Details of the gravitational collapse of compressed gas behind an isothermal shock are presented. The growth rate of an unstable mode is given as a function of its wavelength for a single-parameter family of layers. Simple formulas for the maximum growth rates, for the wavelength at which this maximum occurs, and for the critically unstable wavelength are given as functions of the model parameter. The astrophysical implications of these results are discussed. The minimum unstable mass in the plane-parallel case was found to be stable, if it collapses to a pressure-bounded isothermal sphere. The mass of the most rapidly growing instability is also stable as a sphere, unless the column density in this layer is sufficiently large in proportion to the square root of the external sphere. A criterion is derived for this dominant, plane-parallel instability to lead to an unstable sphere. It is possible that this criterion marks the beginning of rapid star formation behind a shock.
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Elmegreen et al. (1978) studied this question.