We propose an analytical framework to model the effect of single and multiple mechanical surface oscillators on the dynamics of vertically polarized elastic waves propagating in a half-space. The formulation extends the canonical Lamb’s problem, originally developed to obtain the wavefield induced by a harmonic line source in an elastic half-space. In short, our approach utilizes the solution of the classical Lamb’s problem as Green’s function to formulate the multiple scattered fields generated by a cluster of mechanical resonators attached to the surface. For an arbitrary number of resonators, arranged atop the elastic half-space in an arbitrary configuration, the displacement fields are obtained in closed form and validated with numerics developed in a finite element environment. We demonstrate that our approach can correctly model elastic waves interacting with single and couples of resonators, and capture complex dynamics phenomena such as wave conversion and wave localization induced by arrays of resonators, also known as metasurfaces.

我们提出了一个分析框架来模拟单个和多个机械表面振荡器对在半空间中传播的垂直极化弹性波的动力学的影响。该公式扩展了经典 Lamb 问题，该问题最初是为了获得由弹性半空间中的谐波线源引起的波场而开发的。简而言之，我们的方法利用经典兰姆问题的解决方案作为格林函数来制定由连接到表面的一组机械谐振器产生的多个散射场。对于以任意配置排列在弹性半空间顶部的任意数量的谐振器，位移场以封闭形式获得，并使用在有限元环境中开发的数值进行验证。