China Ocean Eng., 2017, Vol. 31, No. 6, P. 781–787 DOI: 10.1007/s13344-017-0089-z, ISSN 0890-5487 http://www.chinaoceanengin.cn/ E-mail: [email protected] Third-Order Stokes Wave Solutions of the Free Surface Capillary- Gravity Wave and the Interfacial Internal Wave MENG Rui-jun†, CUI Ji-feng†, CHEN Xiao-gang*, ZHANG Bao-le, ZHANG Hong-bo College of Science, Inner Mongolia University of Technology, Hohhot 010051, China Received September 7, 2016; revised June 21, 2017; accepted August 10, 2017 ©2017 Chinese Ocean Engineering Society and Springer-Verlag GmbH Germany, part of Springer Nature Abstract 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. Key words: surface tension, capillary-gravity waves, Stokes wave theory, third-order Stokes wave solutions Citation: Meng, R. J., Cui, J. F., Chen, X. G., Zhang, B. L., Zhang, H. B., 2017. Third-order Stokes wave solutions of the free surface capillary- gravity wave and the interfacial internal wave. China Ocean Eng., 31(6): 781–787, doi: 10.1007/s13344-017-0089-z 1 Introduction (2009), using elliptic integral of the first kind and so on. For In recent years, people have paid more attention to the internal waves, Stokes (1847) established a two-layer ocean importance of interfacial internal waves and capillary grav- internal wave theory; Benjamin (1966, 1967), Davis and ity waves. Firstly, investigation on them is very important Acrivos (1967), Ono (1975), Joseph (1977), Kubota et al. not only to understand the phenomena related to themselves (1978), Choi and Camassa (1996) derived the general evolu- but also to find out the knowledge about the interactions tion equations of two dimensional weakly nonlinear intern- between internal waves and many other multi-scale ocean al waves in two-fluid system. Song (2004) and Song and Sun waves and so on. Secondly, the remote sensing technology (2006) derived the second-order solutions for the random has become an effective means of the observation on sea interfacial waves in a two-layer fluid system, respectively. conditions. Owing to the Bragg scattering mechanisms, ca- Chen et al. (2005) studied the second-order Stokes solu- pillary waves have a significant effect on the sea microwave tions for internal waves in three-layer density-stratified fluid. (Wilton, 1915; Hogan, 1979, 1980, 1981; Maxworthy, In this paper, we investigate the capillary-gravity wave 1979; Song and Li, 1989, Cummins et al., 2003; Duda, and the interfacial internal wave in two-layer constant dep- 2004). th’s fluid system. Capillary-gravity waves and internal waves have very interesting properties. For example, for capillary-gravity 2 Basic equation and boundary conditions waves, the evidence of multiple solutions was first shown The interfacial waves propagating at the interface of two-layer fluid system are shown in Fig. 1. by Harrison (1909) and Wilton (1915) who included sur- The upper surface of the system is assumed to be free and face tension in Stoker’s classical expansion for pure gravity with the fluid in each layer homogeneous, inviscid, incom- waves. It was later extended by Lamb (1932), Defant pressible, and irrotational. The velocity potential Φ(1) in the up- (1961), Umeyama (2000, 2002) and others. Steady periodic per and Φ(2) in the lower layer satisfy the Laplace equations: capillary-gravity waves were calculated by Aider and Debi- 2 (1) 2 (1) ane (2006) using Neutrino’s method. Small-amplitude capil- ∂ Φ ∂ Φ + = 0; z1 + η 6 z 6 ζ; (1) lary-gravity waves were considered by Ionescu-Kruse ∂x2 ∂z2 Foundation item: The project was financially supported by the Science Research Project of Inner Mongolia University of Technology, China (Grant No. ZD201613). *Corresponding author. E-mail: [email protected] †The authors contributed equally. 782 MENG Rui-jun et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 781–787 ∂2Φ(2) ∂2Φ(2) where k is the wave number, ω is the angular frequency, and + = 0; z 6 z 6 z + η; (2) ∂x2 ∂z2 2 1 H is the characteristics of amplitude. where the horizontal coordinate axis x is fixed on the undis- Substituting Eq. (9) into Eqs. (1)–(8), the dimensionless turbed free surface. The vertical axis z is positive upwards equations can then be obtained as follows (for simplicity, “′” (Fig. 1); t is time; h1 and h2 are the depth of the upper and is neglected in the dimensionless quantity): lower layers, respectively; η(x, t) is the vertical displace- ∂2Φ(1) ∂2Φ(1) ment of the density interface; and ζ(x, t) is the vertical dis- + = 0; z1 + εη 6 z 6 εζ; (10) ∂x2 ∂z2 placement of the free surface measured from z=z0. ∂2Φ(2) ∂2Φ(2) + = 0; z 6 z 6 z + εη; (11) ∂x2 ∂z2 2 1 ∂Φ(1) ∂ζ ∂ζ ∂Φ(1) = ω + ε ; z = εζ; (12) ∂z ∂t ∂x ∂x 8 2 ! ! 39 > (1) 6 (1) 2 (1) 27> < ∂Φ 1 6 ∂Φ ∂Φ 7= ρ >ζ + ω + ε46 + 57> 1 : ∂t 2 ∂x ∂z ; 2 ! 3− 3 ∂2ζ 6 ∂ζ 27 2 = Γ 461 + ε 57 ; z = εζ; (13) ∂x2 ∂x Fig. 1. Sketch map of the two-layered density-stratified fluid system. ∂Φ(2) The boundary conditions at the free surface and the = 0; z = z ; (14) ∂z 2 density interface are 8 2 ! ! 39 (1) (1) > (1) 6 (1) 2 (1) 27> ∂Φ ∂η ∂η ∂Φ <∂Φ 1 6 ∂Φ ∂Φ 7= = ω + ε ; z = z1 + εη; (15) ρ > + gζ + 46 + 57> ∂z ∂t ∂x ∂x 1 : ∂t 2 ∂x ∂z ; (2) (2) 2 ! 3− 3 ∂Φ ∂η ∂η ∂Φ 2 2 = + ; = + ∂2ζ 6 ∂ζ 7 ω ε z z1 εη; (16) − Γ 461 + 57 = 0; z = ζ; (3) ∂z ∂t ∂x ∂x ∂x2 ∂x 8 2 ! ! 39 > (2) 6 (2) 2 (2) 27> < ∂Φ 1 6 ∂Φ ∂Φ 7= ∂Φ(1) ∂ζ ∂ζ ∂Φ(1) ρ >η + ω + ε46 + 57> = + ; z = ζ; (4) 2 : ∂t 2 ∂x ∂z ; ∂z ∂t ∂x ∂x 8 2 ! ! 39 <> (1) 6 (1) 2 (1) 27=> (1) (1) = + ∂Φ + 1 6 ∂Φ + ∂Φ 7 ; = + ; ∂Φ ∂η ∂η ∂Φ ρ1 >η ω ε4 5> z z1 εη = + ; z = η + z1; (5) : ∂t 2 ∂x ∂z ; ∂z ∂t ∂x ∂x ∂Φ(2) ∂η ∂η ∂Φ(2) (17) = + ; z = η + z ; (6) ∂z ∂t ∂x ∂x 1 kH 8 2 ! ! 39 where ε = ≪ 1 (H ≪ L, H is the characteristics of wave > (2) 6 (2) 2 (2) 27> 2 < ∂Φ 1 6 ∂Φ ∂Φ 7= ρ >gη + + 46 + 57> height, and L is the characteristics of wavelength). 2 : ∂t 2 ∂x ∂z ; 8 2 ! ! 39 > (1) 6 (1) 2 (1) 27> 3 Third-order Stokes-wave solutions with expansion < ∂Φ 1 6 ∂Φ ∂Φ 7= = ρ >gη+ + 46 + 57>; z = η + z1; (7) technology 1 : ∂t 2 ∂x ∂z ; Φ(i) (i=1, 2), ζ, η and ω in power series of an ordering where g is the acceleration of gravity, and Γ is the surface parameter ε (similar to Song and Sun (2006) and Umeyama tension which is a constant. (2000)) are expanded: The boundary condition at the bottom is n o X3 n o ( ) (2) (i); ; ; = (i); ; ; n−1 + 3 ; ∂Φ Φ ζ η ω Φn ζn ηn ωn ε O ε (18) = 0; z = z2: (8) ∂z n=1 Dimensionless quantity is introduced by choosing x′ as where O is the order symbol, and subscripts 1, 2 and 3 de- the unit horizontal length, z′ as the unit vertical length and t′ note quantities corresponding to the first-order, second-or- as the unit time. Now we define: der, and third-order perturbation solutions. 8 0 = ; 0 = ; 0 = ; > x kx z kz t sωt By substituting Eq. (18) into Eqs. (10)–(17), the govern- > (i) > 0 ing equations for each Φ , ζ , η , ω (i=1, 2, 3) are ob- <> (i) 0 0 0 2 k (i) 0 n n n n Φ (x ;z ;t ) = Φ (x;z;t); hi = khi(i = 1;2); > H g (9) tained by equating power of ε as follows. > > 0 0 0 2 0 0 0 2 The first-order equations and the boundary conditions : ζ (x ;t ) = ζ(x;t); η (x ;t ) = η(x;t): H H are MENG Rui-jun et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 781–787 783 (1) (1) (2) (2) ∂2Φ ∂2Φ ∂Φ ∂2Φ ∂η ∂η 1 + 1 = 0; z 6 z 6 ζ; (19) 2 + η 1 = ω 2 + ω 1 ∂x2 ∂z2 1 ∂z 1 ∂z2 1 ∂t 2 ∂t (2) 2 (2) 2 (2) ∂Φ ∂ Φ ∂ Φ + ∂η1 1 ; = 1 + 1 = 0; z 6 z 6 z ; (20) z z1; (33) ∂x2 ∂z2 2 1 ∂x ∂x (1) 8 ∂Φ > (2) (2) 2 (2) 1 = ∂ζ1 ; = <> ∂Φ ∂Φ ∂ Φ ω1 z 0; (21) > + 2 + 1 + 1 ∂z ∂t ρ2 :>η2 ω1 ω2 ω1η1 2 3 ∂t ∂t ∂t∂z 6 (1) 7 2 20 1 0 1 39 6 ∂Φ 7 ∂ ζ 6 (2) 2 (2) 27> 6 + 1 7 = 1 ; = 6B∂Φ C B∂Φ C 7> ρ1 46ζ1 ω1 57 Γ z 0; (22) 1 6B 1 C B 1 C 7= ∂t ∂x2 + 6@B AC + @B AC 7> 2 4 ∂x ∂z 5;> (2) 8 > (1) (1) 2 (1) ∂Φ1 <> ∂Φ ∂Φ ∂ Φ = 0; z = z ; (23) 2 1 1 2 = ρ >η + ω1 + ω2 + ω1η ∂z 1 : 2 ∂t ∂t 1 ∂t∂z (1) 20 1 0 1 39 ∂Φ 6 (1) 2 (1) 27> 1 = ∂η1 ; = 6B∂Φ C B∂Φ C 7=> ω1 z z1; (24) 1 6B 1 C B 1 C 7 ∂z ∂t + 6@B AC + @B AC 7>; z = z1: (34) 2 4 ∂x ∂z 5;> (2) ∂Φ ∂η 1 = ω 1 ; z = z ; (25) The third-order equations and boundary conditions are ∂z 1 ∂t 1 2 3 2 3 ∂2Φ(1) ∂2Φ(1) 6 ∂Φ(2) 7 6 ∂Φ(1) 7 3 + 3 = 0; z 6 z 6 z ; (35) 6 1 7 6 1 7 2 2 1 0 ρ 46η + ω1 57 = ρ 46η + ω1 57; z = z1: (26) ∂x ∂z 2 1 ∂t 1 1 ∂t ∂2Φ(1) ∂2Φ(1) The second-order equations and boundary conditions are 3 + 3 = 0; z 6 z 6 z ; (36) 2 2 2 1 2 (1) 2 (1) ∂x ∂z ∂ Φ2 ∂ Φ2 + = 0; z1 6 z 6 z0; (27) (1) (1) (1) (1) ∂x2 ∂z2 ∂Φ ∂2Φ ∂2Φ 1 ∂3Φ 3 + ζ 2 + ζ 1 + ζ2 1 1 2 2 2 1 3 ∂2Φ(2) ∂2Φ(2) ∂z ∂z ∂z 2 ∂z 2 + 2 = 0; z 6 z 6 z ; (28) (1) 2 2 2 1 ∂ζ ∂ζ ∂ζ ∂ζ ∂Φ ∂x ∂z = ω I + ω 2 + ω 3 + 1 2 3 ∂t 2 ∂t 1 ∂t ∂x ∂x (1) 2 (1) ∂Φ ∂ Φ (1) (1) 2 + 1 = ∂ζ2 ∂ζ ∂Φ ∂ζ ∂2Φ ζ1 ω1 + 2 1 + 1 1 ; = z 2 t ζ1 z 0; (37) ∂ ∂z ∂ ∂x ∂x ∂x ∂x∂z (1) ∂ζ1 ∂ζ1 ∂Φ1 + ω2 + ; z = 0; (29) 2 0 1 ∂t ∂x ∂x 6 ∂Φ(1) B ∂2Φ(1) ∂2Φ(1) ∂3Φ(1) C 8