Tune and Stability of High Intensity Bunch Trains in the CERN SPS and LHC

The complex resistive wall impedance generates a non-uniform transverse focussing along a bunch train. This was first computed by V. Balbekov [1] for the incoherent effect for the UNK machine. Similarly there exists a coherent effect for non centered beams. Its origin is discussed and then quantified for bunch trains in the CERN SPS and future LHC. Then the transverse resistive wall instability of bunch trains is examined and compared with the well known case of the uniformly filled machine. It is shown that the frequency of the most unstable mode is centred around the inverse of the length of the bunch train with a growth rate that is larger than the one expected from linear extrapolation of a full machine