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very simple longitudinal approximation which, in the classical case, has been found worthy over the full wave-vectors spectrum except ne ar the perpendicular direction [16, 17]. The dispersion equation for longitudinal waves read s [18], 

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where f ± is the distribution function of each b eam. With f ± ( v ) = δ ( v$_{x}$ ) δ ( v$_{y}$ ) δ ( v$_{z}$ ∓ v 0 ), the result reads ϵ (Ω) = 1 − F (Ω) = 0, where 

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in terms of the non-dimensional parameters, 

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where ω$_{p}$ = 4 πn 0 q $^{2}$/m is the electron plasm a frequency (note that the plasma frequency is here defined for a density n 0 different fro m the beam electronic densities n 0 / 2). The 

