Spectral volumetric integral equation methods for acoustic medium scattering in a 3D waveguide

Abstract

The Lippmann–Schwinger integral equation describes the scattering of acoustic waves from an inhomogeneous medium. For scattering problems in free space, Vainikko proposed a fast spectral solution method exploiting the convolution structure of this equation's integral operator and the fast Fourier transform. Although the integral operator of the Lippmann–Schwinger integral equation for scattering in a planar three-dimensional waveguide is not a convolution, we show in this paper that the separable structure of the kernel allows to construct fast spectral collocation methods. The numerical analysis of this method requires smooth material parameters; for discontinuous materials there is no theoretical convergence statement. Therefore, we construct a Galerkin variant of Vainikko's method avoiding this drawback. For several distant scattering objects inside the three-dimensional waveguide this discretization technique would lead to a computational domain consisting of one large box containing all scatterers and hence many unnecessary unknowns. However, the integral equation can be reformulated as a coupled system with unknowns defined on the different parts of the scatterer. Discretizing this coupled system by a combined spectral/multipole approach yields an efficient method for waveguide scattering from multiple objects.