Coherent betatron oscillation modes due to beam-beam interaction

A simple consideration is given of the eigenmodes of the coherent betatron oscillation excited by a series of sinusoidal kicks and influenced by the beam-beam interaction in e+e- colliding storage rings. The differences between tunes corresponding to the eigenmodes are just half of what is expected from rigid point like bunches. The discussion is mainly limited to the case where the beam-beam force is not so strong so that the bunches remain Gaussian and the linear approximation of the force is valid. The nonlinear effect is also discussed briefly. (orig.)