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Evidence for wave-like corrugations are well established in the Milky Way and in nearby disc galaxies. These were originally detected as a displacement of the interstellar medium about the midplane, either in terms of vertical distance or vertical velocity. Over the past decade, similar patterns have emerged in the Milky Way's stellar disc. We investigate how these vertical waves are triggered by a passing satellite. Using high-resolution N-body/hydrodynamical simulations, we systematically study how the corrugations set up and evolve jointly in the stellar and gaseous discs. We find that the gas corrugations follow the stellar corrugations, i.e. they are initially in phase although, after a few rotation periods (500-700 Myr), the distinct waves separate and thereafter evolve in different ways. The spatial and kinematic amplitudes (and thus the energy) of the corrugations dampen with time, with the gaseous corrugation settling at a faster rate (~800 Myr versus ~1 Gyr). In contrast, the vertical energy of individual disc stars is fairly constant throughout the galaxy's evolution. This difference arises because corrugations are an emergent phenomenon supported by the collective, ordered motions of co-spatial ensembles of stars. We show that the damping of the stellar corrugations can be understood as a consequence of incomplete phase mixing, while the damping of the gaseous corrugations is a natural consequence of the dissipative nature of the gas. We suggest that - in the absence of further, strong perturbations - the degree of correlation between the stellar and gaseous waves may help to age-date the phenomenon.