Prediction of sound field generated by an infinitely long periodic structure is often required in engineering. One of the examples is the sound field created by vibration of the rail of a slab railway track, of which the radiating and scattering boundaries are periodic in the track direction due to the rail fasteners. To provide a proper computational tool for such problems, we develop the periodic acoustic boundary element method (PABEM) which only requires a three-dimensional (3D) boundary element mesh for a single cell. The development of the PABEM involves Floquet-transformation of the acoustic boundary integral equation of the infinite periodic structure, exploration of the Floquet-transformed Green's functions, and solution of the boundary integral equation using the boundary element method (BEM). Although the last step is largely the same as the conventional BEM, differences do exist and need to be carefully treated. Special attentions are given to the singularities of the Floquet-transformed Green's functions and to the numerical treatment of such singularities. The PABEM is shown to be directly applicable to sound radiation from a propagating vibration wave in an infinite periodic structure. This makes the PABEM useful for railway noise study since the response of the rail is often expressed as the sum of propagating waves at particular wavenumbers.

中文翻译：

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无限长周期结构声场建模的周期性声学边界元法


工程中经常需要预测无限长周期结构产生的声场。例子之一是板式铁路轨道的钢轨振动产生的声场，由于钢轨紧固件的影响，其辐射和散射边界在轨道方向上呈周期性。为了为此类问题提供合适的计算工具，我们开发了周期性声学边界元方法（PABEM），该方法仅需要单个单元的三维（3D）边界元网格。 PABEM的发展涉及无限周期结构的声学边界积分方程的Floquet变换、Floquet变换格林函数的探索以及使用边界元法(BEM)求解边界积分方程。尽管最后一步与传统边界元法基本相同，但差异确实存在，需要仔细对待。特别关注 Floquet 变换格林函数的奇点以及此类奇点的数值处理。 PABEM 被证明可直接应用于无限周期结构中传播的振动波的声辐射。这使得 PABEM 对于铁路噪声研究非常有用，因为铁路的响应通常表示为特定波数下传播波的总和。