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Using magnetohydrodynamic (MHD) theory, we investigated wave propagation in a magnetic slab system representing a simple waveguide model applicable to a wide range of solar phenomena as a first approximation. While some recent advances in modelling solar features using Cartesian geometry studied the effects of introducing several sources (plasma, magnetic, flow) asymmetry into the slab system, we focus our modelling efforts on closing additional boundaries of the slab in order to better encapsulate the finite nature of solar atmospheric structures. In previous approaches, the MHD waves were described with inhomogeneity in a single direction of structuring perpendicular to the direction of the background magnetic field, while the configuration was left semi-infinite in the third direction perpendicular both to the structuring and the magnetic field.This earlier approximation describes the characteristics of magnetohydrodynamic waves in a two-dimensional model. Expanding upon this model, our study presented here, however, delved into all three dimensions of the problem. While we still focused on waves propagating along the magnetic field, we closed the slab in both directions perpendicular to the background magnetic field and left it open only in the direction parallel to the field lines. We derived the system of new governing equations of this complex and generalised problem and determined their solutions in order to obtain a dispersion relation modelling MHD wave propagation applicable to solar phenomena such as e.g., magnetic bright points, light bridges, or prominences.

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