Effective anisotropy of periodic acoustic and elastic composites

TL;DR

This paper derives formulas to estimate the anisotropic effective velocities of acoustic and elastic waves in periodic composites, revealing conditions for strong anisotropy and slow quasi-longitudinal waves.

Contribution

It introduces explicit formulas for effective velocity surfaces in periodic composites, generalizing the Christoffel equation for elastic waves.

Findings

Identification of strongly anisotropic sonic and phononic crystals.

Conditions under which quasi-longitudinal waves are slower than shear waves.

Explicit formulas for effective velocities in periodic composites.

Abstract

The propagation of acoustic or elastic waves in artificial crystals, including the case of phononic and sonic crystals, is inherently anisotropic. As is known from the theory of periodic composites, anisotropy is directly dictated by the space group of the unit cell of the crystal and the rank of the elastic tensor. Here, we examine effective velocities in the long wavelength limit of periodic acoustic and elastic composites as a function of the direction of propagation. We derive explicit and efficient formulas for estimating the effective velocity surfaces, based on second-order perturbation theory, generalizing the Christofell equation for elastic waves in solids. We identify strongly anisotropic sonic crystals for scalar acoustic waves and strongly anisotropic phononic crystals for vector elastic waves. Furthermore, we observe that under specific conditions, quasi-longitudinal waves can be made much slower than shear waves propagating in the same direction.