Diffraction of Waves on Square Lattice by Semi-Infinite Crack

Abstract

In this paper, a discrete analogue of Sommerfeld half-plane diffraction is investigated. The two-dimensional problem of diffraction on a square lattice, of a plane (transverse) wave by a semi-infinite crack, is solved. The discrete Wiener--Hopf method has been used to obtain the exact solution of the discrete Helmholtz equation, with input data prescribed on the crack boundary sites due to a time harmonic incident wave. It is established that there exists a unique saddle point for the diffraction integral and its properties are characterized. An asymptotic approximation of the solution in the far field is provided and, for some values of the frequency it is compared with the numerical solution, of the diffraction problem, using a finite grid. A low frequency approximation of the solution in integral form recovers the classical continuum solution. At sufficiently large frequency in the pass band, the effects due to discreteness and anisotropy appear. The analysis is relevant to a 5-point discretization based numerical solution of the two-dimensional Helmholtz equation. Applications include scattering of an $H$-polarized electromagnetic wave by a conducting half-plane, or its three-dimensional acoustic equivalent.