## Appendix A: calculation of the WKB solutionsThe phase mixing equation for the perturbed velocity in a gravitationally stratified atmosphere with a radially diverging background magnetic field is given by where and . Now set so that Setting with and only taking into account second order derivatives in the damping term we get To obtain a second order partial differential equation with constant coefficients, we eliminate the first order derivative by settings its coefficient equal to zero: To leading order, the coefficient of where we have chosen the positive square root to match the outgoing wave condition. We then find Neglecting the second term in the coefficient of To obtain a solution to the non-zero dissipation case, we set where and . From Eq. (A.5) we then get and with we find, The full solution for the perturbed velocity then becomes: with and . The solution for the perturbed magnetic field can be obtained in a similar manner. © European Southern Observatory (ESO) 2000 Online publication: January 31, 2000 |