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Thermoelastic attenuation of P waves is due to energy conversion to the heat mode, which is diffusive at low frequencies and wave-like at high frequencies, behaving similarly to the Biot slow mode. The conversion is strong in highly heterogeneous media. We consider a layered medium with a random distribution of thermal properties, specifically the Grüneisen ratio, and obtain the phase velocity and quality factor. The relaxation peak of the random medium is wider than those of a periodic sequence of layers and the Zener mechanical model. Indeed, a Cole–Cole fractional model is needed to obtain a good match. These approximations are required to compute wave fields in heterogeneous media. Moreover, the solutions are helpful for studying the physics of thermoelasticity and testing numerical algorithms for wave propagation.

Analytical solution of thermoelastic attenuation in fine layering for random variations of the Grüneisen ratio