Discrete scattering by two staggered semi-infinite defects: reduction of matrix Wiener-Hopf problem

TL;DR

This paper addresses the complex matrix Wiener-Hopf problem in discrete lattice scattering by two staggered semi-infinite defects, proposing a reduction to finite linear equations using scalar Wiener-Hopf factorization.

Contribution

It introduces a novel reformulation of the discrete Wiener-Hopf problem for staggered defects, enabling reduction to finite algebraic equations.

Findings

Reduction of the matrix Wiener-Hopf problem to finite linear equations

Application of scalar Wiener-Hopf factorization to the discrete problem

Relevance to mechanics and physics at small scales

Abstract

As an extension of the discrete Sommerfeld problems on lattices, the scattering of a time harmonic wave is considered on an infinite square lattice when there exists a pair of semi-infinite cracks or rigid constraints. Due to the presence of stagger, also called offset, in the alignment of the defect edges the asymmetry in the problem leads to a matrix Wiener-Hopf kernel that cannot be reduced to scalar Wiener-Hopf in any known way. In the corresponding continuum model the same problem is a well known formidable one which possesses certain special structure with exponentially growing elements on the diagonal of kernel. From this viewpoint the present paper tackles a discrete analogue of the same by reformulating the Wiener-Hopf problem and reducing it to a finite set of linear algebraic equations; the coefficients of which can be found by an application of the scalar Wiener-Hopf factorization. The considered discrete paradigm involving lattice waves is relevant for modern applications of mechanics and physics at small length scales.