Symmetry issues in the hybridization of multi-mode waves with resonators: an example with Lamb waves metamaterial

Locally resonant metamaterials derive their effective properties from hybridization between their resonant unit cells and the incoming wave. This phenomenon is well understood in the case of plane waves that propagate in media where the unit cell respects the symmetry of the incident field. However, in many systems, several modes with orthogonal symmetries can coexist at a given frequency, while the resonant unit cells themselves can have asymmetric scattering cross-sections. In this paper we are interested in the influence of symmetry breaking on the hybridization of a wave field that includes multiple propagative modes. The A0 and S0 Lamb waves that propagate in a thin plate are good candidates for this study, as they are either anti-symmetric or symmetric. First we designed an experimental setup with an asymmetric metamaterial made of long rods glued to one side of a metallic plate. We show that the flexural resonances of the rods induce a break of the orthogonality between the A0/S0 modes of the free-plate. Finally, based on numerical simulations we show that the orthogonality is preserved in the case of a symmetric metamaterial leading to the presence of two independent polariton curves in the dispersion relation.