Magnetohydrodynamic waves in a sheared potential coronal arcade

We study the effects of magnetic field shear (By) and longitudinal propagation of perturbations ($k_y\neq 0$) on the linear and adiabatic magnetohydrodynamic (MHD) normal modes of oscillation of a potential coronal arcade. In a cold plasma, the inclusion of these two effects produces the linear coupling of discrete fast modes, characterised by a discrete spectrum of frequencies and a global velocity structure, and Alfvén continuum modes, characterised by a continuous spectrum of frequencies and with a velocity perturbation confined to given magnetic surfaces in such a way that modes with mixed properties arise [CITE]. The wave equations governing the velocity perturbations have been solved numerically and our results show that the couplings between fast and Alfvén modes are governed by some parity rules for the symmetry of the eigenfunctions of fast and Alfvén modes in the direction along the equilibrium magnetic field. The nature of the coupling between fast and Alfvén modes can be resonant or non-resonant depending on the location of the fast mode frequency within the different Alfvén continua. Also, an important result is that in this kind of configurations coupled modes could be difficult to observe since when both magnetic field shear and longitudinal propagation are present the spatial distribution of the velocity may not be confined to low heights in the solar corona.