A simple, flexible technique for RF cavity wake-field calculations
It is typical in the accelerator field to model machine components, especially RF cavities, as parallel RLC resonators. In the interest of simulating and diagnosing beam instabilities, knowledge of the time-domain voltage waveform over an equivalent resonator by a bunch current often proves useful. This waveform may be found by convolving the bunch current with the RLC resonator impulse response. While analytical and quasi-analytical expressions are available in this regime for common distributions such as the Gaussian, analogous results for less standard distributions are difficult to obtain using direct methods, which opens the door for the development of a more generalized technique. In this paper, a formulation is created that allows for the simple computation of the time-domain voltage waveform of and RLC resonator. The formulation uses the Cauchy Residue Theorem to extract the convolution result from the Fourier Domain, and it only requires that the current distribution Fourier Transform be holomorphic and known at one specific evaluation point. This greatly simplifies the computation of the time domain voltage for a large amount of bunch distributions both common and uncommon.