The propagation of groups of waves in dispersive media, with application to waves on water produced by a travelling disturbance

Abstract

The object of this paper is to illustrate the main features of wave propagation in dispersive media. In the case of surface waves on deep water it has been remarked that the earlier investigators considered the more difficult problem of the propagation of an arbitrary initial disturbance as expressed by a Fourier integral, ignoring the simpler theory developed subsequently by considering the propagation of a single element of their integrals, namely an unending train of simple harmonic waves. The point of view on which stress is laid here consists of a return to the Fourier integral, with the idea that the element of disturbance is not a simple harmonic wave-train, but a simple group, an aggregate of simple wave-trains clustering around a given central period. In many cases it is then possible to select from the integral die few simple groups that are important, and hence to isolate the chief regular features, if any, in the phenomena. In certain of the following sections well-known results appear; the aim has been to develop these from the present point of view, and so illustrate die dependence of the phenomena upon the character of the velocity function, In the other sections it is hoped that progress has been made in the theory if the propagation of an arbitrary initial group of waves, and also of the character of the wave pattern diverging from a point impulse travelling on die surface.