Random driven fast waves in coronal loops I. Without coupling to Alfvén waves

In this paper we study the time evolution of fast MHD waves in a coronal loop driven by footpoint motions in linear ideal MHD. We restrict the analysis to footpoint motions polarized normal to the magnetic flux surfaces such that the fast waves are driven directly. By supposing the azimuthal wave number ky to be zero, the fast waves are decoupled from the Alfvén waves. As a first step to real stochastic driving, we consider the loop to be driven by a train of identical pulses with random time intervals in between. The solution is written as a superposition of eigenmodes whose excitation is determined by the time dependence of the footpoint motion through a convolution and by the spatial dependence of the footpoint motion through a scalar product. An important result from the simulations is that the amount of kinetic energy associated with the body modes is much larger than the amount corresponding to the leaky modes. This means that most of the input energy is stored within the loop. For ky / = 0, body modes can resonantly couple to Alfvén waves at certain magnetic surfaces and hence the energy of the body modes can then be dissipated around the resonant magnetic surfaces. Using a gamma distribution for the time intervals between the successive pulses, we analytically derive a relation between the mean value of the kinetic energy contribution of each eigenmode, the eigenfrequency, the number of pulses and the width of the pulses. The larger the variance of the distribution, the less the power spectrum reveals fine structure, peaks around certain preferred frequencies. The analytical results confirm the output from the numerical simulations.