The excitation of density waves at the Lindblad and corotation resonances by an external potential

We calculate the linear response of a differentially rotating two-dimensional gas disk to a rigidly rotating external potential. The main assumptions are that the sound speed is much smaller than the orbital velocity and that the external potential varies on the scale of the disk
\nradius. We investigate disks both with and without self-gravity. The external potential exerts torques on the disk only at the Lindblad and corotation resonances. The torque is positive at the outer Lindblad resonance and negative at the inner Lindblad resonance; at corotation the torque has the sign of the radial gradient of vorticity
\nper unit surface density. The torques are of the same order of magnitude at both types of resonance and are independent of the sound speed in the disk. The external potential also excites density waves in the vicinity of the Lindblad and corotation resonances. The long trailing wave is excited at a Lindblad resonance. It transports away from the resonance all of the angular momentum which is deposited there by the external torque. Short trailing waves are excited at the corotation resonance. The amplitudes of the excited waves are the same on both sides of the resonance and are small unless the disk is almost gravitationally unstable. No net angular momentum is transported away from the corotation
\nregion by the waves. Thus the angular momentum deposited there by the external torque accumulates in the gas.
\nWe briefly discuss the behavior of particle disks and prove that the external torques on particle disks are identical to those on gas disks.