The stability of a waveguide enclosed relativistic electron beam in a rippled magnetic field has been theoretically investigated. The presence of the ripple can couple negative energy beam modes to positive energy waveguide modes to produce instability and thereby generate microwaves. In the experiment, the thickness of the beam is small compared with all other relevant lengths. Therefore, the nonlinear dynamics of an infinitesimally thin sheet beam are analyzed. The system is described by the orbit equations for the particles, Maxwell's equations, a set of nonlinear boundary conditions across the moving surface, and a continuity equation for the moving surface.