Resonant stellar orbits in spiral galaxies. IV. Hydrodynamic flows in the regions of the lindblad resonances

A theory of the steady hydrodynamic flow of the interstellar gas in the regions of the Lindblad resonances of a spiral galaxy is developed. The Lagrangian equation of motion is solved as an orbital problem with the aid of an epicyclic theory of resonant stellar orbits, and the equation of continuity is solved in terms of a stream function. Applications are made to (1) the linear theory of density waves, (2) nonlinearities in the flow which are intrinsic to the resonance phenomenon, (3) the relationship between the streamlines of the flow and the corresponding periodic stellar orbits, and (4) the conditions for the occurrence of shocks in the flow