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

TL;DR

This paper investigates wave propagation from finite line segment sources in a 2D triangular lattice, establishing theoretical foundations and proposing an efficient numerical method for applications like metamaterials.

Contribution

It introduces a Green's function evaluation approach, analyzes the cone condition's role, and develops a numerical method for wave propagation in lattice structures.

Findings

Validated the Green's function computation method

Established unique solvability and representation formulas

Demonstrated numerical efficiency in metamaterial examples

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.