Acoustic wave diffraction by a quadrant of sound-soft scatterers

TL;DR

This paper introduces an exact analytic method for solving the complex problem of acoustic wave diffraction by a quarter lattice of sound-soft scatterers, with numerical validation and comparison to existing approaches.

Contribution

The authors develop a novel analytic solution technique for a two-variable Wiener--Hopf equation in acoustic scattering, advancing the understanding of wave interactions with periodic structures.

Findings

Exact solution to the quarter lattice scattering problem

Numerical results validate the analytic method

Comparison shows advantages over previous semi-infinite decomposition methods

Abstract

Motivated by research in metamaterials, we consider the challenging problem of acoustic wave scattering by a doubly periodic quadrant of sound-soft scatterers arranged in a square formation, which we have dubbed the quarter lattice. This leads to a Wiener--Hopf equation in two complex variables with three unknown functions for which we can reduce and solve exactly using a new analytic method. After some suitable truncations, the resulting linear system is inverted using elementary matrix arithmetic and the solution can be numerically computed. This solution is also critically compared to a numerical least squares collocation approach and to our previous method where we decomposed the lattice into semi-infinite rows or columns.