This paper presents a novel boundary integral equation (BIE) formulation for the two-dimensional time-harmonic water-waves problem. It utilizes a complex-scaled Laplace's free-space Green's function, resulting in a BIE posed on the infinite boundaries of the domain. The perfectly matched layer (PML) coordinate stretching that is used to render propagating waves exponentially decaying, allows for the effective truncation and discretization of the BIE unbounded domain. We show through a variety of numerical examples that, despite the logarithmic growth of the complex-scaled Laplace's free-space Green's function, the truncation errors are exponentially small with respect to the truncation length. Our formulation uses only simple function evaluations (e.g. complex logarithms and square roots), hence avoiding the need to compute the involved water-wave Green's function. Finally, we show that the proposed approach can also be used to find complex resonances through a \emph{linear} eigenvalue problem since the Green's function is frequency-independent.

翻译：本文提出了一种用于二维时谐水波问题的新型边界积分方程（BIE）公式。该公式利用复缩放拉普拉斯自由空间格林函数，在域的无界边界上构建边界积分方程。通过完美匹配层（PML）坐标拉伸，将传播波转化为指数衰减波，从而实现对无界域边界积分方程的有效截断和离散化。我们通过多种数值算例表明，尽管复缩放拉普拉斯自由空间格林函数具有对数增长特性，但其截断误差关于截断长度呈指数级微小。该公式仅需简单函数求值（如复对数与平方根），从而避免了计算复杂水波格林函数。最后，我们证明由于格林函数与频率无关，该方法还可通过线性特征值问题求解复共振。