A new formulation of longitudinal coherent instabilities

The quadrature response of longitudinal impedance is shown to be the effective impedance for the beam instability. The results of the application of this formulation are compared with that obtained using the Robinson-Pedersen approach and the Sacherer integral equation. The formulation is further generalized to the rigid bunch motion using signal analysis method, where a form factor shows up naturally. Finally, the formulation is applied to solve the coupled bunch instabilities. Examples of the AGS Booster and the AGS coupled bunch instabilities are used to illustrate the applications of the formulation.<
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