Effect of Radiative Transfer on a Sound Wave Traveling in a Gas having γ near One

It is shown that the assumption of specific heat ratio near one simplifies the analysis of plane sound waves subject to energy addition dependent on temperature, and in the case of radiative energy exchange in a fairly transparent gas, a third-order differential equation is obtained. Analytical solutions for a hot and a cool gas are presented which give the complete history of velocity and temperature profiles when a wave is produced by an impulsively started piston, with either a black or reflecting face. In a very hot gas, an optically thin wave which is thermally dominated by the piston face (or the undisturbed gas, if the face is perfectly reflecting) disperses as though by relaxation and is followed by an optically thick wave, which is thermally independent of the piston and disperses as though by conduction. Transition between thin and thick waves is accomplished in a temperature boundary layer. In a cool gas, dispersion occurs as though by simple radiative heat conduction. Finally, it is remarked that dispersion in a transparent gas would be observed on the scale of a meter only if the gas is very hot.