Half-plane diffraction problems on a triangular lattice

Abstract

We investigate thin-slit diffraction problems for two-dimensional lattice waves. Namely, Dirichlet problems for the two-dimensional discrete Helmholtz equation on the triangular lattice in a half-plane are studied. Using the notion of the radiating solution, we prove the existence and uniqueness of a solution for the real wave number $$k\in (0,3)\backslash \{2\sqrt{2}\}$$ without passing to the complex wave number. Besides, an exact representation formula for the solution is derived. Here, we develop a numerical calculation method and demonstrate by example the effectiveness of our approach related to the propagation of wavefronts in metamaterials through two small openings.