  We present solutions for the self-similar propagation of a shock wave in a
hydrostatic, adiabatic medium with a point mass gravitational field. In
contrast to the well-known, Sedov-Taylor blastwave, these solutions apply to
the case when the shock Mach number is of order a few, and the energy of the
shocked fluid is not conserved but self-consistently modified by the binding
energy of the ambient medium that is swept up by the passage of the shock.
Furthermore, we show that there is one solution (for a given ambient density
profile) that smoothly passes through a sonic point in the post-shock flow and
results in accretion onto the central object; in analogy with the Bondi
problem, we propose that these solutions are the ones that are most relevant in
astrophysical environments. We apply these accreting models to failed
supernovae, in which neutron star formation does not unbind the envelope, but a
weak shock is still generated in the outer layers of the star from
neutrino-induced mass loss. We find excellent agreement between the predictions
of our self-similar, shock propagation model and numerical simulations of the
collapse of a yellow supergiant; the self-similar solutions reproduce the
overall scaling of the shock speed, the time and space-dependent evolution of
the velocity, density, and pressure behind the shock, and the accretion rate
onto the black hole. Our results have important implications for the fallback
and ejection of material in failed supernovae.
