Diffraction of waves on square lattice by semi-infinite rigid constraint

• Exact solution for diffraction of time-harmonic lattice wave by a semi-infinite defect.
• Far-field asymptotic approximation of the exact solution.
• Graphical comparison with numerical solution for a set of frequencies in the pass band.
• Low frequency limit of the exact solution to obtain continuous integral form.
• Relevance to five point discretization of the two-dimensional Helmholtz equation.

The problem of diffraction of a time harmonic lattice wave in a two-dimensional square lattice, by a semi-infinite rigid constraint, is investigated as a discrete analogue of diffraction by a Sommerfeld ‘soft’ half plane. The discrete Helmholtz equation, with input data prescribed on a semi-infinite row of lattice sites, is solved exactly using the discrete Wiener–Hopf method. The far-field asymptotic approximation of exact solution is provided. The scattered wave, in far field, is compared with a numerical solution of the problem for a set of frequencies in the pass band. The low frequency approximation of the exact solution is derived and it coincides with the Sommerfeld’s solution in its integral form. The results and discussion associated with the discrete Sommerfeld problem are relevant to numerical methods based on a 5-point discretization of the two-dimensional Helmholtz equation. In addition to the mechanics of waves in lattices, other physical applications of the latter concern the scattering of an EE-polarized electromagnetic wave by a conducting half plane as well as its acoustic counterpart.