Simple progressive solutions of the wave equation

Abstract

The object of this paper is to determine all the solutions of the wave equation which are of the simple form where F denotes an arbitrary function. It will be shown that, in addition to the obvious cases of plane or spherical progressive waves, such solutions exist only when the wave fronts are certain algebraic surfaces of the fourth order, the cyclides of Dupin. These include, as degenerate cases, the sphere, the plane, the cylinder, the cone, and the torus.