We present a high-order boundary integral equation (BIE) method for the frequency-domain acoustic scattering of a point source by a singly-periodic, infinite, corrugated boundary. We apply it to the accurate numerical study of acoustic radiation in the neighborhood of a sound-hard two-dimensional staircase modeled after the El Castillo pyramid. Such staircases support trapped waves which travel along the surface and decay exponentially away from it. We use the array scanning method (Floquet--Bloch transform) to recover the scattered field as an integral over the family of quasiperiodic solutions parameterized by their on-surface wavenumber. Each such BIE solution requires the quasiperiodic Green's function, which we evaluate using an efficient integral representation of lattice sum coefficients. We avoid the singularities and branch cuts present in the array scanning integral by complex contour deformation. For each frequency, this enables a solution accurate to around 10 digits in a couple of seconds. We propose a residue method to extract the limiting powers carried by trapped modes far from the source. Finally, by computing the trapped mode dispersion relation, we use a simple ray model to explain an observed acoustic "raindrop" effect (chirp-like time-domain response).

翻译：我们提出了一种用于单周期、无限长波纹边界上点源频域声散射的高精度边界积分方程方法。将此方法应用于以埃尔卡斯蒂略金字塔为模型建立的声硬二维阶梯附近声辐射的精确数值研究。此类阶梯支持沿表面传播、远离表面呈指数衰减的困陷波。我们采用阵列扫描方法（Floquet-Bloch变换），通过以表面波数为参数的全族准周期解的积分重构散射场。每个边界积分方程解均需计算准周期格林函数，我们采用高效的晶格求和系数积分表示实现该计算。通过复平面围道变形避免了阵列扫描积分中的奇点和支割线。对每个频率，该方法可在数秒内获得约10位精度的解。我们提出留数方法提取远离声源的困陷模态携带的极限功率。最后，通过计算困陷模态色散关系，采用简单射线模型解释了观测到的声学"雨滴"效应（类似啁啾信号的时域响应）。