On the Exact Solution of Boundary Value Problems with Non-separable Geometries, Like a Rough Surface

Abstract

A Dirichlet problem is considered for the Helmholtz equation and for a class of geometries for which the Helmholts operator does not separate (e.g. a rough surface). It is shown that, contrary to the widely held view, it is possible to obtain the solution of this problem in a closed form which resembles closely the solutions obtained for separable geometries—expansions generated by Sturm-Liouville theory. As with separable geometries, we show in particular that the expansion coefficients can be written explicitly as integrals containing a priori known functions—that matrix inversion is not required for the determination of the expansion coefficients. The problem includes as a special case the scattering of electromagnetic waves from a rough cylindrical surface which is the boundary of a perfect conductor. The method is very general and can be used for much more complicated boundary value problems, such as scattering by dielectric interface.