Discrete scattering by a pair of parallel defects

TL;DR

This paper provides an exact analytical solution for wave scattering caused by pairs of defects in a square lattice, relevant for nanoscale electronic and thermal transport studies.

Contribution

It introduces a novel reduction of coupled equations to scalar Wiener-Hopf equations for analyzing defect scattering in lattices.

Findings

Exact solutions for wave scattering by defect pairs

Reduction to scalar Wiener-Hopf equations

Application to nanoscale transport phenomena

Abstract

Scattering of a time harmonic anti-plane shear wave due to either a pair of crack tips or a pair of rigid constraint tips on square lattice is considered. The two problems correspond to the so called zero-offset case of scattering due to a pair of identical Sommerfeld screens. The peculiar structural symmetry allows the reduction of coupled equations to two scalar Wiener-Hopf equations and a total of four geometrically reduced problems on lattice half-plane. Exact solution of each problem for incidence from the bulk lattice, as well as from an associated lattice waveguide, is constructed. A suitable superposition of the four expressions is used to construct the solution of the main problem. The discrete paradigm involving the wave mode incident from the waveguide is relevant for modern applications where an investigation of mechanisms of electronic and thermal transport at nanoscale remains an interesting problem.