Stability and plane wave analyses for propagation in thermally radiating gases under Green–Naghdi’s (1995) theory of dissipationless flow

We examine the propagation of acoustic and second-sound waves in a thermally radiating, but otherwise lossless, gas under a theory of fluids proposed by Green and Naghdi in 1995. After recalling the corresponding classical Stokes model, we first compare the two theories through Whitham-type stability criteria and show that both are stable in the parameter ranges considered. We then carry out a detailed harmonic plane-wave analysis of the Green–Naghdi model, deriving the exact dispersion relation, the associated wave branches, and amplitude ratios under boundary conditions suitable for detecting second sound. Low-frequency and small-radiation asymptotic results are obtained, revealing a branch-switching phenomenon as the ratio of acoustic to thermal-wave speed crosses unity. Finally, in the non-radiative limit, we show that the resulting dispersion relation is closely related to that arising in a monofluid description of helium II, and point out an even closer correspondence with inertial theory. These results support the view that Green–Naghdi theory provides a useful framework for second-sound propagation in radiating gases.