Waves and ripples in water, air, and æther

by Sir J. A. Fleming

Fig. 13.—Water-wave produced in a tank.

From the motion of the bran we can see that the water swings backwards and forwards in a horizontal line with a pendulum-like motion, but its up-and-down or vertical motion is much more restricted. A wave of this kind travels along a canal with a speed which depends upon the depth of the canal. If waves of this kind are started in a very long trough, the wave-length being large compared with the depth of the trough,[8] it can be shown that the speed of the wave is equal to the velocity which would be gained by a stone or other heavy body in falling through half the depth of the canal. Hence, the deeper the water, the quicker the wave travels. This can be shown as an experimental fact as follows: Let two galvanized iron tanks be provided, each about 6 feet long and 1 foot wide and deep.

At one end of each tank a hollow cylinder, such as a coffee-canister or ball made water-tight, is floated, and it may be prevented from moving from its place by being attached to a hinged rod like the ball-cock of a cistern. The two tanks are placed side by side, and one is filled to a depth of 6 inches, and the other to a depth of 3 inches, with water. Two pieces of wood are then provided and joined together as in [Fig. 14], so as to form a double paddle. By pushing this through the water simultaneously in both tanks at the end opposite to that at which the floating cylinders are placed, it is possible to start two solitary waves, one in each tank, at the same instant. These waves rush up to the other end and cause the floats to bob up. It will easily be seen that the float on the deeper water bobs up first, thus showing that the wave on the deeper water has travelled along the tank more quickly than the wave on the shallower water.

Fig. 14.

In order to calculate the speed of the waves, we must call to mind the law governing the speed of falling bodies. If a stone falls from a height its speed increases as it falls. It can be shown that the speed in feet per second after falling from any height is obtained by multiplying together the number 8 and a number which is the square root of the height in feet.

Thus, for instance, if we desire to know the speed attained by falling from a height of 25 feet above the earth’s surface, we multiply 8 by 5, this last number being the square root of 25. Accordingly, we find the velocity to be 40 feet per second, or about 26 miles an hour.

The force of the blow which a body administers and suffers on striking the ground depends on the energy of motion it has acquired during the fall, and as this varies as the square of the speed, it varies also as the height fallen through.

Let us apply these rules to calculate the speed of a long wave in a canal having water 8 feet deep in it. The half-depth of the canal is therefore 4 feet. The square root of 4 is 2; hence the speed of the wave is that of a body which has fallen from a height of 4 feet, and is therefore 16 feet per second, or nearly 11 miles an hour. When we come to consider the question of waves made by ships, in the next chapter, a story will be related of a scientific discovery made by a horse employed in dragging canal-boats, which depended on the fact that the speed of long waves in this canal was nearly the same as the trotting speed of the horse.