Half-plane diffraction problems on a triangular lattice

TL;DR

This paper studies diffraction problems on a triangular lattice, providing new existence, uniqueness, and explicit solution formulas for discrete Helmholtz equations, along with a numerical method demonstrated on metamaterial wave propagation.

Contribution

It offers novel results for real wave numbers without complexification and introduces an exact solution representation for half-plane lattice diffraction problems.

Findings

Proved existence and uniqueness for certain real wave numbers.

Derived an explicit solution formula for the discrete Helmholtz equation.

Demonstrated numerical efficiency with an example on wave propagation in metamaterials.

Abstract

We investigate thin-slit diffraction problems for two-dimensional lattice waves. The peculiar structure allows us to consider the problems on the semi-infinite triangular lattice, consequently, we study Dirichlet problems for the two-dimensional discrete Helmholtz equation in a half-plane. In view of the existence and uniqueness of the solution, we provide new results for the real wave number $k\in (0,3)\backslash\{2\sqrt{2}\}$ without passing to the complex wave number and derive an exact representation formula for the solution. For this purpose, we use the notion of the radiating solution. Finally, we propose a method for numerical calculation. The efficiency of our approach is demonstrated in an example related to the propagation of wave fronts in metamaterials through two small openings.