Edge diffraction on triangular and hexagonal lattices: Existence, uniqueness, and finite section

Abstract

The diffraction of lattice waves by two distinct types of defects, namely a rigid constraint and a crack, in the triangular and hexagonal (honeycomb) lattice models is analyzed. A semi-analytical solution of the discrete Helmholtz equation is provided everywhere in the lattice using the lattice Green’s function, when the length of the defect is finite, in all four problems. For the case of a semi-infinite defect, each problem involves the inversion of a Toeplitz operator on ℓp (1≤p≤∞), a truncation of which appears in the finite defect problem. It is shown that the symbol of the Toeplitz operator satisfies the Krein conditions and is continuous on the unit circle in the complex plane. The existence and uniqueness of solution of the discrete Wiener–Hopf equation in ℓp, corresponding to each of the four problems, follow and the method of finite section is applicable. It is established that the solution of each of the four discrete Sommerfeld problems is unique in ℓ2 and for the case of finite defect the displacement field, near any one tip, is approximated by its semi-infinite counterpart. The paper includes several graphical illustrations supplementary to the mathematical analysis presented and the main results established.