With the appliances here described many beautiful effects can be shown, illustrating the independence of different wave-trains and their _interference_. If we hurl two stones into a lake a little way apart, and thus create two sets of circular ripples (see Fig. 20), we shall notice that these two ripple-trains pass freely through each other, and each behave as if the other did not exist. A careful examination will, however, show that at some places the water-surface is not elevated or disturbed at all, and at others that the disturbance is increased. [Illustration: FIG. 20.—Intersecting ripples produced on a lake by throwing in simultaneously two stones.] 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. [Illustration: 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). [Illustration: FIG. 22.]