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

TL;DR

This paper compares two third-order analytical solutions for surface gravity waves in finite water depth, demonstrating the advantages of the Hamiltonian approach over perturbation methods in accuracy and handling resonances.

Contribution

It introduces and compares perturbation and Hamiltonian solutions for third-order surface gravity waves, highlighting the Hamiltonian method's ability to overcome perturbation limitations.

Findings

Hamiltonian solutions are more accurate and handle resonances better.

Both solutions agree up to second order, diverging at third order.

Hamiltonian approach conserves energy-related quantities.

Abstract

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.