Seismic metasurfaces: Sub-wavelength resonators and Rayleigh wave interaction

TL;DR

This paper analyzes how arrays of sub-wavelength resonators on elastic substrates interact with Rayleigh waves, creating effects like band-gaps and wave transformation, with analytical and numerical methods revealing complex wave behavior.

Contribution

It provides an analytical dispersion relation for resonator arrays on elastic substrates and demonstrates wave mode transitions through combined analysis and simulations.

Findings

Resonator arrays induce effective band-gaps for surface waves.

Wave dispersion can transition from Rayleigh to shear wave behavior.

Numerical simulations confirm analytical predictions of wave mode changes.

Abstract

We consider the canonical problem of an array of rods, which act as resonators, placed on an elastic substrate; the substrate being either a thin elastic plate or an elastic half-space. In both cases the flexural plate, or Rayleigh surface, waves in the substrate interact with the resonators to create interesting effects such as effective band-gaps for surface waves or filters that transform surface waves into bulk waves; these effects have parallels in the 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 dispersion relations analytically thereby examining the influence of both the flexural and compressional resonances on the propagating wave. For an array of resonators atop an elastic half-space we augment the analysis with numerical simulations. Amongst other effects, we demonstrate the striking effect of a dispersion curve that transitions from Rayleigh wave-like to shear wave-like behaviour and the resultant change in displacement from surface to bulk waves.