Periodic Structures, Irreducible Brillouin Zone, Dispersion Relations and the Plane Wave Expansion Method

Abstract

The Plane Wave Expansion (PWE) method allows the calculation of dispersion curves, i.e., the relation linking the frequency to the wave number for any propagating mode of periodic structures made of elastic materials such as phononic crystals. The method is relatively easy to implement numerically but presents some limitations. After recalling some fundamental aspects of crystallography that are necessary to the study of periodic structures, the PWE method described in detail for the case of bulk phononic crystals, i.e., structures of infinite extent, and its advantages and drawbacks are discussed. It is also shown that the method can be used for calculating the band structure of phononic crystals of finite thickness and for analysing the evanescent waves within the phononic band gaps.