Addition of Waves.—The superposition of two waves so as to obtain the effect of both waves at the same place is carried out very simply. The displacements at any point due to the two waves separately are algebraically added together, and this sum is the actual displacement. In Fig. 13 the dotted lines represent two simple waves, one of which has double the wave-length of the other. At any point P on the solid line, the displacement PN is equal to the algebraic sum of the displacement NQ due to one of the waves and NR due to the other. The solid line, therefore, represents the resulting wave. We may repeat this process for any number of simple waves, and by suitably choosing the wave-length and amplitude of the simple waves we may build up any desired form of wave. The mathematician Fourier has shown that any form of wave, even the single irregular disturbance, can thus be expressed as the sum of a series of simple waves and that the wave-lengths of these simple waves are equal to the original wave-length, one-half of it, one-third, one-quarter, one-fifth, and so on in an infinite series. Fourier has also shown that only one such series is possible for any particular form of wave.
FIG. 13.
The importance of this mathematical expression lies in the fact that in a number of ways Fourier's series of simple waves is manufactured from the original wave and the different members of the series become separated. Thus the most useful way in which we can represent any wave is, not to draw the actual form of a wave, but to represent what simple waves go to form it and to show how much energy there is in each particular simple wave.
Energy—Wave-length Curve.—This can be done quite simply as in Fig. 14. The distance PN from the line OA being drawn to scale to represent the energy in the simple wave whose length is represented by ON.
FIG. 14.