Bound States in the Continuum in Elastic Metamaterials*

Diffraction of elastic waves is considered for a system consisting of two parallel arrays of thin (subwavelength) cylinders that are arranged periodically. The embedding media supports waves with all polarizations, one longitudinal and two transverse, having different dispersion relations. An interaction with defects mixes the in-plane longitudinal and transverse modes. It is shown that the system supports bound states in the continuum (BSC) that have no specific polarization, that is, there are standing waves localized in the scattering structure whose wave numbers lies in the first open diffraction channels for both longitudinal and transverse modes. BSCs are shown to exists only for specific distances between the arrays and for specific values of the wave vector component along the array. An analytic solution is obtained for BSCs containing coupled elastic waves with different dispersion relations. For distances between the parallel arrays much larger than the wavelength, the existence of BSCs is explained by a destructive interference similar to the interference in a Fabri-Perot interferometer.