The mutual interaction between intense relativistic electron bunches and a cavity was investigated. It was found that this interaction generated an oscilating voltage in the cavity with an amplitude and a phase that depended on (1) the frequency of the modulated electron beam, (2) the characteristic frequency of the cavity, (3) the characteristic impedance of the cavity, and (4) peak beam current. This voltage caused energy exchange between electrons within a bunch resulting in modifications of the shape and peak current of the bunches. For a ‘‘critical’’ cavity geometry the voltage increased to a high amplitude, drastically reducing beam peak and average currents.