Beam pipe Chao (1993) Collective Instabilities in Wakefield Coupled Bunches Objective - OCS6 Damping Ring - Transverse Growth Rates Kai Hock Liverpool Accelerator Group Meeting, Cockcroft 14 February 2007

Uniform Resistive Wall Transverse Force No wakefield this side Chao (1993) Wake potential

v Equation of motion y s = c No wakefield Wakefield from bunch ahead n n+1 n+2

Damping Ring y0y0 y1y1 ymym y M-1 - y n = transverse displacement - periodic nature modes

Trial solution Modes Eigenvector / Mode Circulant Matrix (Gray 2006)

Characteristic Equation e.g. 2 bunches, Mode 0 Multiple solutions: If assume dominated by betatron oscillation … a = 1 b 1 = 0.1 b 2 … = 0 tau = 1 | Left hand side – right hand side |

Growth Rate … derive analytic expression for small wakefield Chao (1993) Mode y n (0) FFT

Simulation Method - Integrate over one time interval between slices - Repeat for next interval SHMKick

OCS6 Damping Ring

Mode Amplitudes High frequency oscillations? - Mode amplitude ~ exp( t/ ) - Growth rate 1/ ~ initial gradient

OCS6 Growth Rate Assume constant beta for analytic curve.

Problems not complete. May also be Non-exponential behaviour? not general. Could be Growth rate ? (Wright 1948)