  We obtain self-similar solutions that describe the gravitational collapse of
nonrotating, isothermal, magnetic molecular cloud cores. We use simplifying
assumptions but explicitly include the induction equation, and the semianalytic
solutions we derive are the first to account for the effects of ambipolar
diffusion following the formation of a central point mass. Our results
demonstrate that, after the protostar first forms, ambipolar diffusion causes
the magnetic flux to decouple in a growing region around the center. The
decoupled field lines remain approximately stationary and drive a hydromagnetic
C-shock that moves outward at a fraction of the speed of sound (typically a few
tenths of a kilometer per second), reaching a distance of a few thousand AU at
the end of the main accretion phase for a solar-mass star. We also show that,
in the absence of field diffusivity, a contracting core will not give rise to a
shock if, as is likely to be the case, the inflow speed near the origin is
nonzero at the time of point-mass formation. Although the evolution of
realistic molecular cloud cores will not be exactly self-similar, our results
reproduce the main qualitative features found in detailed core-collapse
simulations (Ciolek & Konigl 1998)
