Complexity of band structures: Semi-analytical finite element analysis of one-dimensional surface phononic crystals

Band structures of surface phononic crystals with one-dimensional periodicity are investigated in this paper. Real and complex dispersion relations of surface acoustic waves are calculated by a semi-analytical finite element method in the frequency domain. The model applies 2Dnine-noded elements in the periodic direction to discretize a unit cell with finite depth. Propagation perpendicular to the periodic direction is modeled by analytic functions. Two eigenproblems are obtained from the method which lead to real or to complex dispersion relations of the one-dimensional phononic crystal. The model results in a Lamb wave band structure where the surface modes are identified by their displacement and elastic energy distribution. The complex band structure includes imaginary and complex modes in addition to the real dispersion relation describing also standing and evanescent modes. Such evanescent waves not only include the folded surface waves, but it is also shown that an evanescent mode is present within the stop band.