## 3. Mathematical approachAt first we represent the footpoint motions by inhomogeneous
boundary conditions for Eqs. (4) and (5) at the We have assumed for mathematical simplicity that the dependencies
on Apart from these restrictions the functions 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 we include the footpoint motions as driving terms in the equations.
The boundary conditions then become homogeneous which allows us to
expand , 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 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 where 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 By inverting the Laplace transformation, recalling the definition
of and the sine transformations, we obtain the
following solution for and
as function of
and is the scalar product of 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 © European Southern Observatory (ESO) 1998 Online publication: June 12, 1998 |