Fig. 5.—A section of the same picture enlarged.
(From Grimsehl, Lehrbuch der Physik.)
Let us suppose that we are in a boat which is anchored on a body of water and let us watch the regular waves which pass us. If there is neither wind nor current, a light body like a cork, lying on the surface, rises with the wave crests and sinks with the troughs, going forward slightly with the former and backward with the latter, but remaining, on the whole, in the same spot. Since the cork follows the surrounding water particles, it shows their movements, and we thus see that the individual particles are in oscillation, or more accurately, in circulation, one circulation being completed during the time in which the wave motion advances a wave-length, i.e., the distance from one crest to the next. This interval of time is called the time of oscillation, or the period. If the number of crests passed in a given time is counted, the oscillations of the individual particles in the same time can be determined. The number of oscillations in the unit of time, which we here may take to be one minute, is called the frequency. If the frequency is forty and the wave-length is three metres, the wave progresses 3 × 40 = 120 metres in one minute. The velocity with which the wave motion advances, or in other words its velocity of propagation, is then 120 metres per minute. We thus have the rule that velocity of propagation is equal to the product of frequency and wave-length ([cf. Fig. 8]).
On the surface of a body of water there may exist at the same time several wave systems; large waves created by winds which have themselves perhaps died down, small ripples produced by breezes and running over the larger waves, and waves from ships, etc. The form of the surface and the changes of form may thus be very complicated; but the problem is simplified by combining the motions of the individual wave systems at any given point. If one system at a given time gives a crest and another at the same instant also gives a crest at the same point, the two together produce a higher crest. Similarly, the resultant of two simultaneous troughs is a deeper trough; a crest from one system and a simultaneous trough from the other partially destroy or neutralize each other. A very interesting yet simple case of such “interference” of two wave systems is obtained when the systems have equal wave-lengths and equal amplitudes. Such an interference can be produced by throwing two stones, as much alike as possible, into the water at the same time, at a short distance from each other. When the two sets of wave rings meet there is created a network of crests and troughs. Figs. 4 and 5 show photographs of such an interference, produced by setting in oscillation two spheres which were suspended over a body of water.
Fig. 6.—Schematic representation of an interference.
In [Fig. 6] there is a schematic representation of an interference of the same nature. Let us examine the situation at points along the lower boundary line. At 0, which is equidistant from the two wave centres, there is evidently a wave crest in each system; therefore there is a resultant crest of double the amplitude of a single crest if the two systems have the same amplitude. Half a period later there is a trough in each system with a resultant trough of twice the amplitude of a single trough. Thus higher crests and deeper troughs alternate. The same situation is found at point 2, a wave-length farther from the left than from the right wave centre; in fact, these results are found at all points such as 2, 2′, 4 and 4′, where the difference in distance from the two wave centres is an even number of wave-lengths. At the point 1, on the other hand, where the difference between the distance from the centres is one-half a wave-length, a crest from one system meets a trough from the other, and the resultant is neither crest nor trough but zero. There is the same result at points 1′, 3, 3′, 5, 5′, etc., where the difference between the distances from the two wave centres is an odd number of half wave-lengths. By throwing a stone into the water in front of a smooth wall an interference is obtained, similar to the one described above. The waves are reflected from the wall as if they came from a centre at a point behind the wall and symmetrically placed with respect to the point where the stone actually falls.
Fig. 7.—Waves which are reflected by a board and pass through a hole in it.