Wave scattering on lattice structures involving an array of cracks

Abstract

<jats:p>Scattering of waves as a result of a vertical array of equally spaced cracks on a square lattice is studied. The convenience of Floquet periodicity reduces the study to that of scattering of a specific wave-mode from a single crack in a waveguide. The discrete Green’s function, for the waveguide, is used to obtain the semi-analytical solution for the scattering problem in the case of finite cracks whereas the limiting case of semi-infinite cracks is tackled by an application of the Wiener–Hopf technique. Reflectance and transmittance of such an array of cracks, in terms of incident wave parameters, is analysed. Potential applications include construction of tunable atomic-scale interfaces to control energy transmission at different frequencies.</jats:p>

Scattering of surface waves by inhomogeneities in crystalline structures

“…A discrete model of scalar surface wave propagation in an elastic half-space, representing the surface of a crystalline structure, is analysed. The problem of scattering of surface waves due to interface, associated with piece-wise constant surface properties, is solved against the backdrop of discrete scattering problems involving atomically sharp crack tips and rigid constraints [21,[26][27][28]. The exact solution of the lattice half-plane problem was deferred in [25] and instead it was provided for a lattice strip (see the paragraph preceding §3 in [25]); in fact, the same can be easily obtained after symbolic changes of the solution in the present article.…”

Phase recovery from phaseless scattering data for discrete Schrödinger operators

“…In particular, the case d = 2 corresponds to a discrete analogue of anti-plane shear waves in elastic continuum. Examples of forward analysis of such equations, in the case d = 2, with an exact solution of scattering of time harmonic lattice waves by atomically sharp crack tips and rigid constraints, can be found in [11][12][13][14][15][16][17][18]. The physical literature concerning the discrete Schrödinger equation also includes, in particular, [19].…”