Diffraction by an impedance strip II. Solving Riemann-Hilbert problems by OE-equation method

TL;DR

This paper introduces a novel OE-equation method to solve Riemann-Hilbert problems arising in acoustic wave diffraction by impedance segments, providing a new numerical approach for these complex boundary value problems.

Contribution

The paper develops and applies the OE-equation method to solve Riemann-Hilbert problems in diffraction, advancing analytical and numerical techniques for impedance boundary problems.

Findings

Successfully formulates the OE-equation for the Riemann-Hilbert problem

Demonstrates the numerical solution of the OE-equation

Provides a new approach for acoustic wave diffraction analysis

Abstract

The current paper is the second part of a series of two papers dedicated to 2D problem of diffraction of acoustic waves by a segment bearing impedance boundary conditions. In the first part some preliminary steps were made, namely, the problem was reduced to two matrix Riemann-Hilbert problem. Here the Riemann-Hilbert problems are solved with the help of a novel method of OE-equations. Each Riemann-Hilbert problem is embedded into a family of similar problems with the same coefficient and growth condition, but with some other cuts. The family is indexed by an artificial parameter. It is proven that the dependence of the solution on this parameter can be described by a simple ordinary differential equation (ODE1). The boundary conditions for this equation are known and the inverse problem of reconstruction of the coefficient of ODE1 from the boundary conditions is formulated. This problem is called the OE-equation. The OE-equation is solved by a simple numerical algorithm.