Energy and plane waves in linear viscoelastic media

Abstract

The mathematical framework for describing plane waves in elastic and linear anelastic media is presented. Theoretical results suggest that the nature of plane waves in anelastic materials is distinctly different from the nature of plane waves in elastic materials. In elastic media the only type of inhomogeneous plane wave (P or S) that can propagate is one for which planes of constant phase are perpendicular to planes of constant amplitude. However, in anelastic media this is the only type of inhomogeneous wave that cannot propagate. For an inhomogeneous P or S plane wave the particle motion is elliptical, the velocity is less than that of a corresponding homogeneous wave, the maximum attenuation is greater than that of a corresponding homogeneous wave, and the direction of maximum energy flow is not the direction of phase propagation. Expressions for the energy flux, energy densities, dissipated energy, stored energy, and Q−1 are derived from an explicit energy conservation relation, valid for an arbitrary steady state viscoelastic radiation field. Each energy expression is valid for homogeneous or inhomogeneous P or S plane waves in elastic or linear anelastic media.