We present results from numerical evolutions of single black holes distorte
d by axisymmetric, but equatorially asymmetric, gravitational (Brill) waves
. Net radiated energies, apparent horizon embeddings, and recoil velocities
are shown for a range of Brill wave parameters, including both even and od
d parity distortions of Schwarzschild black holes. We find that a wave pack
et initially concentrated on the black hole throat, a likely model also for
highly asymmetric stellar collapse and late stage binary mergers, can gene
rate a maximum recoil velocity of about 150 (23) km/sec for even (odd) pari
ty perturbations, significantly less than that required to eject black hole
s from galactic cores. [S0556-2821(99)10418-1].