Towards an explanation of features in the diagnostic diagram of a model atmosphere - I. Linear wave equations with convenient invariants

New standard forms of the time-independent linear adiabatic wave equation of plane atmospheres are presented. The main objective is to obtain equations with invariants as simple as possible so that oscillation theorems can be applied effectively. By transformations of both the independent and the dependent variables, equations with simple invariants are formulated. We present a standard form of the wave equation the invariant of which depends only on the first derivative of the equilibrium density, as opposed to the common standard form the invariant of which depends also on second derivatives. Further, we discuss a procedure which replaces the wave equation by a system of two simple second order differential equations. In this case we try to draw conclusions on the general behavior of solutions by use of oscillation theorems. In addition, a re-formulation of the wave equation is presented, which eliminates terms with first derivatives of atmospheric quantities. The independent variable of the resulting equation depends not only on the geometrical height but also on the ratio omega/k. In this case, it is necessary to use a diagnostic diagram the axes of which are given by omega/k and ! instead of the common k-omega diagram. Therefore we discuss the meaning of the parameter omega/k for the representation of dispersion curves. Finally, for the VAL-atmosphere (Vernazza et al. 1981), regions of certainly nonoscillatory waves are considered.