Bending Waves in Flattened Stellar Systems a

Warping of disk galaxies is much more evident in the gas than in stars. The warp in the HI layer of the Milky Way was detected from 21-cm observations in the 1 9 5 0 ~ . ' ~ ~ Then, in the 1970s, it became apparent that warps are a common feature of the H1 layers of external g a l a x i e ~ . ~ . ~ Warps of H1 layers become strong only where they extend far beyond the optical image,5 and the optical evidence for the occurrence of warps in stellar disks is much less strong. There are clear-cut cases such as M31,6 but van der Kruit and S e a ~ l e ~ ~ detected only one very marginal warp in the seven edge-on galaxies that they observed. Although Sanchez-Saavedra et a/." claimed to detect warps in 42 of the 86 highly inclined spirals that they studied, they did classify 23 of the 42 as barely perceptible. Two of the mechanisms that have been proposed for explaining warps, an intergalactic wind" or an infall of matter," are likely to affect gaseous disks more strongly than stellar disks. Other proposed mechanisms apply equally well to stellar as to gaseous disks. It was clear to the first detectors of the warp in the Milky Way that this warp is much too large to be attributed to the tide currently being raised by any near neighbor. This explanation is similarly unsatisfactory for the warps in many external galaxies. Past close encounters could raise larger tides. Both this and other possibilities such as Lynden-Bell'~'~ proposal that the warp is a free mode of oscillation, require an understanding of dynamics of bent disks and the waves that they can support. It is this dynamics, rather than the explanation of galactic warps, that is the topic of this paper. Massive dark halos may very well play a major role in galactic