A NOTE ON HAMILTONIAN FOR LONG WATER-WAVES IN VARYING DEPTH

The Hamiltonian for two-dimensional long waves over a slowly varying d epth is derived. The vertical variation of the velocity field is obtai ned by using a perturbation method in terms of velocity potential. Emp loying the canonical theorem, the conventional Boussinesq equations ar e recovered. The Hamiltonian becomes negative when the wavelength beco mes short. A modified Hamiltonian is constructed so that it remains po sitive and finite for short waves. The corresponding Boussinesq-type e quations are then given.