On the determination of the boundary impedance from the far field pattern

TL;DR

This paper presents two methods for determining boundary impedance from far field patterns in Helmholtz problems, using Fourier analysis and WKB approximation, with numerical validation and discussion on approximation validity.

Contribution

It introduces explicit and WKB-based methods for impedance reconstruction from far field data, advancing inverse boundary value problem techniques.

Findings

Methods successfully reconstruct boundary impedance.

Numerical examples demonstrate approach efficiency.

Discussion on WKB approximation validity.

Abstract

We consider the Helmholtz equation in the half space and suggest two methods for determining the boundary impedance from knowledge of the far field pattern of the time-harmonic incident wave. We introduce a potential for which the far field patterns in specially selected directions represent its Fourier coefficients. The boundary impedance is then calculated from the potential by an explicit formula or from the WKB approximation. Numerical examples are given to demonstrate efficiency of the approaches. We also discuss the validity of the WKB approximation in determining the impedance of an obstacle.