Self-consistent solution for gaseous shock and stellar density wave

In this paper, the self-consistent density wave theory containing both a gaseous shock and a linear stellar density wave is studied, and a quasi-stable, tightly-wound, two-arm solution is obtained. The solution is convergent if the incomplete, linearized hydrodynamic equations are used, and the solution then gives the same dispersion relation as the local, asympotic solution, but the density and field profiles will be non-sinusoidal. The stellar wave will be unstable if the complete, linearized hydrodynamic equations are used.