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Radiative shocks are known to be subject to an oscillatory overstability in one dimension. A linear stability analysis suggests that non-planar modes may also be unstable (Bertschinger, ApJ, 304, 154). We have explored the stability of initially planar radiative shocks using two-dimensional hydrodynamic simulations. Although the growth of some non-planar modes was seen in the radiative cooling region of the shock, the most noticeable consequence of the initial perturbations was the influence on the cold, dense gas layer down stream of the cooling region. Two sources of instability are identified in the interface between the hot, cooling gas and the cold gas layer: the influence of unstable oscillatory modes in the cooling region, and the Richtmeyer-Meshkov instability generated when the oscillations of the shock front drive shock waves through the cold gas layer. The resulting perturbations in the cold gas layer are dominated by spatial wavelengths, $\lambda \sim L_{c}$, the cooling length of the shock. The action of this instability insures that interstellar radiative shocks will not be smooth on length scales of order the local cooling length, unless some other factor (such as magnetic fields) acts to suppress this instability.
