THE COURANT-HILBERT SOLUTIONS OF THE WAVE-EQUATION

The Bateman transformations which generate many new solutions from a particular solution of the scalar wave equation in a homogeneous medium are discussed. The Bateman transformations are then used to classify the Courant-Hilbert solutions of the wave equation. The latter are the product of an arbitrary function F of the phase u, a solution of the characteristic equation, and of a specific attenuation factor f that does not depend on the particular form of F. It is shown that for a given phase u, f is not unique. Consequently, there exist spherical waves F(r+/-ct) with an attenuation factor different from r-1.