Diffraction of waves on triangular lattice by a semi-infinite rigid constraint and crack

Abstract

An exact solution is provided for a discrete analog of each of the two Sommerfeld diffraction problems using a triangular lattice model, with nearest neighbor interactions, deforming in the anti-plane direction. The discrete Helmholtz equation with time-harmonic data prescribed on semi-infinite row(s) of lattice sites is solved using the discrete Wiener–Hopf method. An asymptotic expression of the exact solution, given in the form of a contour integral, has been obtained in far field using the standard approximation of the diffraction integrals based on the method of stationary phase. In case of the rigid constraint diffraction, the displacement of particles on the semi-infinite row complementing the constrained lattice sites, as well as on the adjacent row, is presented in closed form as a discrete convolution. For the crack diffraction problem, the length of both types of slant bonds on the semi-infinite row complementing the crack, as well as the crack opening displacement, is given in a similar form but in terms of the Fourier coefficients, of the associated Wiener–Hopf kernel, which are not available in closed form. The displacement field associated with the scattered waves at sites far from the defect tip, as well as few sites near the tip, is compared graphically with that of a numerical solution on a finite grid. Both discrete Sommerfeld problems are naturally relevant to their continuous counterparts, involving the traditional Helmholtz equation, that model the diffraction of electromagnetic and acoustic waves by a semi-infinite screen, based on 7-point discretization on a triangular grid.