Wave Excitation in Three-Dimensional Disks by External Potential
  We study the excitation of density and bending waves and the associated angular momentum transfer in gaseous disks with finite thickness by a rotating external potential. The disk is assumed to be isothermal in the vertical direction and has no self-gravity. The disk perturbations are decomposed into different modes, each characterized by the azimuthal index m and the vertical index n, which specifies the nodal number of the density perturbation along the disk normal direction. The n=0 modes correspond to the two-dimensional density waves previously studied by Goldreich & Tremaine and others. In a three-dimensional disk, waves can be excited at both Lindblad resonances (for modes with n=0,1,2...) and vertical resonances (for the $n\ge 1$ modes only). The torque on the disk is positive for waves excited at outer Linblad/vertical resonances and negative at inner Lindblad/vertical resonances. While the n=0 modes are evanescent around corotation, the $n\ge 1$ modes can propagate into the corotation region where they are damped and deposit their angular momenta. We have derived analytical expressions for the amplitudes of different wave modes excited at Lindblad and/or vertical resonances and the resulting torques on the disk. It is found that for $n\ge 1$, angular momentum transfer through vertical resonances is much more efficient than Lindblad resonances. This implies that in some situations (e.g., a circumstellar disk perturbed by a planet in an inclined orbit), vertical resonances may be an important channel of angular momentum transfer between the disk and the external potential. We have also derived new formulae for the angular momentum deposition at corotation and studied wave excitations at disk boundaries. 
Hang Zhang and Dong Lai