  The Cauchy problem is considered for the scalar wave equation in the Kerr
geometry. We prove that by choosing a suitable wave packet as initial data, one
can extract energy from the black hole, thereby putting supperradiance, the
wave analogue of the Penrose process, into a rigorous mathematical framework.
We quantify the maximal energy gain. We also compute the infinitesimal change
of mass and angular momentum of the black hole, in agreement with
Christodoulou's result for the Penrose process. The main mathematical tool is
our previously derived integral representation of the wave propagator.
