Diffraction by an impedance strip I. Reducing diffraction problem to Riemann-Hilbert problems

TL;DR

This paper reduces a 2D acoustic wave diffraction problem involving an impedance strip to two matrix Riemann-Hilbert problems, laying groundwork for solving complex scattering issues through advanced mathematical techniques.

Contribution

It introduces a method to transform the diffraction problem into Riemann-Hilbert problems using Wiener--Hopf and Hurd's methods, facilitating future solutions.

Findings

Formulation of diffraction problem as Riemann-Hilbert problems

Application of Wiener--Hopf and Hurd's methods for problem reduction

Preparation for solving Riemann-Hilbert problems with a novel OE-equation method

Abstract

A 2D problem of acoustic wave scattering by a segment bearing impedance boundary conditions is considered. In the current paper (the first part of a series of two) some preliminary steps are made, namely, the diffraction problem is reduced to two matrix Riemann-Hilbert problems with exponential growth of unknown functions (for the symmetrical part and for the antisymmetrical part). For this, the Wiener--Hopf problems are formulated, they are reduced to auxiliary functional problems by applying the embedding formula, and finally the Riemann-Hilbert problems are formulated by applying the Hurd's method. In the second part the Riemann-Hilbert problems will be solved by a novel method of OE-equation.