A self-consistent semi-analytical theory of beam loading in inhomogeneous accelerating structures based on the generalized theory of coupled modes is proposed. A single-mode approximation was used when the fields are represented as a sum of two components, one of which is associated with the right travelling eigen wave, and the second with the left. However, this second component is not always a left travelling. When a field is excited by an electron beam it can have complex spatial distribution. The results of calculation of the distribution of electric fields excited by a relativistic electron beam are presented.
M. I. Ayzatsky (2025) studied this question.