Scattering of a plane scalar wave by fundamental surfaces with mixed boundary conditions

Abstract

The problem of scattering of an arbitrarily directed scalar incident plane wave by a fundamental surfacE, of orthogonality with mixed linear boundary conditions is solved for arbitrary separable coordinate system. From the Fredholm integral equation of the second kind, derived from the Helmholtz solution, a simple linear relationship is found between the unknown coefficients in the expansions of the wave potential function and its normal derivative on the fundamental surface of orthogonality, in terms of the eigenfunctions of the separable coordinate system. By applying the mixed linear boundary conditions on the fundamental surface of orthogonality another set of simple linear relationships between the unknown coefficients is found. From the two sets of linear equations, the unknown coefficients may be determined and the scattered field calculated. The above method may be applied to both Laplace and Helmholtz equations. Since we require for the solution only the free space Green’s function expansion, the problem of finding explicit representations of the various specialized Green’s functions is avoided.