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Astron. Astrophys. 335, 329-340 (1998)

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3. Mathematical approach

At first we represent the footpoint motions by inhomogeneous boundary conditions for Eqs. (4) and (5) at the z and the z boundary planes:

[EQUATION]

We have assumed for mathematical simplicity that the dependencies on x and t of the footpoint motions are separable. In order to avoid complications with initial conditions we assume in addition that at t, [FORMULA], [FORMULA] and both their time derivatives are zero and as a consequence:

[EQUATION]

Apart from these restrictions the functions R and T can be chosen completely arbitrarily.

A convenient way to solve the coupled partial differential Eqs. (4-5) is to get rid of as many derivatives as possible. First of all we can remove the z-derivatives. By introducing the function

[EQUATION]

we include the footpoint motions as driving terms in the equations. The boundary conditions then become homogeneous which allows us to expand [FORMULA] , i and (1) in a series of sines.

With the use of these sine transforms the coupled partial differential Eqs. (4) and (5) are replaced by an infinite set of coupled partial differential equations for X and Y, the 'nth' coefficients in the series of sines of [FORMULA] and i respectively. There is no coupling between different n-modes.

By taking the Laplace transform of these coupled equations, we also remove the time-derivative in the equations. We then are left with two coupled equations for each n :

[EQUATION]

where A only contains derivatives with respect to x and B is dependent on T and R. Eq. (7) can be solved by inversion of the operator A. Formally this is not a problem, because A corresponds to the ideal MHD force operator which is Hermitian with respect to the following scalar product

[EQUATION]

It is well known that the eigenfunctions form a complete set so that the solution to Eq. (7) can be written as a spectral representation:

[EQUATION]

with orthonormal discrete and continuum eigenfunctions satisfying the orthonormal relationships :

[EQUATION]

The boundary conditions in the x-direction and the form of the Eqs. (4-5) suggest that the first component and the second component of the eigenfunction can be expanded in series of sines and cosines respectively.

By inverting the Laplace transformation, recalling the definition of [FORMULA] and the sine transformations, we obtain the following solution for [FORMULA] and [FORMULA] as function of x, z and t:

[EQUATION]


In these equations, T is the time convolution:

[EQUATION]

and [FORMULA] is the scalar product of R and the eigenfunction [FORMULA] without weight function:

[EQUATION]

[FORMULA] and [FORMULA] are the coefficients of the sine and cosine expansions of the two components of the eigenfunctions respectively.

This approach is described in more detail in Tirry, Berghmans & Goossens (1997) and enables us to obtain an expression which describes the generation of linear MHD waves (coupled fast-Alfvén waves) by radially polarized footpoint motions. Expressions (9) and (10) can be easily evaluated numerically at any time with the structure of the waves fully resolved as long as a sufficiently large numbers of sines in both x and z directions are taken into account.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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