  Third-order approximate solutions for surface gravity waves in the finite
water depth are studied in the context of potential flow theory. This solution
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
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 to 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 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 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, and excess kurtosis.
