Analytical formulae for the wakefields produced by the nonrelativistic charged particles in periodic disk-loaded structures

In this paper we consider the wakefields induced in periodic disk-loaded cavities by charged particles of velocities less than that of light. In frequency domain the particle velocity-dependent wakefields can be calculated by generalizing the analytical formulae given in J. Gao (Nucl. Instr. and Meth. A 381 (1996) 174; or LAL/RT 95-01, April 1995). The physical picture of this effect can be drawn so that the frequencies of the excited modes in cavities felt by the particles are increased by a factor of 1/β (β is the normalized particleʹs velocity). Some examples are given to demonstrate the utility of these formulae in obtaining the quantities, such as loss factors, short-range and long-range wakefields as functions of cavity dimension, bunch length and particle velocity.