On the propagation of linear 3-D hydrodynamic waves in plane non-isothermal atmospheres

Closed solutions of the time independent wave equation of 3-D linear adiabatic waves in non-isothermal atmospheres are presented and discussed. Three temperature stratifications are considered applying the full wave equation without any approximations: a continuous temperature step between two asymptotically isothermal layers, an exponentially decreasing, and an exponentially increasing temperature stratification. In this first paper we present the fundamental systems, discuss general properties, and present some tools for more detailed investigations. For the exponential temperature stratifications, the wave equation is transformed to the hypergeometric equation. Linear transformation formulas are used to study details of the solutions. For the exponentially decreasing temperature stratification the second solution of the fundamental mode is calculated, and the convective and Rayleigh-Taylor instabilities are considered. In the case of the continuous temperature step the wave equation can be reduced to Heun's differential equation. Various representations of the solutions are presented and the continuation of the solutions is discussed.