  We use time-dependent, axisymmetric, hydrodynamic simulations to study the
linear stability of the stalled, spherical accretion shock that arises in the
post-bounce phase of core-collapse supernovae. We show that this accretion
shock is stable to radial modes, with decay rates and oscillation frequencies
in close agreement with the linear stability analysis of Houck and Chevalier.
For non-spherical perturbations we find that the l=1 mode is always unstable
for parameters appropriate to core-collapse supernovae. We also find that the
l=2 mode is unstable, but typically has a growth rate smaller than that for
l=1. Furthermore, the l=1 mode is the only mode found to transition into a
nonlinear stage in our simulations. This result provides a possible explanation
for the dominance of an l=1 'sloshing' mode seen in many two-dimensional
simulations of core-collapse supernovae.
