Coronal Loop Oscillations Driven by Footpoint Motions: Analytical Results for a Model Problem

We study the excitation of MHD waves in a coronal loop as its field-line footpoints are forced to follow the photospheric convective motions. Due to the large photo spheric-coronal density ratio, these waves are reflected back and forth in the loop and thus build up a global oscillation. By focusing on a specific case the problem is reduced to the simplest situation conceivable which still contains the relevant physics of wave propagation, reflections, resonances, and phase mixing. This model problem allows for a fully analytical treatment which greatly helps to understand earlier results found in the literature. It is shown that the classical asymptotic state assumption, in which all perturbed quantities have the same frequency as the driver, is justified only when a wave damping mechanism, such as leakage, is present. Special attention is given to footpoint motions with resonant frequencies and to footpoint motions with finite lifetimes.