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点源在单周期无限波纹边界上频域声散射的高阶边界积分方程（BIE）方法。该方法应用于以埃尔卡斯蒂略金字塔为模型建模的硬声二维阶梯结构附近声辐射的精确数值研究。此类阶梯结构支持沿表面传播并随距离指数衰减的束缚波。我们采用阵列扫描方法（弗洛凯-布洛赫变换）将散射场重构为依赖于表面波数的准周期解族积分。每个BIE解需使用准周期格林函数，通过晶格求和系数的有效积分表示进行求值。通过复变形技术避免了阵列扫描积分中的奇点和支割线，使得每个频率可在数秒内实现约10位精度的解。我们提出留数法提取远离源区域束缚模式携带的极限功率。最后，通过计算束缚模式色散关系，利用简单射线模型解释了观测到的声学"雨滴"效应（类似啁啾的时域响应）。