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

• The diffraction problem of a time harmonic lattice wave by a semi-infinite or finite rigid constraint as well as semi-infinite or finite crack, in a two-dimensional triangular lattice and hexagonal lattice, is solved using the lattice Green’s functions.
• The symbol of the relevant Toeplitz operators is provided enabling the application of certain rigorous methods.
• The existence and uniqueness of solution for the semi-infinite defect problem is established using Krein conditions.
• The approximation theorem relating the finite defect problem to semi-infinite defect problem is established.

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 ℓpℓp (1≤p≤∞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 ℓpℓp, 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 ℓ2ℓ2 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.