waves. The size of the clumps is determined by the size of the regions with negative density gradient. The electron distribution function of beam electrons remains smooth as in the previous case with sine-like density oscillations.

![The spectral energy density of Langmuir waves $W(v,x,t)$, the energy density of plasma waves $E(x)$ at $t=5.0$s, and the local plasma frequency $f_p(x)$ (\[random\]) as functions of distance. Numerical solution of kinetic equations with random density fluctuations (\[random\]) $n'=100$cm$^{-3}$, $v_0=1.0\times 10^{10}$cm s$^{-1}$.[]{data-label="fig7"}](ms1121f7a.eps "fig:"){width="90mm"} ![The spectral energy density of Langmuir waves $W(v,x,t)$, the energy density of plasma waves $E(x)$ at $t=5.0$s, and the local plasma frequency $f_p(x)$ (\[random\]) as functions of distance. Numerical solution of kinetic equations with random density fluctuations (\[random\]) $n'=100$cm$^{-3}$, $v_0=1.0\times 10^{10}$cm s$^{-1}$.[]{data-label="fig7"}](ms1121f7b.eps "fig:"){width="90mm"}

The dependence of wave energy density on the amplitude of the density fluctuations is of special interest. From equation (\[eqk2\]) it follows that a Langmuir wave propagating with the group velocity $v_{gr}=3v_{Te}^2/v$ over the distance $\Delta l$ experiences a shift of phase velocity $$\label{delta_v}
  \Delta v \approx \frac{v^2}{Lv_{gr}}\Delta l.$$ Using the density profile (\[sin\]) and estimating $L\approx
\Delta l/\alpha$ one derives that $$\label{delta_v2}
  \frac{\Delta v}{v} \approx \frac{1}{3}\left(\frac{v}{v_{Te}}\right)^2\alpha$$ where we obtain, for our parameters, a phase velocity shift $\leq
