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Aims. Motivated by recent observations of oscillations in coronal arcades, we investigate analytically the influence of longitudinal structuring on the modes of oscillation of a straight coronal loop arcade. As a first step towards more complicated models, we use a simple structure to obtain analytical solutions. Methods. A partial differential equation is derived for the total pressure perturbation of the fast modes in a zero beta plasma and it is solved analytically. We first recover the results for a homogeneous structure, and then study an equilibrium with an exponentially structured density profile, solving it in terms of Bessel functions of non-integer order and exponential argument, thus obtaining a dispersion relation. The properties of this dispersion relation are discussed and some limits studied, leading to analytical approximations to the eigenfrequencies. Results. The introduction of longitudinal structuring results in a modification to the oscillatory frequencies of the modes of oscillation in such structures when compared with the uniform case. Regarding the oscillatory periods Pn, n = 1, 2 , . .. , the period ratios P1/2P2 and P1/3P3 are both shifted from unity. Other properties described in structured coronal loops are also found in an arcade: the occurrence of avoided crossings in the dispersion diagram and the displacement of the extrema towards the footpoints in the spatial structure of the eigenmodes. Conclusions. We show analytically for simple arcade modes that the shift in the fundamental period proves to be small, but the ratio P1/2P2 depends strongly on the density structure. Moreover, transversal propagation also shifts the ratio P1/2P2 from unity, so it can be used in the coronal seismology of arcades in which transversal propagation is present. We use the currently available observational data to illustrate this application.

Effect of longitudinal density structure on a straight magnetic field modelling coronal arcade oscillations