Parabolic decomposition of the separable Helmholtz equation

Abstract

The solution of the separable Helmholtz equation is represented as the convolution of solutions of two related (formally) parabolic partial differential equations. Separable means that the square of the index of refraction is a function of depth plus a function of range and that the depth of the ocean is assumed to be constant. This representation decomposes the separable Helmholtz equation into two parabolic partial differential equations. One equation is the usual Fock‐Tappert parabolic equation for sound propagation. The solutions of the second parabolic equation may be regarded as candidate kernels of special integral transformations, or transmutations. All of these equations have many solutions, of course, so we present initial and/or boundary conditions that will determine solutions of the parabolic equations so that their convolution will give the Green's function for the Helmholtz equation. Examples of such transmutation representations will be presented.