The effects of truncation on surface-wave Hamiltonians

The Hamiltonian form of the equations for surface waves can generate very nonlinear, realistic-looking solutions even when the Hamiltonian is truncated to low order – two or three terms – in its slope expansion. A perturbation analysis of these equations shows that most of the basic fluid behaviour is retained in the low-order terms; however, the lowest-order nonlinear equations become dramatically unstable at wavenumbers greater than g/w2, where w is the local vertical surface velocity. One more term in the Hamiltonian mitigates this instability, extending the regime of stable slopes and wavenumbers.