Exterior diffraction problems for a triangular lattice

TL;DR

This paper investigates the existence, uniqueness, and numerical computation of solutions to exterior Dirichlet problems for lattice waves on a semi-infinite triangular lattice, focusing on the discrete Helmholtz equation with real wave numbers.

Contribution

It provides new theoretical results on solution existence and uniqueness for real wave numbers without complexification, and introduces a Green's formula and numerical method for these problems.

Findings

Proved existence and uniqueness of solutions for real wave numbers in (0, 2√2).

Derived Green's representation formula using difference potentials.

Developed a numerical method for solving the lattice wave problems.

Abstract

Exterior Dirichlet problems for two-dimensional lattice waves on the semi-infinite triangular lattice are considered. Namely, we study Dirichlet problems for the two-dimensional discrete Helmholtz equation in a plane with a hole. New results are obtained for the existence and uniqueness of the solution in the case of the real wave number $k \in (0, 2\sqrt{2})$ without passing to a complex wave number. Besides, Green's representation formula for the solution is derived with the help of difference potentials. To demonstrate the results, we propose a method for numerical calculation.