An exact Hamiltonian coupled-mode system with application to extreme design waves over variable bathymetry

A novel, exact, Hamiltonian system of two nonlinear evolution equations, coupled with a time-independent system of horizontal differential equations providing the Dirichlet-to-Neumann operator for any bathymetry, is applied to the study of the evolution of wave trains in finite depth, aiming at the identification of nonlinear high waves in finite depth, and over a sloping bottom. The vertical structure of the wave field is exactly represented up to the instantaneous free surface, by means of an appropriately constructed, rapidly convergent, local vertical series expansion of the wave potential. This Hamiltonian system is used for studying the fully nonlinear refocusing of transient wave groups, obtained by linear backpropagation of high-amplitude wave trains constructed by the theory of quasi-determinism. The results presented give a first quantification of the effects of sloping bottom and of spectral bandwidth on rogue-wave dynamics and kinematics, in finite depth.