正方形网格上两个交错半无限裂纹的散射：渐近Wiener-Hopf分解的应用
Scattering by two staggered semi-infinite cracks on square lattice: an application of asymptotic Wiener-Hopf factorization
摘要：
本文研究了正方形网格上两平行的半无限排相互交错的边界对时谐平面波的散射。施加在半无限行上的条件是Neumann边界条件的离散模拟。一个物理解释假设的平面外位移的颗粒排列在一个正方形的晶格的形式，并与最近的邻居相互作用，相关联的散射问题，晶格波散射由于存在两个交错的，但平行的裂纹尖端。利用傅立叶变换，将离散散射问题归结为复平面圆环上的一对维纳-霍普夫方程的研究。由于裂纹边缘之间的偏移，Wiener-Hopf核，一个2x2矩阵，是不适合因式分解在一个理想的形式和渐近方法进行了调整。此外，在远场的近似进行了使用固定相位法。给出了基于渐近Wiener-Hopf方法的远场近似与数值解的图形比较。还包括一个图形说明的低频近似，它已被发现，散射问题的数值解与众所周知的强大的解决方案在连续框架相一致。
英文摘要：
Scattering of time-harmonic plane wave by two parallel semi-infinite rows, but with staggered edges, is considered on square lattice. The condition imposed on the semi-infinite rows is a discrete analogue of Neumann boundary condition. A physical interpretation assuming an out-of-plane displacement for the particles arranged in the form of a square lattice and interacting with nearest-neighbours, associates the scattering problem to lattice wave scattering due to the presence of two staggered but parallel crack tips. The discrete scattering problem is reduced to the study of a pair of Wiener-Hopf equation on an annulus in complex plane, using Fourier transforms. Due to the offset between the crack edges, the Wiener-Hopf kernel, a 2x2 matrix, is not amenable to factorization in a desirable form and an asymptotic method is adapted. Further, an approximation in the far field is carried out using the stationary phase method. A graphical comparison between the far-field approximation based on asymptotic Wiener-Hopf method and that obtained by a numerical solution is provided. Also included is a graphical illustration of the low frequency approximation, where it has been found that the numerical solution of the scattering problem coincides with the well known formidable solution in the continuum framework.
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