On two approaches to the third-order solution of surface gravity waves

A third-order approximate solution for surface gravity waves in the finite water depth is studied in the context of potential flow theory. This solution corresponds to the steady-state part of a multidirectional irregular wave field and provides explicit expressions for the surface elevation, free-surface velocity potential, and velocity potential. The amplitude dispersion relation is also provided. Two approaches are used to derive the third-order analytical solution, resulting in two types of approximate solutions: the perturbation solution and the Hamiltonian solution. The perturbation solution is obtained by a classical perturbation technique in which the time variable is expanded in multiscale to eliminate secular terms. The Hamiltonian solution is derived from the canonical transformation in the Hamiltonian theory of water waves. By comparing the two types of solutions, it is found that they are completely equivalent for the first- and second-order solutions and the nonlinear dispersion, but for the third-order part only the sum–sum terms are the same. Due to the canonical transformation that could completely separate the dynamic and bound harmonics, the Hamiltonian solutions break through the difficulty that the perturbation theory breaks down due to singularities in the transfer functions when the quartet resonance criterion is satisfied. Furthermore, it is also found that some time-averaged quantities based on the Hamiltonian solution, such as mean potential energy and mean kinetic energy, are equal to those in the initial state in which the sea surface is assumed to be a Gaussian random process. This is because there are associated conserved quantities in the Hamiltonian form. All of these show that the Hamiltonian solution is more reasonable and accurate to describe the third-order steady-state wave field. Finally, based on the Hamiltonian solution, some statistics are given such as the volume flux, skewness, excess kurtosis, and non-uniqueness of the induced mean flow and mean surface.