Unsteady undular bores in fully nonlinear shallow-water theory

We consider unsteady undular bores in the fully nonlinear dissipationless shallow- water dynamics described by the Green-Naghdi system. We use the Whitham modula- tion theory to obtain an asymptotic analytical description for a full range of the depth ratio across the bore. The positions of the edges of the undular bore and the amplitude of the leading solitary wave are found as functions of this depth ratio. The formation of a partial undular bore with a rapidly-varying finite-amplitude rear wave front is predicted for depth ratios across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with the full numerical solution for the development of an undular bore in the Green-Naghdi system.