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

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 analog 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.