Transverse and longitudinal coupled bunch instabilities in trains of closely spaced bunches

It is noted that damping rings for the next generation of linear collider may need to contain several bunch trains within which the bunches are quite closely spaced (1 or 2 RF wavelengths). Methods are presented for studying the transverse and longitudinal coupled bunch instabilities applicable to this problem and to other cases in which the placement of the bunches is not necessarily symmetric. It is shown that, given the coherent frequencies and normal modes, the Laplace transform can be used to obtain the motion of the bunches, taking the initial conditions correctly into account. A computer tracking method has been used in the transverse case to obtain the offset of each bunch as a function of time. It is pointed out that at least one of these methods is always applicable, and in many cases both of them are