In this paper we develop second kind integral formulations for flexural wave scattering problems involving the clamped, free, and supported plate boundary conditions. While the clamped plate problem can be solved with layer potentials previously developed for the biharmonic equation [1], the free plate problem is more difficult due to the complex nature of the boundary conditions. In this paper we describe a representation for the free plate problem that uses the Hilbert transform to cancel singularities of certain layer potentials, ultimately leading to a Fredholm integral equation of the second kind. Additionally, for the supported plate problem, we improve on an existing representation to obtain a second kind integral equation. With these representations, it is possible to solve flexural wave scattering problems with high-order-accurate methods, examine the far-field patterns of scattering objects, and solve large problems involving multiple scatterers.

翻译：本文针对涉及夹支、自由及简支板边界条件的弯曲波散射问题，建立了第二类积分公式。虽然夹支板问题可利用先前针对双调和方程建立的层势方法求解[1]，但自由板问题由于边界条件的复杂性而更具挑战性。本文提出一种针对自由板问题的表示方法，该方法利用希尔伯特变换消除特定层势的奇异性，最终导出一个第二类弗雷德霍姆积分方程。此外，对于简支板问题，我们改进了现有表示方法，从而获得第二类积分方程。基于这些表示方法，可采用高阶精度数值方法求解弯曲波散射问题，分析散射体的远场模式，并解决涉及多重散射体的大规模问题。