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Astron. Astrophys. 330, 726-738 (1998)

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4. Linear wave equation

In order to gain some understanding on the properties of nonlinear fast magnetosonic waves, we consider the following expression governing the propagation of fast waves in the linear regime (cf. Oliver et al. 1993),

[EQUATION]

where [FORMULA] and [FORMULA] are given by expressions (11) and (8), respectively.

Note that after taking [FORMULA], the `force' on the right-hand side of this expression lies in the normal direction (to the equilibrium B). Hence, parallel motions do not propagate in the present conditions and the slow mode is absent.

Moreover, Eq. (15) multiplied by [FORMULA] reduces to

[EQUATION]

where the Laplacian operator on the right-hand side indicates that the magnetic pressure gradient is responsible for the propagation of fast MHD perturbances in the present model.

Following ade & Ballester (1995b), analytical solutions to Eq. (16) with [FORMULA] are obtained for the variable v defined as follows,

[EQUATION]

where [FORMULA] is the normal velocity component. In terms of v Eq. (16) reduces to

[EQUATION]

which can be solved analytically using Fourier transformation techniques once the initial velocity

[EQUATION]

and its time derivative

[EQUATION]

are imposed.

The function [FORMULA] follows from the expression

[EQUATION]

with

[EQUATION]

[EQUATION]

and

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: January 16, 1998
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