for the initial distribution function (\[g\_0\]) $$\label{u_max}
  u(t\rightarrow \infty) =\sqrt{3/2}v_0$$

![The electron distribution function $f(v,t)$ and the spectral energy density of Langmuir waves $W(v,t)$ at various times for the case where the plasma density increases with distance, $L_0=5\times 10^9$cm. Numerical solution of kinetic equations (\[k1\],\[k2\]) $n'=100$cm$^{-3}$, $v_0=1.0\times 10^{10}$cm s$^{-1}$.[]{data-label="fig2"}](ms1121f2.eps){width="90mm"}

As predicted, the numerical solution tends to the maximum velocity $\approx 1.22v_0$ (fig. \[fig2\]). As in the previous case, at $t=0.1$s we have the result of quasilinear relaxation - a plateau in the electron distribution function and a high level of Langmuir waves. For times $t>0.1$s, the drift of Langmuir waves and consequent acceleration of electrons is observable. At $t=1$s almost all plasma waves are observed near the leading edge of the plateau and the maximum plateau velocity is close to the value given by (\[u\_max\]).

Propagation of an electron cloud
================================

In this section the numerical results of the evolution of the electron beam in the plasma with density fluctuations are presented. We begin with the case where the ambient density fluctuations in the plasma are periodic and sine-like. The dependency of plasma density on distance is $$\label{sin}
  n(x)= n_0(1+\alpha \mbox{sin}(x/\Delta x))$$ where $\Delta x$ defines the period of the 