  We consider the evolution of white dwarfs with compact object companions
(specifically black holes with masses up to 10^6 solar masses, neutron stars,
and other white dwarfs). We suppose that the orbits are initially quite
elliptical and then shrink and circularise under the action of gravitational
radiation. During this evolution, the white dwarfs will pass through resonances
when harmonics of the orbital frequency match the stellar oscillation
eigenfrequencies. As a star passes through these resonances, the associated
modes will be excited and can be driven to amplitudes that are so large that
there is a back reaction on the orbit which, in turn, limits the growth of the
modes. A formalism is presented for describing this dynamical interaction for a
non-rotating star in the linear approximation when the orbit can be treated as
non-relativistic. A semi-analytical expression is found for computing the
resonant energy transfer as a function of stellar and orbital parameters for
the regime where back reaction may be neglected. This is used to calculate the
results of passage through a sequence of resonances for several hypothetical
systems. It is found that the amplitude of the l=m=2 f-mode can be driven into
the non-linear regime for appropriate initial conditions. We also discuss where
the no back reaction approximation is expected to fail, and the qualitative
effects of back reaction.
