Acoustic-gravity waves in a viscous and thermally conducting isothermal atmosphere (part III: for arbitrary Prandtl number)

In this paper we will investigate the combined effect of Newtonian cooling, viscosity and thermal condition on upward propagating acoustic waves in an isothermal atmosphere. In part one of this series we considered the case of large Prandtl number, while in part two we investigated the case of small Prandtl number. In those parts we examined only the limiting cases, i.e. the cases of small and large Prandtl number, and it is more interesting to consider the case of arbitrary Prandtl number, which is the subject of this paper, because it is a better representative model. It is shown that if the Newtonian cooling coefficient is small compared to the frequency of the wave, the effect of the thermal conduction is dominated by that of the viscosity. Moreover, the solution can be written as a linear combination of an upward and a downward propagating wave with equal wavelengths and equal damping factors. On the other hand if Newtonian cooling is large compared to the frequency of the wave the effect of thermal conduction will be eliminated completely and the atmosphere will be transformed from the adiabatic form to an isothermal. In addition, all the linear relations among the perturbations quantities will be modified. It follows from the above conclusions and those of the first two parts, that when the effect of Newtonian cooling is negligible thermal conduction influences the propagation of the wave only in the case of small Prandtl number.