# An asymptotic hyperbolic-elliptic model for flexural-seismic metasurfaces

## TL;DR

This paper develops an asymptotic hyperbolic-elliptic model for elastic metasurfaces with flexural resonators, simplifying dispersion relations and revealing how junction conditions influence wave propagation and stop bands.

## Contribution

It introduces a new asymptotic scalar hyperbolic model for flexural resonators on elastic surfaces, extending previous compressional models and highlighting the impact of junction conditions.

## Key findings

- The model provides explicit dispersion relations with closed-form solutions.
- Flexural resonators significantly alter wave stop bands.
- Junction conditions dramatically affect wave propagation features.

## Abstract

We consider a periodic array of resonators, formed from Euler-Bernoulli beams, attached to the surface of an elastic half-space. Earlier studies of such systems have concentrated on compressional resonators. In this paper we consider the effect of the flexural motion of the resonators, adapting a recently established asymptotic methodology that leads to an explicit scalar hyperbolic equation governing the propagation of Rayleigh-like waves. Compared with classical approaches, the asymptotic model yields a significantly simpler dispersion relation, with closed form solutions, shown to be accurate for surface wave-speeds close to that of the Rayleigh wave. Special attention is devoted to the effect of various junction conditions joining the beams to the elastic half-space which arise from considering flexural motion and are not present for the case of purely compressional resonators. Such effects are shown to provide significant and interesting features and, in particular, the choice of junction conditions dramatically changes the distribution and sizes of stop bands. Given that flexural vibrations in thin beams are excited more readily than compressional modes and the ability to model elastic surface waves using the scalar wave equation (i.e. waves on a membrane), the paper provides new pathways toward novel experimental set-ups for elastic metasurfaces.