Mathematical Analysis of Subwavelength Resonance in Elastic Metascreen

Abstract

The aim of this paper is to provide a comprehensive and mathematically rigorous analysis on determining the existence of subwavelength resonance in elastic metascreen and resonance frequency calculation based on asymptotic analysis of quasi-periodic layer potential operators. An elastic metascreen is a thin sheet with subwavelength structures, which nevertheless has a significant effect on elastic wave propagation at specific frequencies. Periodic subwavelength elastic scatterers positioned on a reflective plane are considered in this paper. Firstly an explicit formula of quasi-periodic Green's function of Lam\'{e} system with Dirichlet boundary condition is derived for the first time. The subsequent discussion is twofold. In the first part where the shear modulus of scatterers is assumed to tend to infinity, the subwavelength resonance frequencies are given and approximated field inside inclusions and far-away from metascreen are calculated to demonstrate the dramatic change of scattered field due to subwavelength resonance. In the second part where the shear modulus of background is assumed to go to infinity, the absence of subwavelength resonance is proved. Without imposing conditions on the material parameters, the discussion in this paper provides the necessary condition for the occurrence of subwavelength resonance.