Astron. Astrophys. 332, 314-324 (1998)

2. Basic equations

An 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 x. All equilibrium variables are independent of the vertical coordinate z. The set of linearized hydrodynamical equations is

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 t and z is introduced

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 u is obtained

which in terms of the new dependent variable

reads

where is a phase velocity. Eq. (15) defines the dependence on x of the amplitude of an acoustic wave with frequency and vertical phase velocity . In the limit of a uniform atmosphere, where and , the equation is reduced to the dispersion relation , where is the "horizontal" wavenumber. In the general case of a structured atmosphere the dispersion equation for acoustic waves could be defined by the condition that the wave amplitude is bounded at infinity. This condition corresponds to the requirement that in the case of the uniform atmosphere.

In this paper I treat only the case where the equilibrium temperature, density and vertical velocity are periodic functions of x, with spatial period :

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
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