Aims. Signals of stationary slow modes have been detected in observational data and modelled through numerical computations, comparing these results with the modes of a homogeneous tube. Here we explore the effect of structure along the magnetic field on the modes of oscillation of a coronal loop. Methods. We present a limit in which the slow mode is decoupled from the other magnetohydrodynamic modes, describing its behaviour in terms of a relatively simple partial differential equation. This

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... uation is solved analytically and numerically for various longitudinal profiles. Results. For low density contrast between footpoints and apex, the modes of the structured tube are similar to the modes of the homogeneous tube, evolving regularly from them, with small modifications in frequency and spatial structure. As the density contrast is increased, the extrema are displaced towards the dense layers and the frequencies of the higher harmonics are strongly modified. Finally, as the ratio is increased further, two types of modes appear: modes approximately line-tied in the dense layer and modes with high amplitude in them (with avoided crossings between them in the dispersion diagrams). Conclusions. Different regimes can be identified, depending on the density contrast between the loop footpoints and its apex. This allows us to compare apparently different numerical results and understand their various features. Our analytical results are in accordance with current numerical simulations. Article published by EDP Sciences and available at