Influence of nonlinearity of the phonon dispersion relation on wave velocities in the four-moment maximum entropy phonon hydrodynamics

This paper analyzes the propagation of the waves of weak discontinuity in a phonon gas described by the four-moment maximum entropy phonon hydrodynamics involving a nonlinear isotropic phonon dispersion relation. For the considered hyperbolic equations of phonon gas hydrodynamics, the eigenvalue problem is analyzed and the condition of genuine nonlinearity is discussed. The speed of the wave front propagating into the region in thermal equilibrium is first determined in terms of the integral formula dependent on the phonon dispersion relation and subsequently explicitly calculated for the Dubey dispersion-relation model: | k | = ω c − 1 ( 1 + b ω 2 ) . The specification of the parameters c and b for sodium fluoride (NaF) and semimetallic bismuth (Bi) then makes it possible to compare the calculated dependence of the wave-front speed on the sample’s temperature with the empirical relations of Coleman and Newman (1988) describing for NaF and Bi the variation of the second-sound speed with temperature. It is demonstrated that the calculated temperature dependence of the wave-front speed resembles the empirical relation and that the parameters c and b obtained from fitting respectively the empirical relation and the original material parameters of Dubey (1973) are of the same order of magnitude, the difference being in the values of the numerical factors. It is also shown that the calculated temperature dependence is in good agreement with the predictions of Hardy and Jaswal’s theory (Hardy and Jaswal, 1971) on second-sound propagation. This suggests that the nonlinearity of a phonon dispersion relation should be taken into account in the theories aiming at the description of the wave-type phonon heat transport and that the Dubey nonlinear isotropic dispersion-relation model can be very useful for this purpose.