0.1 v$. Expression (\[delta\_v2\]) also demonstrates that the shift of the wave phase velocity linearly depends on the amplitude of the density fluctuations. Therefore, in the case with an arbitrary amplitude of the density fluctuations, the higher the amplitude of the plasma inhomogeneity the larger the variations of the wave energy distribution. This tendency can be observed in fig. \[fig7\].

Main results and discussion
===========================

From a physical point of view it is interesting to consider the physical processes which lead to the reported results. As we see, the main physical effect, which leads to a complex spatial distribution of waves, is the shift of the phase velocity $\Delta
v$ due to the wave movement. The growth rate of beam-plasma instability $$\gamma (x)=\frac{\pi
\omega_{pe}}{n}v^2\frac{\partial f}{\partial v}, \label{increment}$$ also depends on distance. However, this dependency of the instability increment on local plasma density is negligible. At every spatial point we have a plateau with $\partial f/\partial
v\approx 0$ and the value of $\partial f/\partial v$ is determined by the dynamics of a BPS not by the local plasma density. Therefore, the shift in phase velocity dominates the effect of instability increment dependency on distance. Indeed, if we manually exclude the terms 