  Solar flare accelerated electron beams propagating away from the Sun can
interact with the turbulent interplanetary media, producing plasma waves and
type III radio emission. These electron beams are detected near the Earth with
a double power-law energy spectrum. We simulate electron beam propagation from
the Sun to the Earth in the weak turbulent regime taking into account the
self-consistent generation of plasma waves and subsequent wave interaction with
density fluctuations from low frequency MHD turbulence. The rate at which
plasma waves are induced by an unstable electron beam is reduced by background
density fluctuations, most acutely when fluctuations have large amplitudes or
small wavelengths. This suppression of plasma waves alters the wave
distribution which changes the electron beam transport. Assuming a 5/3
Kolmogorov-type power density spectrum of fluctuations often observed near the
Earth, we investigate the corresponding energy spectrum of the electron beam
after it has propagated 1 AU. We find a direct correlation between the spectrum
of the double power-law below the break energy and the turbulent intensity of
the background plasma. For an initial spectral index of 3.5, we find a range of
spectra below the break energy between 1.6-2.1, with higher levels of
turbulence corresponding to higher spectral indices.
