On the electron flow movement in the decelerated wave field with exponentially positive-going amplitude

Initially homogeneous electron beam movement in longitudinal electric field of the decelerated electromagnetic wave is discussed. The wave amplitude exponentially increases along the electron’s trajectory. Comparison of solutions obtained numerically and analytically by disturbance method for small amplitude as a sixth approximation is made. It is shown that the approximate solution’s error is small when transit angle is small and the error starts catastrophically rising at some angle’s value. The problem is urgent for building the non-linear clinotron’s theory.