Closed Analytical Expression for the Electric Field Profile in a Loaded RF Structure with Arbitrarily Varying $v_{g}$ and $R'/Q$

The design of a detuned and damped accelerating structure implies variations in the geometry which induce in turn a variation of the group velocity v^g and of the impedance per unit length R', divided by the quality factor Q. The resulting differential equation for the longitudinal electric field (fundamental mode) contains coefficients that depend on the distance z along the structure. This report describes a possible method to solve this nonlinear, first order differential equation analytically and how to obtain approximate closed algebraic forms, by using the sequence of Gauss integration methods. Analytical expressions of the longitudinal field profile in a loaded or unloaded accelerating section is deduced for both linear and arbitrary variations of v^g and R'/Q = Q. Simple relations between the average field < E > and the field at the entrance of the structure E(0) make it possible to provide the dependence of the field function E(z) on the design value for < E >? and on the structure parameters. The results are in good agreement with the direct numerical integration. Applications are presented for particular structure designs.