Aims.A model of a elliptically shaped coronal arcade with piecewise constant density is discussed to explore the effects of curvature on radially polarised fast modes. It is important to test whether the main results in the straight and cylindrical geometries can be extrapolated to these more complex equilibria. Methods: .An equilibrium model for a force-free, line-tied elliptical arcade is introduced and a partial differential equation is derived for the velocity perturbation of the fast modes, which is solved analytically. The properties of the modes are studied in terms of the dispersion relation, which depends on the eccentricity, the arcade width, and the density contrast. Results: .Modes mainly contained in the cavity below the arcade are also present, and have avoided crossings with the modes of the arcade. Even the fundamental mode becomes leaky due to curvature. Approximated relations are deduced for the frequency of the modes and the spatial structure is discussed, focusing on the different families through which a rich mode spectrum can be classified. Conclusions: .The different types of modes of the spectrum are described and its relevance to observations is discussed. The periods obtained in Cartesian geometry provide a reasonable approximation, but this geometry lacks some other key ingredients: the damping rates are different and some types of modes present in the elliptical geometry are not sustained in the straight slab.