Let us consider, in the first place, the effect of sending out into the air two sets of air waves of slightly different wave lengths. These waves both travel at the same rate, hence we shall not affect the combined effects of the waves upon the air if we consider both sets of waves to stand still. For the sake of simplicity, we will consider that the wave-length of one train is 20 inches, and that of the other is 21. Moreover, let the two wave-trains be so placed relatively to one another that they both start from one point in the same phase of movement; that is, let their zero points, or their humps or hollows, coincide. Then if we draw two wavy lines (see Fig. 56) to represent these two trains, it will be evident that, since the wave-length of one is 1 inch longer than that of the other—that is, a distance equal to twenty wave-lengths—one wave-train will have gained a whole wave-length upon the other, and in a distance equal to ten wave-lengths, one wave train will have gained half a wave-length upon the other. If we therefore imagine the two wave-trains superimposed, we shall find, on looking along the line of propagation, an alternate doubling or destruction of wave-effect at regular intervals. In other words, the effect of superimposing two trains of waves of slightly different wave-lengths is to produce a resultant wave-train in which the wave-amplitude increases up to a certain point, and then dies away again nearly to nothing, as shown in the lowest of the three wave-lines in Fig. 56. [Illustration: FIG. 56.—The formation of beats by two wave-trains.] We must, then, determine how far apart these points of maximum wave-amplitude or points of no wave effect lie. If the wave-length of one train is, as stated, 20 inches, then a length of ten wave-lengths is 200 inches, and this must be, therefore, the distance from a place of maximum combined wave-effect to a place of zero wave-effect. Accordingly, the distance between two places where the two wave trains help one another must be 400 inches, and this must also be the distance between two adjacent places of wave-destruction. If, therefore, we look along the wavy line representing the resultant wave, every 400 inches we shall find a maximum wave-amplitude, and every 400 inches a place where the waves have destroyed each other. We may call this distance _a wave-train length_, and it is obviously equal to the product of the constituent wave-lengths divided by the difference of the two constituent wave-lengths.