Energy flux and dissipation of inhomogeneous plane wave forms in Kelvin-Voigt viscoelasticity

Inhomogeneous plane waves of complex frequency ω and slowness S are propagated through a (linear) Kelvin-Voigt viscoelastic solid. The energy density, energy flux and dissipation are quadratic in the small quantities (namely, the displacement gradient, velocity and velocity gradient) and so give rise to (attenuated) constant terms as well as to inhomogeneous plane waves of frequency 2ω. The latter- terms are usually removed by time averaging, but here we retain them as they are necessarily of a magnitude comparable with those of the time-averaged quantities. A relationship is derived that connects the amplitudes of the terms of the energy density, energy flux and dissipation that have frequency 2ω. It is shown that the complex group velocity is related to these amplitudes rather than to the attenuated constant terms as it is for homogeneous waves in conservative materials. The results are specialized to isotropic viscoelasticity and to viscous fluids.