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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

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

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

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

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

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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 |

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

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

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OCS6 Damping Ring

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

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OCS6 Growth Rate Assume constant beta for analytic curve.

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

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