1. ILC Damping Ring Wolski (2007) Dynamics with Transverse Coupled-Bunch Wake Fields in the ILC Damping Rings Kai Hock and Andy Wolski

2. Objective: 1. Effect of varying beta function in macroparticle model 2. Analytic solution for constant beta function. Previous work – for constant beta function : 1. Equation of motion decouple into Fourier modes (Chao 1993). 2. Modes decay / grown exponentially, analytic formula available. 3. Analytic formula for bunch trajectories available but incomplete (Thompson and Ruth 1991). Current work: 1.Varying beta function – modes remain coupled, decay modes grow. 2.Analytic formula for bunch trajectories completed with error bounds.

3. Example of an application

4. v A Macroparticle Model y No wake fieldResistive wake force n n+1 n+2 Equation of motion Sands (1969), Thompson and Ruth (1991), … History

5. Normal Modes where coupled decoupled (Fourier modes) DFT Chao (1993)

6. If beta function is constant OCS6 Damping Ring: Uniform Fill

7. A few surprises …

8. Mode Frequency Spectra Turn 90 Parametric force High frequency oscillation comes from …

9. Mode Coupling Decay modes grow because … Growth mode Decay mode Decay mode ?

10. Bunch Trajectories Equation of motion Analytic solution (Thompson, Ruth 1991) Incomplete – Zero wake field limit is SHM, need – Else cannot have arbitrary

11. Characteristic Equation Elementary solution Contour plot General solution If beta function is constant:

12. Integration of Depends on initial history

13. If wake field very small Conjecture Depends on initial condition at t = 0 Depends on initial history Just like SHM

14. An analytic solution Conjecture validated

15. H(  ) Rogers (2006)  Feedback system bandwidth Frequency  mode number Bandwidth must cover growth modes If decay modes grow unexpectedly …