Chapter 3 - Isotropic Anelastic Media

Abstract

The space dimension reveals other properties of anelastic (viscoelastic) wave fields. There is a distinct difference between the inhomogeneous waves of lossless media (interface waves) and those of viscoelastic media (body waves). In the former case, the direction of attenuation is normal to the direction of propagation, whereas for inhomogeneous viscoelastic waves, that angle must be less than π/2. Furthermore, for viscoelastic inhomogeneous waves, the energy does not propagate in the direction of the slowness vector and the particle motion is elliptical in general. The phase velocity is less than that of the corresponding homogeneous wave (for which planes of constant phase coincide with planes of constant amplitude); critical angles do not exist in general, and, unlike the case of lossless media, the phase velocity and the attenuation factor of the transmitted waves depend on the angle of incidence. There is one more degree of freedom, since the attenuation vector is playing a role at the same level as the wavenumber vector. Snell law, for instance, implies continuity of the tangential components of both vectors at the interface of discontinuity. For homogeneous plane waves, the energy-velocity vector is equal to the phase-velocity vector. The last part of the chapter analyzes the viscoelastic wave equation expressed in terms of fractional time derivatives, and provides expressions of the reflection and transmission coefficients corresponding to a partially welded interface.