If two sets of waves set out from different origins and arrive simultaneously at the same spot, then it is clear that if the crests or hollows of both waves reach that point at the same instant, the agitation of the water will be increased. If, however, the crest of a wave from one source reaches it at the same time as the hollow of another equal wave from the other origin, then it is not difficult to see that the two waves will obliterate each other. This mutual destruction of wave by wave is called interference, and it is a very important fact in connection with wave-motion. It is not too much to say that whenever we can prove the existence of interference, that alone is an almost crucial proof that we are dealing with wave-motion. The conditions under which interference can take place must be examined a little more closely. Let us suppose that two wave-trains, having equal velocity, equal wave-length, and equal amplitude or wave-height, are started from two points, A and B ([see Fig. 21]). Consider any point, P. What is the condition that the waves from the two sources shall destroy each other at that point? Obviously it is that the difference of the distances AP and BP shall be an odd number of half wave-lengths. For if in the length AP there are 100 waves, and in the distance BP there are 100¹⁄₂ waves, or 101¹⁄₂ or 103¹⁄₂, etc., waves, then the crest of a wave from A will reach P at the same time as the hollow of a wave from B, and there will be no wave at all at the point P. This is true for all such positions of P that the difference of its distances from A and B are constant.
Fig. 21.
But again, we may choose a point, Q, such that the difference of its distances from A and B is equal to an even number of half wave-lengths, so that whilst in the length AQ there are, say, 100 waves, in the distance BQ there are 101, 102, 103, etc., waves. When this is the case, the wave-effects will conspire or assist each other at Q, and the wave-height will be doubled. If, then, we have any two points, A and B, which are origins of equal waves, we can mark out curved lines such that the difference of the distances of all points on these lines from these origins is constant. These curves are called hyperbolas ([see Fig. 22]).
Fig. 22.
All along each hyperbola the disturbance due to the combined effect of the waves is either doubled or annulled when compared with that due to each wave-train separately. With the apparatus described, we can arrange to create and adjust two sets of similar water ripples from origins not far apart, and on looking at the complicated shadow-pattern due to the interference of the waves, we shall be able to trace out certain white lines along which the waves are annulled, these lines being hyperbolic curves ([see Fig. 23]). With the same appliances another characteristic of wave-motion, which is equally important, can be well shown.
Fig. 23.—Interfering ripples on a mercury surface, showing interference along hyperbolic lines (Vincent).
We make one half of the circular tank in which the ripples are generated much more shallow than the other half, by placing in it a thick semicircular plate of glass. It has already been explained that the speed with which long waves travel in a canal increases with the depth of the water in the canal. The same is true, with certain restrictions, of ripples produced in a confined space or tank, one part of which is much shallower than the rest. If waves are made by dropping water on to the water-surface in the deeper part of the tank, they will travel more quickly in this deeper part than in the shallower portion. We can then adjust the water-dropping jet in such a position that it creates circular ripples which originate in deep water, but at certain places pass over a boundary into a region of shallower water ([see Fig. 24]). The left-hand side of the circular tank represented in the diagram is more shallow than the right-hand side.