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Abstract Using N-body simulations of the Milky Way interacting with a satellite similar to the Sagittarius dwarf galaxy, we quantitatively analyse the vertical response of the Galactic disc to the satellite’s repeated impacts. We approximate the vertical distortion of the Galactic disc as the sum of the first three Fourier azimuthal terms m = 0, 1 and 2, and observe their evolution in different dynamical regimes of interaction. After the first interaction, the m = 0 term manifests itself as outgoing ring-like vertical distortions. The m = 1 term (S-shape warp) is prograde when the impacts of the satellite are more frequent, or in general close to an interaction, whereas it is slowly retrograde in the most quiescent phases. The m = 2 term is typically prograde, and close to an interaction it couples with the m = 1 term. Finally, we find that the vertical response of the disc can be recovered in an unbiased way using the instantaneous positions and velocities of stars in a limited volume of the Galactic disc, analogous to real data, and that the measured vertical pattern speeds have a constraining power in the context of a Milky Way-satellite interaction.