Guided elastic waves at a periodic array of thin coplanar cavities in a solid

Abstract

This paper is concerned with guided elastic waves at a periodic array of thin coplanar cavities in a solid and the role they play in the scattering of incident longitudinal and transverse waves in the plane perpendicular to the cavities. It is treated as a mixed boundary-condition plane strain problem pertaining to two elastic half spaces which are joined together within infinitely long regularly spaced strips and unattached in between. It provides an approach to modeling wave interactions at an array of coplanar cracks, partially bonded surfaces, mining stopes, and other analogous physical situations, and it is of relevance in the topical field of phononic crystals. The method of analysis brought to bear on this problem involves smoothing out the discontinuities in the boundary conditions, invoking a truncated Fourier series representation of the wave field and applying the boundary conditions at a discrete set of points within the repeat interval at the interface to determine the Fourier coefficients. The dispersion relation for interfacial waves is obtained, and it is shown how (in the supersonic domain with respect to bulk transverse waves) there are branches associated with leaky interfacial waves, which at certain isolated points in $k$ space uncouple from the bulk wave continuum to exist as secluded supersonic interfacial waves. These observations are able to explain striking resonant features in the scattering of bulk waves at the interface.