  We propose a model for radio supernovae (RSN) based on the synchrotron
emission from relativistic electrons which are diffusively accelerated at the
expanding supernova shock. This model was originally developed for application
to the optically thin emission observed from SN1987A. Here we generalise it by
including the effects of free-free absorption from both an external screen and
from material internal to the source, and by relaxing the restriction to an
azimuthal B-field. We find a good fit to the entire set of radio data for the
best observed highly-luminous RSN -- SN1986J -- with a reduced \chi^2 of 3.85.
Applying the new model to SN1988Z, another intrinsically bright RSN, also
yields a good fit (\chi^2_{\rm red}\approx 2) but this is less significant,
because of the limited data on this distant (z=0.02) source. These fits suggest
that the shock expands at constant speed, that the magnetic field within the
source decreases with time according to t^{-2}, and that the compression ratio
of the shock front is close to the value expected of a strong shock in an ideal
gas of adiabatic index 5/3 -- indicating a relatively low value of the cosmic
ray pressure compared with SN1987A. In the case of SN1986J we derive an
explosion date in August/September 1982, a magnetic field at the position of
the shock 1000 days after explosion of 4nT and a spatial diffusion coefficient
of the electrons of 4 10^{19} cm^2/s, four orders of magnitude greater than the
Bohm value. In addition, we obtain the optical depths to external and internal
absorption, and derive an estimate of the mass-loss rate.
