Langmuir wave energy density in the plasma frequency – time plane. The electron beam initial parameters are as in Figs. 3, 4 except that the background plasma includes density fluctuations. Again, the coefficient that models the magnetic flux tube expansion is β = 3.5 (top left), β = 3.0 (top right), β = 2.5 (bottom left), β = 2.0 (bottom right). Note the different values of the frequency and time axis between each graph.