One-dimensional periodic flows with a shock transition - Application to the density wave theory of spiral structure

The problem of one-dimensional gas flow through a sinusoidal gravitational potential with a series of equally spaced shock fronts is considered. For time-independent and spatially periodic flows, an integral equation relates the flow velocity to the gravitational potential and source functions of energy and momentum densities. It is suggested that this problem simulates some of the dynamical effects of the azimuthal flow on a parcel of gas at a fixed radius in a galactic disk that supports a spiral density wave structure. It omits the radial motions. In this context, a typical azimuthal velocity across a spiral arm is specified and the resultant source functions are obtained. The implications of this calculation suggest heating of the gaseous disk by O and B stars behind the shock front, and subsequent interarm cooling is a plausible mechanism for maintaining the periodic cycling of the gaseous component of the galactic disk.