Small-Amplitude Disturbances In A Radiating And Scattering Grey Medium Ii. Solutions Of Given Real Wave Number k

Abstract

We reanalyze the propagation of one-dimensional small-amplitude disturbances of given real frequency ω in a radiating and scattering grey medium using the Eddington approximation, which has been studied previously by us (Kaneko et al., 1976). Numerical results reveals three frequency regimes to be distinguished, and two wave modes always appear in each frequency regime. The governing equations and analytic solutions are derived for all wave modes using Whitham's method modified into quadratic form and approximate methods based on radiation thermodynamics. In the high-frequency regime appear the radiation-wave and adiabatic sound modes, which are damped by opacity and radiative cooling, respectively. Wave patterns in the intermediate-frequency and low-frequency regimes depend critically on the importance of radiation, for which the criterion is given in terms of the ratio of total specific heats at constant pressure and constant volume. When the radition overwhelms the matter (radiation-dominated case), the radiative mode in the intermediate-frequency regime is the constant-volume diffusion mode. When the matter overwhelms the radiation (matter-dominated case), damped radiation-wave and damped radiation-diffusion modes newly appear between the radiation-wave and constant-volume diffusion modes. The acoustic mode in the intermediate-frequency regime is the isothermal sound mode,which is damped by radiative cooling at higher frequencies and by radiation-thermal drag force at lower frequencies. Two modes appearing in the low-frequency regime are the isentropic radiation-acoustic and constant-pressure diffusion modes. The absorption coefficient derived for the former is shown to be a radiation-thermodynamic extension of that of Landau and Lifshitz (1987). The transition frequencies between all adjacent two modes are also derived to discuss the implications of them.