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.

The displacement field near a tip of a finite crack, due to the diffraction of a wave on square lattice is studied. The finite section method, in the theory of Toeplitz operators on 2 , is invoked as the semi-infinite crack diffraction problem is shown equivalent to the inversion of a Toeplitz operator, a truncation of which appears in the finite crack diffraction problem for the same incident wave. The existence and uniqueness of the solution in 2 for the semi-infinite crack problem is established by an application of the well known Krein conditions. Continuum limit of the semiinfinite crack diffraction problem is established in a discrete Sobolev space; a graphical illustration of convergence in the relevant Sobolev norm is also included. A low frequency asymptotic approximation of the normalized shear force, in 'vertical' bonds ahead of the crack tip, recovers the classical crack tip singularity. Displacement of particles in the vicinity of the crack tip, a closed form expression of which is provided, is compared, graphically, with that obtained by a numerical solution of the diffraction problem on a finite grid. Graphical results are also included to demonstrate that the normalized shear force in 'horizontal' bonds, along the crack face, approaches the corresponding stress component in the continuum model at sufficiently low frequency. Numerical solutions indicate that the crack opening displacement of a semi-infinite crack approximates that of a finite crack of sufficiently large size, and at sufficiently high frequency of incident wave, away from the neighborhood of the other crack tip, while it differs significantly for low frequencies, as a result of multiple scattering due to the two crack tips.

No abstract

No abstract

No abstract