On wave equations and cut-off frequencies of plane atmospheres

This paper deals with the one-dimensional vertical propagation of linear adiabatic waves in plane atmospheres. In the literature there are various representations of the standard form of the wave equation from which different forms of the so called cut-off frequency are inferred. It is not uncommon that statements concerning the propagation behavior of waves are made which are based on the height dependence of a cutoff frequency. In this paper, first we critically discuss concepts resting on the use of cut-off frequencies. We add a further wave equation to three wave equations previously presented in the literature, yielding an additional cut-off frequency. Comparison among the various cut-off frequencies of the VAL-atmosphere reveals significant differences, which illustrate the difficulties of interpreting a height dependent cut-off frequency. We also discuss the cut-off frequency of the parabolic temperature profile and the behavior of the polytropic atmosphere. The invariants of the four wave equations presented contain first and second derivatives of the adiabatic sound speed. These derivatives cause oscillations and peaks in the space dependent part of the invariants, which unnecessarily complicate the discussion. We therefore present a new form of the wave equation, the invariant of which is extremely simple and does not contain derivatives of the thermodynamic variables. It is valid for any LTE equation of state. It allows us to make effective use of strict oscillation theorems. We calculate the height-dependent part of the invariant of this equation for the VAL-atmosphere including ionization and dissociation. For this real atmosphere, there is no obvious correspondence between the behavior of the invariant and the temperature structure or the sound speed profile. The invariant of the wave equation is nearly constant around the temperature minimum. In the chromosphere, the invariant is almost linear. The case of the wave equation with a linear invariant is studied analytically.