Propagation of Waves from Finite Sources Arranged in Line Segments within an Infinite Triangular Lattice

Abstract

This paper examines the propagation of time-harmonic waves in a two-dimensional triangular lattice with a lattice constant $a=1$. The sources are positioned along line segments within the lattice. Specifically, we investigate the discrete Helmholtz equation with a wavenumber $k\in \left(0,2\sqrt{2}\right)$, where input data is prescribed on finite rows or columns of lattice sites. We focus on two main questions: the efficacy of the numerical methods employed in evaluating the Green's function, and the necessity of the cone condition. Consistent with a continuum theory, we employ the notion of radiating solution and establish a unique solvability result and Green's representation formula using difference potentials. Finally, we propose a numerical computation method and demonstrate its efficiency through examples related to the propagation problems in the left-handed two-dimensional inductor-capacitor metamaterial.

23 Pages, 15 Figures, 2 Tables. Additional information on a two-dimensional infinite triangular lattice has been included. Minor improvements have also been made to the paper