Effect of bunch shape on its TMCI spectrum and threshold with high space charge.

Transverse mode coupling instability of bunched beam is investigated in the paper at different form of the bunches with space charge included. Equation of transverse motion of the bunch in parabolic potential well of synchrotron oscillations is derived and analysed. The bunch of constant density (flat bunch) is examined in detail to make comparison with the square well model. It is shown that both models result in very close instability thresholds of the flat bunch. Then different form bunches are investigated in the parabolic potential well. It is shown that decrease of the bunch r.m.s length leads to the growth of its threshold, that is the flat bunch model gives only a minimal estimation of the threshold. The results are treated in terms of Landau damping due to spread of the space charge tune shift.