Wave Scattering by a Perforated Duct

Abstract

Recently, a number of linear wave scattering problems have been studied in the context of embedding formulae. Such formulae typically express the solution for any incident wave angle in terms of the solutions corresponding to a finite number of angles; they are of interest from a theoretical point of view and because they lead to significant computational savings. Here we investigate the diffraction and radiation of plane waves by a straight, infinitely long, impermeable duct, the walls of which are punctuated by an arbitrary arrangement of finitely many gaps, each of finite extent. This problem is reduced to a matrix integral equation which may be regarded as the canonical form of a wide class of similar problems, and embedding formulae are derived for this extended class. The formulae are illustrated by developing a variational method for the matrix integral equation to generate the minimal solution set required.