Interaction between high intensity beam and structure resonances

With the help of the linearized perturbation theory, the collective beam instability due to the space charge, modelled by the Vlasov-Poisson equation, is well studied in the former research theoretically in a general sense [Chao Li and R. A. Jameson PRAB, 21, 024204 (2018)], where the related instability is depicted by the "structure resonance". As an extension of the theoretical study, this paper aims to verify the validity of the structure resonance with numerical simulations and to show the basic principles which interpret the interaction between structure resonance and beam in a general sense. It is found that the reaction of beam to the structure resonance ($n\sigma\approx 180^\circ$) differs for the cases with $n= 0$ and $n\ne 0$ stop bands. The non-resonant "plasma oscillation" is also revisited; the "phase space mixing", lying in the transient plasma oscillation, is the mechanism that ties the resonant effects, non-resonant effects and macroscopic beam phenomena together.