Analytical methods are fundamental in studying acoustics problems. One of the important tools is the Wiener-Hopf method, which can be used to solve many canonical problems with sharp transitions in boundary conditions on a plane/plate. However, there are some strict limitations to its use, usually the boundary conditions need to be imposed on parallel lines (after a suitable mapping). Such mappings exist for wedges with continuous boundaries, but for discrete boundaries, they have not yet been constructed. In our previous article, we have overcome this limitation and studied the diffraction of acoustic waves by a wedge consisting of point scatterers. Here, the problem is generalised to an arbitrary number of periodic semi-infinite arrays with arbitrary orientations. This is done by constructing several coupled systems of equations (one for every semi-infinite array) which are treated independently. The derived systems of equations are solved using the discrete Wiener-Hopf technique and the resulting matrix equation is inverted using elementary matrix arithmetic. Of course, numerically this matrix needs to be truncated, but we are able to do so such that thousands of scatterers on every array are included in the numerical results. Comparisons with other numerical methods are considered, and their strengths/weaknesses are highlighted.

翻译：解析方法在研究声学问题中具有基础性地位。维纳-霍普夫法作为重要工具之一，可用于求解诸多边界条件在平面/平板上发生突变的典型问题。然而该方法存在严格的应用限制，通常需将边界条件施加于平行直线之上（经适当映射后）。此类映射虽对连续边界楔形存在，但针对离散边界尚未构建。在先前研究中，我们突破了这一限制，研究了由点散射体构成的楔形对声波的衍射现象。本文将该问题推广至任意数量、任意取向的周期性半无限阵列。通过构建若干独立处理的耦合方程组（每个半无限阵列对应一组），实现了这一推广。所导出的方程组采用离散维纳-霍普夫技术求解，所得矩阵方程则通过初等矩阵运算求逆。尽管在数值实现中需对该矩阵进行截断，但我们能保证数值结果包含每个阵列上数千个散射体的贡献。本文还与其它数值方法进行了比较，并指出了各自的优势与不足。