Calculation of the Electromagnetic Field, Frequency and Circuit Parameters of High-Frequency Resonator Cavities

A method for the calculation of the field components of free electromagnetic oscillations in metal cavities of any shape is described with reference to Klystron resonators. Whereas in a previous paper a method for computing solutions of Laplace´s equations was given, the present paper is concerned with the wave equation ¿¿ + (¿2/c2)¿=0. Again, the differential equation is replaced by a system of difference equations, which in the case of free vibrations are homogeneous and soluble only for certain values of the parameter ¿2/c2, the proper values. A method for finding the lowest value of ¿2/c2 without solving a determinantal equation is described. The boundary of Klystron resonators contains sharp corners, often feather edges, which present a special problem. In order to obtain the necessary accuracy for the circuit parameters, with a reasonable amount of computation work, it is essential to take account of the analytic behaviour of the fields near sharp corners. This is done in a manner which is particularly well suited to the relaxation method of solution of the equations, and a great deal of computation work is thereby saved. Once the field components and the resonant frequency are found, the beam impedance and the damping constant are easily determined.