Lamb’s problem for a half-space coupled to a generic distribution of oscillators at the surface

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.