Energy and dissipation of inhomogeneous plane waves in thermoelasticity

Inhomogeneous small-amplitude plane waves of (complex) frequency ω are propagated through a linear dissipative material. For thermoelasticity we derive an energy-dissipation equation that contains all the quadratic dependence on the field quantities, see Eq. (10). In addition, we derive a new energy-dissipation equation (Eq. (22)) involving the total energy density which contains terms linear in the field quantities as well as the usual quadratic terms. The terms quadratic in the small quantities in the energy density, energy flux and dissipation give rise to inhomogeneous plane waves of frequency 2ω and to (attenuated) constant terms. Usually these quadratic quantities are time-averaged and only the attenuated constant terms remain. We derive a new result in thermoelasticity for these terms, see Eq. (54). The present innovation is to retain the terms of frequency 2ω, since they are comparable in magnitude to the attenuated constant terms, and a new result, see Eq. (44), is derived for a general energy-dissipation equation that connects the amplitudes of the terms of the energy density, energy flux and dissipation that have frequency 2ω. Furthermore, for dissipative waves or inhomogeneous conservative waves 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.