Canonical analytical solutions of wave-induced thermoelastic attenuation

Thermoelastic attenuation is similar to wave-induced fluid-flow attenuation (mesoscopic loss) due to conversion of the fast P wave to the slow (Biot) P mode. In the thermoelastic case, the P- and S-wave energies are lost because of thermal diffusion. The thermal mode is diffusive at low frequencies and wave-like at high frequencies, in the same manner as the Biot slow mode. Therefore, at low frequencies, that is, neglecting the inertial terms, a mathematical analogy can be established between the diffusion equations in poroelasticity and thermoelasticity. We study thermoelastic dissipation for spherical and cylindrical cavities (or pores) in 2-D and 3-D, respectively, and a finely layered system, where, in the latter case, only the Gru ̈neisen ratio is allowed to vary. The results show typical quality-factor relaxation curves similar to Zener peaks. There is no dissipation when the radius of the pores tends to zero and the layers have the same properties. Although idealized, these canonical solutions are useful to study the physics of thermoelasticity and test numerical algorithm codes that simulate thermoelastic dissipation.