Non-axisymmetric instability in thin discs

Theoretical evidence is presented for the inevitability of the appearance of instability in dynamically rotating thin disks. The disk is treated as a parallel shear flow of a thin, compressible, uniform density gas sheet with a constant velocity gradient. A parabolic-cylinder differential equation is used to express the perturbations, which include corotation and Lindblad resonances. Eigenmodes are found to arise in forbidden regions between the resonant turning points. A maximum growth rate is characterized for the eigenmodes (WKB modes). A reflecting boundary is determined necessary for formation of a self-sustained oscillation, and will appear if the density at the inner or outer edge of the disk cuts off on a scale shorter than the radial eigenmode wavelength.