We consider the propagation of synchrotron-emitting electrons near the shocks of supernova remnants (SNRs). Using high-resolution radio observations of four Galactic supernova remnants, we set upper limits on the scattering mean free paths of these electrons in front of the supernova shock fronts. It is shown that for the sharpest rims, this mean free path is typically less than one percent of that derived for cosmic rays of similar rigidity in the interstellar medium. This result is interpreted in terms of an enhanced hydromagnetic wave intensity generated by diffusive shock acceleration. The dimensionless intensities of the scattering waves near supernova remnants are shown to be Tau(sub SNR)(5 GV/c) greater than or equal to 6 x 10(exp -4), in contrast to typical interstellar values Tau(sub ISM)(5 GV/c) less than or equal to 10(exp -5), where the intensity Tau = (delta B/B)(exp 2), where delta B is the amplitude of the waves with wavelengths resonant with particles of the specified rigidities. For SNR parameters, these wavelengths are of order 10(exp 12) - 10(exp 13) cm. Higher resolution radio and X-ray observations should provide useful diagnostics of the details of particle acceleration in these environments.