Evolution of Disturbances in Isothermal Self‐gravitating Fields

Abstract

We study the evolution of waves generated by compact sources of turbulence in self-gravitating fields under isothermal conditions. We apply the theory of aerodynamically generated sound waves, originally developed by Lighthill, and solve the linearized wave equation by Green's method. We adopt a uniform and stationary background as the initial condition. We consider two cases for the time development of an added disturbance. In the first, the disturbance is imposed on a compact region in a pulselike form, and the density grows exponentially around the source, producing a density concentration that has a Gaussian density profile centered at the source with increasing amplitude and width. This is understood in terms of the development of Jeans-unstable modes generated by the disturbance. In contrast, if the source of disturbance is oscillating with a frequency of ω0, we first observe oscillations and then an exponential growth of the density concentration. This can be interpreted as follows: Oscillating and growing modes correspond to different Fourier components. At first the amplitude of the former modes is largest, thus producing oscillatory wave patterns in density distribution. After the sound crossing time over the Jeans length, however, the latter dominate over the former in amplitude because of the development of Jeans instability. In conclusion, any compact disturbance in self-gravitating media can generate Jeans-unstable waves, regardless of whether the disturbance is instantaneous or oscillating.