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We consider the canonical problem of an array of rods, which act asresonators, placed on an elastic substrate; the substrate being either a thinelastic plate or an elastic half-space. In both cases the flexural plate, orRayleigh surface, waves in the substrate interact with the resonators to createinteresting effects such as effective band-gaps for surface waves or filtersthat transform surface waves into bulk waves; these effects have parallels inthe field of optics where such sub-wavelength resonators create metamaterials,and metasurfaces, in the bulk and at the surface respectively.  Here we carefully analyse this canonical problem by extracting the dispersionrelations analytically thereby examining the influence of both the flexural andcompressional resonances on the propagating wave. For an array of resonatorsatop an elastic half-space we augment the analysis with numerical simulations.Amongst other effects, we demonstrate the striking effect of a dispersion curvethat transitions from Rayleigh wave-like to shear wave-like behaviour and theresultant change in displacement from surface to bulk waves.

Seismic metasurfaces: Sub-wavelength resonators and Rayleigh wave interaction