In the context of magnetospheric and laboratory beam‐plasma dynamics, there appears to be a need to examine consistently electron beam density development when coupled to a passive resistive background medium. The problem motivating the present study of an electron beam passing through the ionosphere is one case in point; the localized density evacuations predicted may be the forerunner of a low‐frequency, diffusion‐dominated, excitatory mechanism in auroral activity. The contribution offered by this work lies in the technique used to follow the electron beam in space time within a Lagrangian framework via the method of characteristics, consistently feeding back the effect of the field coupling with the resistive medium. This consistency along with a simulation of the influence of a local periodic disturbance are achieved by adjusting the upstream conditions in the characteristics solution. Prediction concerning the nature of the instability from the dispersive equation are shown to agree with those predicted from this generalized technique.