## 2. Basic equationsAn equilibrium atmosphere with vertical hot upflows and cold
downflows is considered. The equilibrium pressure
is constant in the entire atmosphere, while the
equilibrium values of temperature , density
and vertical velocity are
arbitrary functions of The time derivative of the momentum equation is The substitution of Eqs. (2) and (3) into the time derivative of the linearized equation of state (4) gives After substitution of from (1) into (6) and of (6) into (5) the set of Eqs. (1)-(4) is reduced to one vector equation for the velocity Now take the divergence of both sides of Eq. (7) where the notation is used. The substitution of (7) into (8) gives Now a harmonic dependence of the variables on and the components of Eq. (7) and Eq. (9) are rewritten as where the tilde has been omitted. After elimination of
and from
Eqs. (10)-(12) a second-order equation for which in terms of the new dependent variable where is a phase velocity. Eq. (15)
defines the dependence on In this paper I treat only the case where the equilibrium
temperature, density and vertical velocity are periodic functions of
Eq. (15) belongs to the class of equations with periodic coefficients. The treatment of such equations has been developed especially for the Mathieu and Hill equations, which often appear in physical problems (Ince 1944). However, Eq. (15) is more general than the Mathieu and Hill equations, and a more detailed treatment is necessary. For convenience we introduce dimensionless variables is the mean of the sound velocity over the space period . The reason for the unusual definition of the spatial wavenumber is that in this way Eq. (15) is reduced to the form used for the Mathieu and Hill equations. It is thus easier to refer to the properties of equations with periodic coefficients. In terms of the dimensionless variables Eq. (15) reads © European Southern Observatory (ESO) 1998 Online publication: March 10, 1998 |