The discrepancy was resolved in 1994. The gravitational restoring force from a bend is substantially weaker in finite or inhomogeneous galaxies than in infinite sheets and slabs, since there is less matter at large distances to contribute to the restoring force. As a result, the long-wavelength modes are not stabilized by gravity, as implied by the dispersion relation derived above. In these more realistic models, a typical star feels a vertical forcing frequency from a long-wavelength bend that is roughly twice the frequency Ω Ω z of its unperturbed orbital motion along the long axis. Stability to global bending modes then requires that this forcing frequency be greater than Ω Ω z , the frequency of orbital motion parallel to the short axis. The resulting (approximate) condition