  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.
