Beam induced RF cavity transient voltage

We calculate the transient voltage induced in a radio frequency (RF) cavity by the injection of a relativistic bunched beam into a circular accelerator. A simplified model of the beam induced voltage, using a single tone current signal, is generated and compared with the voltage induced by a more realistic model of a point-like bunched beam. The high Q limit of the bunched beam model is shown to be related simply to the simplified model. Both models are shown to induce voltages at the resonant frequency ωr of the cavity and at an integer multiple of the bunch revolution frequency (i.e. the accelerating frequency for powered cavity operation) hω0. The presence of two nearby frequencies in the cavity leads to a modulation of the carrier wave exp(jhω0t). A special emphasis is placed in this paper on studying the modulation function. These models prove useful for computing the transient voltage induced in superconducting RF cavities, which was the motivation behind this research. The modulation of the transient cavity voltage discussed in this paper is the physical basis of the recently observed and explained new kind of longitudinal rigid dipole mode which differs from the conventional Robinson mode.