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An elementary analysis of coupled-bunch instabilities

We reconsider the equations of motion of wake field coupled bunches in the light of recent developments in Delay Differential Equations. In the case of uniform resistive wall in a storage ring, we demonstrate an alternative way to characterize the growth modes. For each multibunch Fourier mode, an infinite number of time domain modes can arise from an exact solution of the equation of motion. The growth rate as it is commonly defined corresponds to only one of them. The amplitude of each Fourier mode can therefore evolve with time in a way that is not a simple exponential. This is a result that has been observed in simulations of wake field coupled bunches.