# Sommerfeld--type integrals for discrete diffraction problems

## TL;DR

This paper develops analytical solutions for discrete diffraction problems related to the Helmholtz equation using Sommerfeld integrals, providing explicit formulas for various geometries through algebraic and elliptic functions.

## Contribution

It introduces a novel application of Sommerfeld integral methods to discrete Helmholtz problems, deriving explicit solutions for complex diffraction geometries.

## Key findings

- Explicit integral representations for discrete diffraction fields
- Solutions expressed via recursive, algebraic, and elliptic functions
- Analytical framework applicable to various diffraction problems

## Abstract

Three problems for a discrete analogue of the Helmholtz equation are studied analytically using the plane wave decomposition and the Sommerfeld integral approach. They are: 1) the problem with a point source on an entire plane; 2) the problem of diffraction by a Dirichlet half-line; 3) the problem of diffraction by a Dirichlet right angle. It is shown that total field can be represented as an integral of an algebraic function over a contour drawn on some manifold. The latter is a torus. As the result, the explicit solutions are obtained in terms of recursive relations (for the Green's function), algebraic functions (for the half-line problem), or elliptic functions (for the right angle problem).