Propagation of waves from finite sources arranged in line segments within an infinite triangular lattice

This paper examines the propagation of time-harmonic waves in a two-dimensional triangular lattice with a lattice constant a = 1 {a=1} . The sources are positioned along line segments within the lattice. Specifically, we investigate the discrete Helmholtz equation with a wavenumber k ∈ ( 0 , 2 ⁢ 2 ) {k\in(0,2\sqrt{2})} , 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.