Nonlocal Equations for the Electromagnetic Field in the Inhomogeneous Accelerating Structures

Abstract

The procedure for obtaining a difference equation, the solution of which is the components of the electric (or magnetic) field at the chosen set of points in the volume of the resonator chain, was developed. We started with the wave equation with boundary conditions and obtained the difference equation without boundary conditions. Boundary conditions were included into the coefficients of the difference equation for the electric field. As the obtained equation connects the field values in different points (in general case, on an infinite set of points), the proposed procedure represents a nonlocal model for field description. Solutions of the difference equation for the electric field were analyzed. It was shown that they coincide with good accuracy with the ones that were obtained by direct summing of relevant series.

Modification of the Coupled Integral Equations Method for Calculation of the Accelerating Structure Characteristics

“…where the operator det  is defined on the base of rules of common determinants that equation (34) does not have spurious solutions as it was for the equation based on a coupled cavities model [10].…”