Dispersion relation governing the propagation of wave pulses in a mixture of interacting longitudinal and transverse phonon gases

The investigations by Koreeda et al. (2009) and Ma (2013) of the effect of second sound in dielectric crystals assume the representation of the phonon system as a mixture composed of the gas of longitudinal phonons and the gas of transverse phonons. The theoretical background of those investigations can be related either to linear phonon gas hydrodynamics based on the moment equations of Dreyer and Struchtrup (1993) or to the heuristic model of energy transport in a phonon gas proposed by Rogers (1971). The relationship between these two approaches is analyzed. Specifically, restricting attention to the one-dimensional flows, the system of linear hyperbolic equations of six-moment phonon gas hydrodynamics is discussed together with the 'equivalent' system of two coupled third-order in time partial differential equations for the energy densities of longitudinal and transverse phonon branches. Given the six-moment system and its equivalent, the dispersion relation governing the propagation of complex plane harmonic waves is studied in detail. It is demonstrated that the interaction between the longitudinal phonon gas and the transverse phonon gas plays an important role when determining how the complex wave number depends on the real angular frequency of plane harmonic waves. The derived dispersion relation is systematically compared with that postulated by Ma (2013). Using the principle of superposition and the real plane harmonic waves as a basis, the manifestly real pulse-like solution to the equations of phonon hydrodynamics is explicitly constructed. It is shown that the computed energy-pulse profiles have the features that agree well with the main features of the energy or temperature profiles observed in the heat-pulse experiments.