Aims. Fast magnetoacoustic waves in curved coronal loops are investigated and the role of lateral leakage in wave damping, which includes the mechanism of wave tunneling, is explored.\ud Methods. A coronal loop is modeled as a curved, magnetic slab in the zero plasma-β limit. In this model and for an arbitrary piece-wise continuous power law equilibrium density profile, the wave equation governing linear vertically polarised fast magnetoacoustic waves is solved\ud analytically. An associated dispersion relation is derived and the frequencies and eigenfunctions of the wave modes are characterised.\ud Results. For some equilibria, the waves are shown to be all damped due to lateral leakage. It is demonstrated that waves either leak straight out into the external medium or have to overcome an evanescent barrier, which is linked to wave tunneling. The wave solutions consist of alternating vertically polarised kink and sausage branches. Fast kink oscillations may have a non-zero density perturbation when\ud averaged across the loop. The calculated damping rate of fast magnetoacoustic kink oscillations is shown to be consistent with related numerical simulations and show that lateral leakage may explain the observed damping of (vertically polarised) fast magnetoacoustic kink oscillations.\u
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