Functional dependence of wakefunctions for v<c

In evaluating the wakefield effects of medium energy particles interacting with a cavity, the integral for the wakefunction must be carried out on the pipe radius in order to avoid an infinity due to the finite space charge effect. To obtain the wakefunctions at other radial positions, a proper extrapolation algorithm is needed. This paper presents the extrapolation method for calculating wakefunctions of v<c. For cases with low energies, the slippage between the particle and the fields induces longitudinal smearing of the wakefunctions, and the wakefunctions inside the beam pipe are found to be weighted averages of the wakefunctions calculated on the pipe radius. The smearing effect for calculating the wakefunctions on the axis is related to R=σ _zγ/α, with the smearing effect negligible for large R. The usual ultrarelativistic assumption is found to be reasonable for R⩾1.5. For cases with R⩽1.5, a weighted average must be taken to calculate the wakes inside the beam pipe