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Hahn-Ramo theory is used to derive a characteristic wave equation for an electron stream whose single-valued velocity is a function of spatial co-ordinates. A means of solving this equation is found, for the particular case of two-dimensional Cartesian coordinates. A specific, but practical, linear velocity distribution is assumed. It is shown that for several types of boundary conditions, the only waves which can be set up in such a beam are purely propagational and not growing, bearing out the result derived by an approximate method by G. Kent. Numerical analysis for a cylindrical beam with potential depression was performed by a digital computer. As before, the results showed absence of any growing waves. In order to check early results of Haeff, which appeared to show the possibility of gain in single beam devices, an experiment was set up whereby a movable pickup cavity measured the amplitude of space-charge waves at a number of points along a drift tube. Outputs were compared at different drift lengths for pulsed and continuous operation. No evidence of growing waves was observed, verifying the analytical results. It was found, however, that operation of the collector electrode at very low potentials created secondary electrons which returned to the gun region, were reflected, and then they flowed back with the primary beam. This double-stream action produced electronic gains up to 30 db.