The far‐field behaviour of Green's function for a triangular lattice and radiation conditions

We consider the Helmholtz equation (Δd + k2)u = f on the triangular lattice, where Δd is the discrete Laplacian, f has finite support, and wave number k belongs to the pass-band. Using the limiting absorption principle, we derive the discrete analogue of the Sommerfeld radiation condition for all values of k ∈ ( 0,3 ) \ { 2 2 }. It turns out that this condition is anisotropic and depends on the value of k. We introduce the notion of a radiating solution and prove the unique solvability result.