Wiener‐hopf equations for waves scattered by a system of parallel sommerfeld half‐planes

Abstract

AbstractA scalar time‐harmonic wave (governed by Helmholtz's equation) impinges on N semi‐infinite half‐planes. The scattered field is sought when first, second, and third‐kind boundary conditions or even general linear transmission conditions on the plates ∑m and their complementary parts ∑ are prescribed. Making use of the Fourier transform a representation formula for H1 (Ω) solutions is presented. The boundary/transmission problem is shown to be equivalent to a (2N × 2N)‐Wiener–Hopf (WH) system for jumps of the Dirichlet–and Neumann–Cauchy data across the semi‐infinite screens ∑m. The (2N × 2N)‐Fourier symbol matrix 𝔖𝒫 contains N block matrices on the diagonal corresponding to Sommerfeld boundary/transmission problems for a single plate. These (2 × 2)‐symbol matrices are factorizable and thus the full WH system is invertible by a perturbation argument for not too small spacings of neighbouring screens ∑m.