Transverse Coherent Instabilities in the Presence of Linear Coupling

Strong coupling between the transverse planes of a particle beam leads to an "equipartition" of the oscillation energy, including the growth rates in the case of coherent instability. The aim of this paper is to give a general formula, which includes linear coupling and which extends to two dimensions the one-dimensional results of transverse coherent instabilities. From this equation, previous results are recovered as expected: (i) Sacherer's formula for the coherent modes of oscillation, (ii) Kohaupt's formula for the mode coupling instability, and (iii) the coupled Landau damping mechanism (transfer of frequency spread), which includes the sharing of the instability growth rates. Measurements have been performed in the CERN PS, which confirm the predicted beneficial effect of coupling by both frequency spread and chromaticity sharing.