On density waves in galaxies. I. Source terms and action conservation

The first in a series directed at mechanisms for the generation and maintenance of density waves in galaxies, this paper examines the problem of how source terms for wave action can actually be introduced into a modified action conservation principle. Generalized versions of the Lin-Shu dispersion relation and the Toomre-Shu conservation principle of wave action are derived which are valid even in the neighborhood of all the Lindblad resonances. Because of the finite radial epicyclic excursions of the resonant stars, each resonance influences a region on the order of one or two epicyclic distances. For our Galaxy, these are annular regions each several kiloparsecs in width. The proposed generalized dispersion relation gives finite wavenumbers throughout the resonance regions. Provided that there is not also a very rapid temporal decay in amplitude, both leading and trailing short waves are found to decay spatially in the direction of their respective group propagation. This decay process can be described physically in terms of the detailed balance of energy and angular momentum through a generalized conservation principle of wave action.