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:
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, , and both their time derivatives are zero and as a consequence:
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
we include the footpoint motions as driving terms in the equations. The boundary conditions then become homogeneous which allows us to expand , 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 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 :
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
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:
with orthonormal discrete and continuum eigenfunctions satisfying the orthonormal relationships :
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.
and is the scalar product of R and the eigenfunction without weight function:
and 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.
© European Southern Observatory (ESO) 1998
Online publication: June 12, 1998