  The radio spectra of many shell-type supernova remnants show deviations from
those expected on theoretical grounds. In this paper we determine the effect of
stochastic reacceleration on the spectra of electrons in the GeV band and at
lower energies, and we investigate whether or not reacceleration can explain
the observed variation of radio spectral indices. We explicitely calculate the
momentum diffusion coefficient for 3 types of turbulence expected downstream of
the forward shock: fast-mode waves, small-scale non-resonant modes, and
large-scale modes arising from turbulent dynamo activity. Noting that
low-energy particles are efficiently coupled to the quasi-thermal plasma, a
simplified cosmic-ray transport equation can be formulated and is numerically
solved. Only fast-mode waves can provide momentum diffusion fast enough to
significantly modify the spectra of particles. Using a synchrotron emissivity
that accurately reflects a highly turbulent magnetic field, we calculate the
radio spectral index and find that soft spectra with index $\alpha\lesssim
-0.6$ can be maintained over more than 2 decades in radio frequency, even if
the electrons experience reacceleration for only one acceleration time. A
spectral hardening is possible but considerably more frequency-dependent. The
spectral modification imposed by stochastic reacceleration downstream of the
forward shock depends only weakly on the initial spectrum provided by, e.g.,
diffusive shock acceleration at the shock itself.
