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

✍ Scribed by E. Meister; K. Rottbrand

• Publisher: John Wiley and Sons
• Year: 1991
• Tongue: English
• Weight: 838 KB
• Volume: 14
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1670140802

✦ Synopsis

Abstract

A 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 H^1^ (Ω) solutions is presented. The boundary/transmission problem is shown to be equivalent to a (2__N__ × 2__N__)‐Wiener–Hopf (WH) system for jumps of the Dirichlet–and Neumann–Cauchy data across the semi‐infinite screens ∑~m~. The (2__N__ × 2__N__)‐Fourier symbol matrix 𝔖~𝒫~ 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~.