wall of water, and works great havoc with boats and shipping which have had the misfortune to be left in its path. To understand more completely how a “bore” is formed, the reader must be reminded of the cause of all tidal phenomena. Any one who lives by the sea or an estuary knows well that the sea-level rises and falls twice every 24 hours, and that the average interval of time between high water and high water is nearly 12¹⁄₂ hours. The cause of this change of level in the water-surface is the attraction exerted by the sun and moon upon the ocean. The earth is, so to speak, clothed with a flexible garment of water, and this garment is pulled out of shape by the attractive force of our luminaries; very roughly speaking, we may say that the ocean-surface is distorted into a shape called an ellipsoid, and that there are therefore two elevations of water which march across the sea-covered regions of the earth as it revolves on its axis. These elevations are called the tidal waves. The effects, however, are much complicated by the fact that the ocean does not cover all parts of the earth. There is no difficulty in showing that, as the tidal wave progresses round the earth across each great ocean, it produces an elevation of the sea-surface which is not simultaneous at all places. The time when the crest of the tidal wave reaches any place is called the “time of high tide.” Thus if we consider an estuary, such as that of the Thames, there is a marked difference between the time of high tide as we ascend the estuary.

Taking three places, Margate, Gravesend, and London Bridge, we find that if the time of high tide at Margate is at noon on any day, then it is high tide at Gravesend at 2.15 p.m., and at London Bridge a little before three o’clock. This difference is due to the time required for the tidal wave to travel up the estuary of the Thames.