Propagation of warps in moderately thick disks

We show that the propagation of warps in gaseous disks can be strongly affected by compressional effects, when the thickness of the disk is taken into account. The physical reason is that, in realistic self-gravitating disks, the sound time through the disk is comparable with the rotation time; thus the vertical hydrostatic equilibrium cannot be maintained adiabatically as the wave propagates (an implicit hypothesis in the thin-disk approximation) and the disk cannot move up and down solidly to follow the warp perturbation. There results, together with the main vertical motion, a strong horizontal one which significantly modifies the dispersion relation. We then turn to the case of a disk composed of two fluids with different temperatures: this can correspond either to the combined motions of the gas and the stars in a galactic disk, or to their coupling with a flattened massive halo (assuming that, as for spiral waves, stars are conveniently represented by a fluid if one stays away from Lindblad resonances). We find, in addition to the usual warps, a short-wavelength wave which might explain the ``corrugations'' observed in many galactic disks. Finally, as a side result of this analysis, we discuss a possible weak amplification of m>1 warps.