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 dynamics vertically polarized elastic waves propagating in a semi-infinite medium. The formulation extends canonical Lamb's problem, originally developed obtain wavefield induced by harmonic line source half-space, scenario where finite cluster vertical is attached medium surface. In short, our approach utilizes solution classical problem as Green's function formulate scattered fields generated resonators. For arbitrary number resonators, arranged atop half-space configuration, displacement are obtained closed-form validated with numerics two-dimensional element environment.