Embedding formulae for diffraction problems on square lattices

TL;DR

This paper introduces a Wiener--Hopf based embedding formula for diffraction problems on square lattices, enabling efficient solutions for various incident waves without re-solving boundary problems.

Contribution

The authors develop a general operator-based embedding formula for arbitrary obstacle configurations on lattices, extending the Wiener--Hopf method to discrete diffraction problems.

Findings

Explicit embedding formulas derived for canonical geometries.

Numerical validation confirms the accuracy of the embedding approach.

Method offers efficient solutions for multiple incident wave directions.

Abstract

We develop embedding formulae for all possible diffraction problems with Dirichlet scatterers on square lattices using the Wiener--Hopf perspective. The embedding formula expresses solutions for arbitrary plane-wave incidence in terms of a finite set of auxiliary problems, eliminating the need to re-solve boundary value problems for each incidence angle. First we derive explicit embedding formulae for canonical geometries including the half-plane, finite strip, and right-angled wedge. We then generalize the method through an operator-based approach, obtaining embedding formula for arbitrary configurations of obstacles on lattices. This general embedding formula is a key difference from the continuous setting where this is currently not possible. To validate the theory, we perform numerical experiments, confirming agreement with the results derived using the embedding formula. The results highlight the efficiency and generality of the Wiener--Hopf approach in discrete diffraction theory, with potential applications in inverse problems and other areas of physics and mathematics.