Motivated by research in metamaterials, we consider the challenging problem of acoustic wave scattering by a doubly periodic quadrant of sound-soft scatterers arranged in a square formation, which we have dubbed the quarter lattice. This leads to a Wiener--Hopf equation in two complex variables with three unknown functions for which we can reduce and solve exactly using a new analytic method. After some suitable truncations, the resulting linear system is inverted using elementary matrix arithmetic and the solution can be numerically computed. This solution is also critically compared to a numerical least squares collocation approach and to our previous method where we decomposed the lattice into semi-infinite rows or columns.

翻译：受超材料研究的启发，我们考虑了一个具有挑战性的问题：声波被按正方形排列的双周期声软散射体四分之一阵列（我们称之为四分之一晶格）散射的问题。这导出了一个具有三个未知函数的双复变量维纳-霍普夫方程，我们能够通过一种新的解析方法对其进行约简并精确求解。经过适当的截断后，所得线性系统通过初等矩阵运算求逆，其解可进行数值计算。该解还与数值最小二乘配置法以及我们先前将晶格分解为半无限行或列的方法进行了关键性比较。