On the linear coupling between fast and slow MHD waves due to line-tying effects

Oscillations in coronal loops are usually interpreted in terms of uncoupled magnetohydrodynamic (MHD) waves. Examples of these waves are standing transverse motions, interpreted as the kink MHD modes, and propagating slow modes, commonly reported at the loop footpoints. Here we study a simple system in which fast and slow MHD waves are coupled. The goal is to understand the fingerprints of the coupling when boundary conditions are imposed in the model. The reflection problem of a fast and slow MHD wave interacting with a rigid boundary, representing the line-tying effect of the photosphere, is analytically investigated. Both propagating and standing waves are analysed and the time-dependent problem of the excitation of these waves is considered. An obliquely incident fast MHD wave on the photosphere inevitably generates a slow mode. The frequency of the generated slow mode at the photosphere is exactly the same as the frequency of the incident fast MHD mode, but its wavelength is much smaller, assuming that the sound speed is smaller than the Alfv\'en speed. The main signatures of the generated slow wave are density fluctuations at the loop footpoints. We have derived a simple formula that relates the velocity amplitude of the transverse standing mode with the density enhancements at the footpoints due to the driven slow modes. Using these results it is shown that there are possible evidences in the observations of the coupling between these two modes.