Third-order stokes wave solutions of the free surface capillary-gravity wave and the interfacial internal wave

Based on the Stokes wave theory, the capillary-gravity wave and the interfacial internal wave in two-layer constant depth’s fluid system are investigated. The fluids are assumed to be incompressible, inviscid and irrotational. The third-order Stokes wave solutions are given by using a perturbation method. The results indicate that the third-order solutions depend on the surface tension, the density and the depth of each layer. As expected, the first-order solutions are the linear theoretical results (the small amplitude wave theoretical results). The second-order and the third-order solutions describe the nonlinear modification and the nonlinear interactions. The nonlinear impact appears not only in the n (n≥2) times’ high frequency components, but also in the low frequency components. It is also noted that the wave velocity depends on the wave number, depth, wave amplitude and surface tension.


Introduction
In recent years, people have paid more attention to the importance of interfacial internal waves and capillary gravity waves. Firstly, investigation on them is very important not only to understand the phenomena related to themselves but also to find out the knowledge about the interactions between internal waves and many other multi-scale ocean waves and so on. Secondly, the remote sensing technology has become an effective means of the observation on sea conditions. Owing to the Bragg scattering mechanisms, capillary waves have a significant effect on the sea microwave (Wilton, 1915;Hogan, 1979Hogan, , 1980Hogan, , 1981Maxworthy, 1979;Song andLi, 1989, Cummins et al., 2003;Duda, 2004).
Capillary-gravity waves and internal waves have very interesting properties. For example, for capillary-gravity waves, the evidence of multiple solutions was first shown by Harrison (1909) and Wilton (1915) who included surface tension in Stoker's classical expansion for pure gravity waves. It was later extended by Lamb (1932), Defant (1961), Umeyama (2000Umeyama ( , 2002 and others. Steady periodic capillary-gravity waves were calculated by Aider and Debiane (2006) using Neutrino's method. Small-amplitude capillary-gravity waves were considered by Ionescu-Kruse (2009), using elliptic integral of the first kind and so on. For internal waves, Stokes (1847) established a two-layer ocean internal wave theory; Benjamin (1966Benjamin ( , 1967, Davis and Acrivos (1967), Ono (1975), Joseph (1977), Kubota et al. (1978), Choi and Camassa (1996) derived the general evolution equations of two dimensional weakly nonlinear internal waves in two-fluid system. Song (2004) and Song and Sun (2006) derived the second-order solutions for the random interfacial waves in a two-layer fluid system, respectively. Chen et al. (2005) studied the second-order Stokes solutions for internal waves in three-layer density-stratified fluid.
In this paper, we investigate the capillary-gravity wave and the interfacial internal wave in two-layer constant depth's fluid system.

Basic equation and boundary conditions
The interfacial waves propagating at the interface of two-layer fluid system are shown in Fig. 1.
The upper surface of the system is assumed to be free and with the fluid in each layer homogeneous, inviscid, incompressible, and irrotational. The velocity potential Φ (1) in the upper and Φ (2) in the lower layer satisfy the Laplace equations: where the horizontal coordinate axis x is fixed on the undisturbed free surface. The vertical axis z is positive upwards (Fig. 1); t is time; h 1 and h 2 are the depth of the upper and lower layers, respectively; η(x, t) is the vertical displacement of the density interface; and ζ(x, t) is the vertical displacement of the free surface measured from z=z 0 .
The boundary conditions at the free surface and the density interface are where g is the acceleration of gravity, and Γ is the surface tension which is a constant.
The boundary condition at the bottom is Dimensionless quantity is introduced by choosing x′ as the unit horizontal length, z′ as the unit vertical length and t′ as the unit time. Now we define: where k is the wave number, ω is the angular frequency, and H is the characteristics of amplitude. Substituting Eq. (9) into Eqs. (1)-(8), the dimensionless equations can then be obtained as follows (for simplicity, "′" is neglected in the dimensionless quantity): , H is the characteristics of wave height, and L is the characteristics of wavelength).

Third-order Stokes-wave solutions with expansion technology
, ζ, η and ω in power series of an ordering parameter ε (similar to Song and Sun (2006) and Umeyama (2000)) are expanded: where O is the order symbol, and subscripts 1, 2 and 3 denote quantities corresponding to the first-order, second-order, and third-order perturbation solutions.
The first-order equations and the boundary conditions are The second-order equations and boundary conditions are The third-order equations and boundary conditions are By solving Eqs. (19)- (42), the first-order, the second-order and the third-order solutions are as follows.

First-order solutions
where and denote the amplitudes of first-order component of the waves at the free surface and the interface between two-layer fluids.

Discussion
It can be found from the Eqs. (43)- (46) and (48) that the first-order solutions are the linear wave solutions (the small amplitude wave theoretical results). Eqs. (49)-(61) show that the second-order solutions are determined by the firstorder solutions, second-order nonlinear modifications and interactions, and the circular frequency of the second-order Stokes solutions is identical with the first-order solutions, so the perturbation expansion of the circular frequency cannot be considered to solve the second-order Stokes solutions. Eqs. (62)-(77) indicate that the third-order solutions are determined by the first-order solutions, second-order solutions and third-order nonlinear modification and interactions. Furthermore, the first, second and third order solu-tions are dependent on the densities, depths and surface tension. In addition, from Eqs. (49), (50), (62) and (63), it is also noticed that the nonlinear impact appears not only in the n (n≥2) times' high frequency components, but also in the low frequency components (Zou, 2005). Moreover, from Eqs. (18) and (66), we can obtained that the wave velocity depends on not only the wave number, surface tension and depths but also the wave amplitude. This is a characteristic of the nonlinear wave, but up to the third-order approximation it can be revealed (Brevik andAas, 1979-1980). Finally, when Γ=0, the dispersion relation to Eq. (47) can be deduced to Eq. (42) derived by Umeyama (2000), and when h 2 =0 or ρ 1 =ρ 2 , Eq. (47) can be further simplified as the dispersion relation of the free surface wave with finite depth, so the results of this paper can degenerate to the classic Stokes theory for constant density fluid.

Conclusions
By using the perturbation method and the Stokes wave theory, the capillary-gravity wave and the interfacial internal wave in two-layer constant depth's fluid system are studied and the third-order Stokes wave solutions are derived. The results are as follows.
(1) The first-order solutions are the linear wave solutions (the small amplitude wave theoretical results) and the dispersion equation is related to the densities, depths and surface tension.
(2) The second-order solutions are determined by the first-order solutions, second-order nonlinear modifications and interactions and the perturbation expansion of the circular frequency cannot be considered for solving the secondorder Stokes solutions.
(3) The third-order solutions are determined by the firstorder solutions, the second-order solutions and the third-order nonlinear modification and interactions.
(4) The wave velocity depends on the wave number, surface tension, depths, and wave amplitude.
(5) The nonlinear effect appears not only in the n (n≥2) times' high frequency components, but also in the low frequency components.
(6) The first-order, second-order and third-order solutions are dependent on the densities, depths of a two-layers and the surface tension.