WKB approach to evaluate series of Mathieu functions in scattering problems

TL;DR

This paper introduces a WKB-based method to accurately evaluate Mathieu function series in wave scattering problems involving elliptic obstacles, especially when standard approximations are inadequate.

Contribution

The paper develops a WKB approach for Mathieu functions that improves the precision of scattering calculations in cases where wavelength and obstacle size are comparable.

Findings

WKB approximations outperform standard methods in certain regimes.

Numerical examples demonstrate improved accuracy in Green function computations.

Method applicable to optics, quantum mechanics, and fluid dynamics scattering problems.

Abstract

The scattering of a wave obeying Helmholtz equation by an elliptic obstacle can be described exactly using series of Mathieu functions. This situation is relevant in optics, quantum mechanics and fluid dynamics. We focus on the case when the wavelength is comparable to the obstacle size, when the most standard approximations fail. The approximations of the radial (or modified) Mathieu functions using WKB method are shown to be especially efficient, in order to precisely evaluate series of such functions. It is illustrated with the numerical computation of the Green function when the wave is scattered by a single slit or a strip (ribbon).