The excitation and evolution of density waves

Abstract

We study the linear oscillations of a thin self-gravitating gas sheet. The unperturbed velocity field of the sheet is a parallel shear flow. A Coriolis acceleration is included to simulate the effects of rotation. The sheet exhibits Lindblad resonances, and it can sustain both short and long wavelength density waves. We derive equations which govern the excitation and evolution of density waves in all regions of space, including the Lindblad resonances and the forbidden region around corotation. These equations are solved in the tight winding limit. An initial disturbance in the form of a wave packet of short leading waves evolves as follows. The packet propagates toward corotation, is reflected at the boundary of the forbidden region, and becomes a packet of long leading waves. It then travels back to the Lindblad resonance, where it is reflected and becomes a packet of long trailing waves. Subsequently, this packet moves toward corotation and is reflected again at the boundary of the forbidden region. The packet is now made up of short trailing waves and propagates away from corotation indefinitely. For sufficiently stable disks, the forbidden region around corotation is wide and density waves are almost completely reflected at its boundaries. For marginally stable disks, some of the incident wave tunnels through the forbidden region and the reflected wave is amplified. The excitation of density waves by an arbitrary external potential is considered. In our model sheet, the sole effect of a barlike potential is to excite the long trailing wave at the Lindblad resonances. The amplitude of the excited wave is calculated.