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.