As an extension of the discrete Sommerfeld problems on lattices, the scattering of a time harmonic wave is considered when there exists a pair of semi-infinite cracks or rigid constraints on an infinite square lattice. Due to 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 ordinary way. In the corresponding continuum model the same problem is well known to possess a certain special structure with exponentially growing elements on diagonal of kernel. The present paper tackles a discrete analogue of the same by reformulating the Wiener-Hopf problem and reducing it to a set of linear algebraic equations; the coefficients of which can be found by an application of scalar Wiener-Hopf factorization. The discrete paradigm involving lattice waves is also relevant for modern applications of mechanics and physics at small length scales.

中文翻译：

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两个交错的半无限缺陷的离散散射：矩阵 Wiener-Hopf 问题的约简

作为离散 Sommerfeld 问题在晶格上的扩展，当无限正方形晶格上存在一对半无限裂纹或刚性约束时，考虑时间谐波的散射。由于在缺陷边缘对齐中存在交错，也称为偏移，问题中的不对称性导致矩阵 Wiener-Hopf 核不能以普通方式减少到标量 Wiener-Hopf。在相应的连续模型中，众所周知，同样的问题具有某种特殊结构，在核对角线上的元素呈指数增长。本论文通过重新表述 Wiener-Hopf 问题并将其简化为一组线性代数方程来解决该问题的离散模拟；其系数可以通过应用标量 Wiener-Hopf 分解来找到。