Novel solutions of the Helmholtz equation and their application to diffraction

Abstract

This paper presents a series of novel representations for the solutions of the Helmholtz equation in a broad class of wedge-like domains including those with curvilinear, non-flat faces. These representations are obtained by an original method which combines ray theory with the probabilistic approach to partial differential equations and uses a specific technique to deal with a need for analytical continuation of the specified boundary function. The main results are reminiscent of the standard Feynman–Kac formula but differ in that the averaging over solutions of stochastic differential equations is replaced by averaging over the trajectories of a new two-scale random motion introduced here. The paper focuses on the development of the solutions and for this reason it includes only a brief outline of numerous applications, consequences, extensions and variations of the method, which include, but are not limited to, problems of diffraction and scattering, problems in three-dimensional domains and problems of wave propagation in non-homogeneous media.