The excitation and evolution of density waves.

The behavior of density waves in a simple analog of a rotating disk is analyzed. The model considered is a thin gas sheet whose unperturbed velocity field is a parallel shear flow; the effects of rotation are simulated by introducing a Coriolis acceleration; the model exhibits Lindblad resonances and includes both long- and short-wavelength density waves among its linear perturbations. The linear perturbation equations are set up in a form appropriate to the model and rewritten in terms of the comoving coordinates of the unperturbed flow. The resulting equations are Fourier transformed and then solved in the tight-winding limit. The excitation of a wave packet by an external potential is calculated along with the packet's subsequent propagation, and analytical solutions are obtained which are valid everywhere, including the Lindblad and corotation resonances. It is found that a barlike potential excites long-wavelength trailing density waves at the Lindblad resonances. The amplification of a density wave incident on the forbidden zone surrounding corotation is also examined.