被困声波与雨滴：周期阶梯局部激励的高阶精确积分方程方法
Trapped acoustic waves and raindrops: high-order accurate integral equation method for localized excitation of a periodic staircase
AI总结：
本文提出高阶边界积分方程法，结合阵列扫描与留数法，精确求解周期阶梯点源散射，并解释被困波导致的声学“雨滴”效应。
摘要：
我们提出了一种高阶边界积分方程（BIE）方法，用于求解点源被单周期、无限、波纹边界散射的频域声学问题。我们将其应用于对仿照埃尔卡斯蒂略金字塔建模的声硬二维阶梯附近声辐射的精确数值研究。此类阶梯支持被困波，这些波沿表面传播并在远离表面时指数衰减。我们使用阵列扫描方法（Floquet-Bloch变换）将散射场恢复为以其表面波数为参数的拟周期解族上的积分。每个这样的BIE解都需要拟周期格林函数，我们通过格和系数的有效积分表示来评估该函数。我们通过复轮廓变形避免了阵列扫描积分中存在的奇异点和分支切割。对于每个频率，这使解在几秒内精确到约10位数字。我们提出了一种留数方法来提取远离源处被困模式携带的极限功率。最后，通过计算被困模式色散关系，我们使用一个简单的射线模型来解释观察到的声学“雨滴”效应（类似啁啾的时域响应）。
英文摘要：
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).
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