This relation, which we shall have frequent occasion to recall, may be stated in another manner. We call the period of a wave the time taken to make one complete movement. The periodic time is therefore inversely proportional to the frequency. Hence we can say that the wave-length, divided by the periodic time, gives us the wave-velocity.
In the case of water waves and ripples, the wave-velocity is determined by the wave-length. This is not the case, as we shall see, with waves in air or waves in æther. In these latter cases, as far as we know, waves of all wave-lengths travel at the same rate. Long sea waves, however, on deep water travel faster than short ones.
A formal and exact proof of the law connecting speed and wave-length for deep-sea waves requires mathematical reasoning of an advanced character; but its results may be expressed in a very simple statement, by saying that, in the case of waves on deep water, the speed with which the waves travel, reckoned in miles per hour, is equal to the square root of 2¹⁄₄ times the wave-length measured in feet. Thus, for instance, if we notice waves on a deep sea which are 100 feet from crest to crest, then the speed with which those waves are travelling, reckoned in miles per hour, is a number obtained by taking the square root of 2¹⁄₄ times 100, viz. 225. Since 15 is the square root of 225 (because 15 times 15 is 225), the speed of these waves is therefore 15 miles an hour.
In the same way it can be found that Atlantic waves 300 feet long would travel at the rate of 26 miles an hour, or as fast as a slow railway train, and much faster than any ordinary ship.[1]
The above rule for the speed of deep-sea waves, viz. wave-velocity = square root of 2¹⁄₄ times the wave-length, combined with the general rule, wave-velocity = wave-length multiplied by frequency, provides us with a useful practical method of finding the speed of deep-sea waves which are passing any fixed point. Suppose that a good way out at sea there is a fixed buoy or rock, and we notice waves racing past it, and desire to know their speed, we may do it as follows: Count the number of waves which pass the fixed point per minute, and divide the number into 198; the quotient is the speed of the waves in miles per hour. Thus, if ten waves per minute race past a fixed buoy, their velocity is very nearly 20 miles an hour.[2]
Waves have been observed by the Challenger 420 to 480 feet long, with a period of 9 seconds. These waves were 18 to 22 feet high. Their speed was therefore 50 feet per second, or nearly 30 knots. Atlantic storm waves are very often 500 to 600 feet long, and have a period of 10 to 11 seconds. Waves have been observed by officers in the French Navy half a mile in length, and with a period of 23 seconds.
It has already been explained that in the case of deep-sea waves the individual particles of water move in circular paths. It can be shown that the diameter of these circular paths decreases very rapidly with the depth of the particle below the surface, so that at a distance below the surface equal only to one wave-length, the diameter of the circle which is described by each water-particle is only ¹⁄₅₃₅ of that at the surface.[3] Hence storm waves on the sea are a purely surface effect. At a few hundred feet down—a distance small compared with the depth of the ocean—the water is quite still, even when the surface is tossed by fearful storms, except in so far as there may be a steady movement due to ocean currents.
By a more elaborate examination of the propagation of wave-motion on a fluid, Sir George Stokes showed, many years ago, that in addition to the circular motion of the water-particles constituting the wave, there is also a transfer of water in the direction in which the wave is moving, the speed of this transfer depending on the depth, and decreasing rapidly as the depth increases. This effect, which is known to sailors as the “heave of the sea,” can clearly be seen on watching waves on not very deep water. For the crest of the wave will be seen to advance more rapidly than the hollow until the wave falls over and breaks; and then a fresh wave is formed behind it, and the process is repeated. Hence waves break if the depth of water under them diminishes; and we know by the presence of breakers at any place that some shallow or sandbank is located there.
It is necessary, in the next place, to point out the difference between a mere wave-motion and a true wave. It has been explained that in a wave-motion each one of a series of contiguous objects executes some identical movement in turn. We have all seen the wind blowing on a breezy day across a cornfield, and producing a sort of dark shadow which sweeps along the field. This is clearly caused by the wind bending down, in turn, each row of cornstalks, and as row after row bows itself and springs up again, we are presented with the appearance of a wave-motion in the form of a rift rushing across the field.
