  We analyze the interaction of a cold fast electron beam with a thermalized
plasma, in the presence of many Langmuir modes. The work aims at characterizing
the deviation of the system behavior from the single mode approximation, both
with respect to a consistent spectral analysis of the most unstable mode
harmonics and with respect to the presence of a dense spectrum, containing
linearly unstable and stable modes. We demonstrate how, on the one hand, the
total energy fraction absorbed by the harmonics is negligible at all (by
evaluating its total amount) and, on the other hand, the additional Langmuir
modes can be excited via an avalanche mechanism, responsible for a transport in
the particle velocity space. In particular, we show that the spectral
broadening outlines a universal shape and the distribution function, associated
to the avalanche mechanism, has an asymptotic plateau, differently from the
coherent structures characterizing the single wave model.
