Cattaneo–Vernotte equation versus Fourier thermoelastic hyperbolic heat equation

The aim of this short note is to point out the inconsistency of the Cattaneo–Vernotte law. This law is assumed to be valid at a scale L of a few mean free paths of phonons lfp. Thus a separation of scales is absent and the Cattaneo–Vernotte law looses its status of equivalent macroscopic law. On an other hand, the classical thermoelastic system, which is well supported by experimental investigations, is valid for L ≫ lfp where a separation of scales is present. In this domain of validity, an acoustic wave is adiabatic. It results in an apparent hyperbolic heat equation which remains valid for L > 10 lfp. This equation shows a similar structure to the Cattaneo–Vernotte equation, however with a different relaxation time.