Surface acoustic waves in laterally periodic superlattices

Abstract

The existence of surface acoustic waves is studied in laterally periodic superlattices, modelled as an elastic half-space of general anisotropy with an arbitrary periodic modulation of material properties in the direction (call it X) parallel to the surface. Unlike a homogeneous half-space, a laterally periodic one may support more than one (subsonic) surface wave. Specifically, it is shown that the maximum number of surface waves travelling along X in any superlattice with a generic (asymmetric) profile of periodic stratification is three. What is rather remarkable is that the same upper bound applies to the total number of waves in two superlattices with mutually inverse profiles or, which is the same, to the number of counter waves travelling in the opposite directions of a given superlattice, i.e. their overall number cannot exceed three either. At least one surface wave exists in one of the two superlattices unless the bulk-wave threshold is of the so-called exceptional type. In the particular case of the symmetric stratification profile, i.e. the one invariant to the inversion of the axis X, the surface wave remains unique, like in the case of a homogeneous half-space. The above general results are illustrated by the perturbation theory derivations under the assumption of weak periodic modulation. Explicit leading-order formulas are obtained for the frequencies of the two quasi-Rayleigh waves evolving from the Rayleigh wave by one for the mutually inverse profiles and for the frequency of a quasibulk wave evolving from the exceptional transonic state for one of the profiles (but not for both).