On the estimation of the wake potential for an ultrarelativistic charge in an accelerating structure

The method to derive the analytic estimations for wake fields of an ultrarelativistic charge in an accelerating structure, that are valid in the range of distances smalller or compared to the effective structure dimensions. The method is based on the approximate space-time domain integrating of the maxwell equations in the Kirchhoff formulation. the method is demonstrated on the examples of obtaining the wake potentials for energy loss of a bunch traversing a scraper, a cavity or periodic iris-loaded structure. Likewise formulae are derived for Green functions that describe transverse force action of wake fields. Simple formulae for the total energy loss evaluation of a bunch with the Gaussian charge density distribution are derived as well. The derived estimations are compared with the computer results and predictions of other models