By Miles Hudson
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Additional resources for Edexcel A Level Science: A2 Physics Students' Book with ActiveBook CD
Both these cases can occur in the above discussed FEL model. For instance, the phase and group velocities of the signal wave in traditional FEL H-ubitron arrangements, as a rule, are coincided [2–4]. However, some other rather exotic physical situations could be realized in some Dopplertron type of FELs. Their characteristic design feature is the use intense electromagnetic waves (including microwaves) as FEL pumping . Therefore the situations with opposite directions of the phase and group velocities can occur really [16, 24, 25].
5) represents the wide class of mathematical objects which are called nonlinear equations. Generalizing, we can say that all equations which contain some nonlinear functions in their right part (in the case above we have the trigonometric sine), could be classified as nonlinear equations. Besides the differential equations the integral and mixed integral–differential nonlinear equations are known also. However, we will take an interest in the differential nonlinear equations only. All physical systems whose dynamics are described by nonlinear equations of any type are called nonlinear dynamical systems.
This characteristic process can be illustrated by nonlinear pendulum. 5). 9). In the second, third, and so on orders of approximation, oscillation harmonics appear in the solution (see further Chapter 5 for more details). The nonlinear pendulum represents the evident example of so called multi-harmonic periodic oscillations. Similarly, higher harmonics are generated really in FEL by pumping and signal oscillation electron phases, simultaneously (see Volume II, for instance). 20)) could be exited, too.