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

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Appendix A: derivation of dispersion equation

A.1. Vertical propagation

Thermally structured atmosphere Substituting the solution (23) into Eq. (22) and collecting the terms with the same space periods we find the recurrence relations for the coefficients [FORMULA]:

[EQUATION]

This infinite set of equations completely defines the solution. A nontrivial solution exists if the infinite Hill determinant is zero:

[EQUATION]

This condition is the dispersion equation for waves with spatial period [FORMULA] and even eigenfunction. It is known that an infinite Hill determinant written in the above form is convergent.

For the even solution with period [FORMULA] the recurrence relation for the coefficients [FORMULA]

[EQUATION]

along with (A2) defines the coefficients of the solution (23). The dispersion equation is

[EQUATION]

Moving isothermal structured atmosphere In this case Eq. (19) reads

[EQUATION]

Eq. (A7) again has the solutions (23) and (24). The even solution of period [FORMULA] is defined by the recurrence relations

[EQUATION]

The recurrence relations (A9) define the Hill determinant. The dispersion equation is

[EQUATION]

General case of a structured atmosphere Eq. (19) is reduced to

[EQUATION]

where

[EQUATION]

The mean flow [FORMULA] is introduced to remove mean plasma flux from the model. It causes a Doppler shift, which is eliminated by replacing the phase velocity in (A10) by [FORMULA] in (A13); the parameter [FORMULA] is replaced by

[EQUATION]

The recurrence relations are

[EQUATION]

and the dispersion relation reads

[EQUATION]

A.2. Oblique propagation

The coefficients of the solution (42) satisfy the recurrence relations

[EQUATION]

where [FORMULA] are defined by (A13). The recurrence relations define an infinite set of linear algebraic equations, which has a non-trivial solution if an infinite Hill determinant is zero. This defines the dispersion equation. For solutions with [FORMULA] this equation is

[EQUATION]

For solutions with [FORMULA] it reads

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998
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