stability ([@Cairns95]). Therefore, we consider the intermediate case of a medium density beam, which is not strong enough to start strong turbulence processes, $$W/nT \ll (k\lambda _{D})^2,
 \label{weakt}$$ but is dense enough to make quasilinear relaxation a dominant process. Here $W$ is the energy density of Langmuir waves generated by the beam, $T$ is the temperature of the surrounding plasma, $k$ is the wave number, and $\lambda _{D}$ is the electron Debye length.

The initial value problem is solved with an initially-unstable electron distribution function, which leads to the formation of a BPS in the case of homogeneous plasma ([@Melnik00a]) $$f(v,x,t=0)=g_0(v)\mbox{exp}(-x^2/d^2),  \label{f_0}$$ where $$g_0(v)=\left\{ \begin{array}{ll} \displaystyle \frac{2n'v}{v_0^2},
&\mbox{$v<v_0$},\\ 0, &\mbox{$v>v_0$}.
\end{array}
\right.
\label{g_0}$$ Here $d$ is the characteristic size of the electron cloud and $v_0$ is the velocity of the electron beam. The initial spectral energy density of Langmuir waves $$W(v,x,t=0)\simeq \frac{T}{2\pi ^2\lambda _{D}^2}, \label{W_0}$$ is of the thermal level and uniformly distributed in space. The electron temperature of the corona is taken to be $T=10^{6}$K, which gives an electron thermal velocity $v_{Te}=\sqrt{3kT/m}\simeq 6.7\times 10^{8}$cm s$^{-1}$.

Ambient density fluctuations
----------------------------

Following common practice in the literature on plasma inhomogeneity Langmuir waves are treated in the approximation of geometrical optics (the 