On the reflection of linear hydrodynamic waves at continuous temperature steps in stellar atmospheres

A simple analytical expression which describes a smooth temperature step is used to study the behavior of vertically propagating adiabatic waves in non-isothermal plane atmospheric layers. With this expression the one-dimensional equation of adiabatic linear atmospheric waves is reduced to a hypergeometric differential equation. From the asymptotic evaluations of the solutions we obtain formulas for reflection coefficients and phases. In the case of acoustic waves the reflection coefficients can be given in terms of simple functions. It is shown how the photospheric temperature decrease and the chromospheric temperature increase of the solar atmosphere can be approximated by the temperature formula. For these approximations various results are presented for both evanescent and acoustic waves. The results of the reflection at continuous steps are compared with those at a temperature discontinuity. The WKB-approximation of the velocity of a wave passing a continuous temperature step is given. A limiting form of the temperature function is used to study the behavior of evanescent waves in a photosphere.