On an analogy between the Wiener--Hopf formulations of discrete and continuous diffraction problems

TL;DR

This paper unifies the Wiener--Hopf framework for discrete and continuous diffraction problems using discrete Green's identity, enabling seamless transition between formulations and validated through classical 2D and 3D diffraction examples.

Contribution

It introduces a formal analogy between discrete and continuous Wiener--Hopf equations, facilitating easier analysis of diffraction problems.

Findings

The analogy is valid for several canonical 2D diffraction problems.

The approach extends naturally to 3D diffraction problems.

It simplifies the derivation of Wiener--Hopf equations across different formulations.

Abstract

This article is dedicated to unifying the framework used to derive the Wiener--Hopf equations arising from some discrete and continuous wave diffraction problems.The main tools are the discrete Green's identity and the appropriate notion of discrete normal derivative. The resulting formal analogy between the Wiener--Hopf equations allows one to effortlessly move between the discrete and continuous formulations. The validity of this novel analogy is illustrated through several famous two-dimensional canonical diffraction problems and extended to three-dimensional problems.