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The Buneman and ion acoustic instabilities are usually associated with different electron and ion drift velocities, in such a way that there is a large current through the plasma. However, due to the recently discovered current-driven Langmuir oscillations (1-3), the relative drift velocity in these configurations will oscillate at the plasma frequency, and with an amplitude of at least the initial drift velocity. In contrast, the textbooks assume a constant drift velocity. Since the growth rates arrived at under that assumption are far less than the plasma frequency, several oscillation periods will take place during the linear growth phase, and this will dampen the instabilities. We provide general theoretical derivations of these oscillations, and show simulation results of the altered behavior of the instabilities. Towards the end, we hypothesize that drift-averaging might be a viable method of calculating the modified growth rates.