Propagation and evolution of waves in a self-gravitating medium

The propagation and evolution of linear waves in a self-gravitating medium are discussed in a comparatively general way. In one special case, the well-known Jeans formula is recovered; in another, an important property of unstable waves in a self-gravitating medium is obtained, namely, the equiphase surfaces of the wave are “frozen” in the moving medium. When the motion of the basic state is one of shear flow (or differential rotation), this property implies that unstable waves would evolve into a quasi-stationary state. This shows that, for a self-gravitating medium, besides the usual non-linear effects, there is a special mechanism within the linear theory which prevents the infinite growth of unstable waves. Possible relation of this effect to the origin of celestial bodies is pointed out.