Beam loading voltage profile of an accelerating section with a linearly varying group velocity

The CLIC Tapered Damped accelerating Structure (TDS) has a 5.4% detuning of the lowest dipole mode. The geometrical variations that produce this detuning range also fix the fundamental mode's group velocity variation - very nearly linear with 0.108c (c is the speed of light) at the structure input to 0.054c at the output. In addition R'/Q also varies approximately linearly, from 22.3 kW/m at the input to 30 kW/m at the output. These variations result in a structure that is neither constant impedance nor constant gradient so the widely used relationships between structure length, input and average accelerating gradient are not applicable. In order to simplify the process of optimizing accelerator parameters an analytic expression for the voltage profile in a structure with a linearly varying group velocity has been derived. A more accurate numerical solution that includes the variation in R'/Q is also presented.