concentrated in this region and the maximum of Langmuir wave density is located at the maximum of electron density $x=27d$. The spectrum of Langmuir waves has a maximum close to $v\approx
0.8v_0$. The spatial profile, averaged over the plasma inhomogeneity period, is close to the result obtained for a homogeneous plasma.

![The electron distribution function $f(v,x,t)$ and the spectral energy density of Langmuir waves $W(v,x,t)$ at $t=5.0$s. Numerical solution of the kinetic equations with sine-like density fluctuations (\[sin\]) $n'=100$cm$^{-3}$, $v_0=1.0\times 10^{10}$cm s$^{-1}$.[]{data-label="fig4"}](ms1121f4a.eps "fig:"){width="90mm"} ![The electron distribution function $f(v,x,t)$ and the spectral energy density of Langmuir waves $W(v,x,t)$ at $t=5.0$s. Numerical solution of the kinetic equations with sine-like density fluctuations (\[sin\]) $n'=100$cm$^{-3}$, $v_0=1.0\times 10^{10}$cm s$^{-1}$.[]{data-label="fig4"}](ms1121f4b.eps "fig:"){width="90mm"}

However, the spatial profile of Langmuir waves has a fine structure that can be seen in fig. \[fig5\]. The Langmuir waves are grouped into clumps (the regions with high level of plasma waves, following the terminology of ([@Smith79])). The size of a clump is determined by the spatial size of the density fluctuations and is equal to half of the density fluctuation period $\pi d/12\approx 0.26 d$. The maxima of Langmuir wave density are located in regions of negative plasma-density gradient and the regions with 