  A novel coupling of the transverse betatron motion to the longitudinal
microwave instability is studied. Besides the radial coherent dipole mode space
charge field, simulation and theoretical studies in this paper show that the
longitudinal coherent dipole mode space charge field due to centroid wiggles
also plays an important role in the isochronous regime, it induces betatron
oscillation frequencies in temporal evolutions of spectra of longitudinal
charge densities, radial centroid offsets and coherent energy deviations of
local centroids.
