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To understand the characteristics of ocean internal waves better, we study the dispersion relation of extended-Korteweg-de Vries (EKdV) equation with quadratic and cubic nonlinear terms in a two-layer fluid by using the Poincaré-Lighthill-Kuo (PLK) method which is one of the perturbation methods. Starting from the partial differential equation, the PLK method can be used to solve the dispersion relation of the equation. In this paper, we use PLK method to solve the equation and derive the dispersion relation of EKdV equation which is related to wave number and amplitude. Based on the dispersion relation obtained in this paper, the expressions of group velocity and phase velocity of the equation are obtained. Under the actual hydrological data, the influence of hydrological parameters on the dispersion relation for descending internal wave is discussed. It is hope that the obtained results will be helpful to the study of energy transfer and other internal wave parameters in the future.

Ocean internal solitary wave is a kind of internal waves, which is the result of dispersion effect and nonlinear effect [

Starting from modal equation, Fliegel and Haskell used Thomson-Haskell method to calculate the dispersion relation of internal waves [

PLK method was first proposed by Poincaré in finding the periodic solutions of the first-order ordinary differential equations. Later, Lighthill made an important promotion in finding the uniformly effective approximate solution of physical problems. Finally, Kuo further extended Lighthill’s original idea in seeking the elegant solution of the incompressible laminar boundary layer of a flat plate and subsequent work [

In this paper, we study the extended-Korteweg-de Vries (EKdV) equation with quadratic and cubic nonlinear terms proposed by T. Sakai and L. G. Redekopp which can better describe large amplitude waves propagation problem [

Grimshaw first describes the weakly nonlinear evolution of interfacial gravity waves on two shallow boundaries with KdV equation [

ζ t + c 0 ( 1 − α 1 ζ − α 2 ζ 2 ) ζ x + β 0 c 0 ζ x x x = 0. (1)

The coefficients of Equation (1) are as follows:

c 0 2 = g ˜ h 1 h 2 h 1 + h 2 , (2)

α 1 = 3 2 h 2 − h 1 h 1 h 2 , (3)

α 2 = 3 8 ( h 2 − h 1 ) 2 + 8 h 1 h 2 ( h 1 h 2 ) 2 , (4)

β 0 = 1 6 h 1 h 2 , (5)

where c 0 is the linear velocity, α 1 is the quadratic nonlinear term, α 2 is the cubic nonlinear term, β 0 is the dispersion coefficient and gravity g ˜ = g ρ 2 − ρ 1 ρ 1 = g Δ ρ ρ 1 , ρ 1 and ρ 2 are the densities of the upper and lower layers of sea water.

When α 2 ≠ 0 , Equation (1) is called KdV2 equation. When α 2 = 0 , Equation (1) is called KdV1 equation. KdV1 equation and KdV2 equation are both KdV families, but they are different [

The dispersion relation of Equation (1) will be derived. Introducing dimensionless variation h 2 − h 1 h 1 h 2 ζ = A , Equation (1) can be transformed into

A t + c 0 A x − 3 2 c 0 A A x − c 0 9 α 2 4 α 1 2 A 2 A x + β 0 c 0 A x x x = 0. (6)

Introducing phase function

ξ = k x − ω t ,

where k and ω are wave number and circular frequency respectively, Equation (6) becomes the following nonlinear ordinary differential equation

− ω d A d ξ + c 0 k d A d ξ − 3 2 c 0 k A d A d ξ − 9 4 c 0 k α 2 α 1 2 A 2 d A d ξ + c 0 β 0 k 3 d 3 A d ξ 3 = 0. (7)

Using the PLK method for Equation (7), the dispersion relation of Equation (1) is obtained. Let η 0 be the amplitude of ζ , and select ε as a small parameter

ε = h 2 − h 1 h 1 h 2 η 0 < 1. (8)

Both A and ω are expanded to power series of ε . Because the dimensionless numbers A and ε are small quantities of the same order, A is expanded from the first order of ε , and ω is expanded from the zero order. The perturbation expansion of A with respect to ε is written as

A = ε A 1 ( ζ ) + ε 2 A 2 ( ζ ) + ε 3 A 3 ( ζ ) + ⋯ , (9)

and the perturbation expansion of the circular frequency ω is written as

ω = ω 0 ( k ) + ε ω 1 ( k ) + ε 2 ω 2 ( k ) + ⋯ . (10)

Substituting Equation (9) and Equation (10) into Equation (7), we can get

− ( ω 0 + ε ω 1 + ε 2 ω 2 ) d ( ε A 1 + ε 2 A 2 + ε 3 A 3 ) d ξ + c 0 k d ( ε A 1 + ε 2 A 2 + ε 3 A 3 ) d ξ − 3 2 c 0 k ( ε A 1 + ε 2 A 2 + ε 3 A 3 ) d ( ε A 1 + ε 2 A 2 + ε 3 A 3 ) d ξ + c 0 β 0 k 3 d 3 ( ε A 1 + ε 2 A 2 + ε 3 A 3 ) d ξ 3 − 9 4 c 0 k α 2 α 1 2 ( ε A 1 + ε 2 A 2 + ε 3 A 3 ) 2 d ( ε A 1 + ε 2 A 2 + ε 3 A 3 ) d ξ = 0. (11)

The first-order to the third-order approximation is respectively

( k c 0 − ω 0 ) d A 1 d ξ + c 0 β 0 k 3 d 3 A 1 d ξ 3 = 0, (12)

− ω 0 d A 2 d ξ − ω 1 d A 1 d ξ + c 0 k d A 2 d ξ − 3 2 c 0 k A 1 d A 1 d ξ + c 0 β 0 k 3 d 3 A 2 d ξ 3 = 0, (13)

− ω 0 d A 3 d ξ − ω 1 d A 2 d ξ − ω 2 d A 1 d ξ + c 0 k d A 3 d ξ − 3 2 c 0 k A 1 d A 2 d ξ − 3 2 c 0 k A 2 d A 1 d ξ − 9 4 c 0 k α 2 α 1 2 A 1 2 d A 1 d ξ + c 0 β 0 k 3 d 3 A 3 d ξ 3 = 0. (14)

The first-order approximation Equation (12) is the second-order oscillation equation for d A 1 d ξ . ε is dimensionless, hence

c 0 k − ω 0 c 0 β 0 k 3 = 1.

The zero order approximation of circular frequency ω is

ω 0 = c 0 k − c 0 β 0 k 3 . (15)

At the same time, the solution of the first-order approximation Equation (12) is obtained

A 1 = sin ξ . (16)

Taking Equation (15) and Equation (16) into the second-order approximation Equation (13), we can get

c 0 β 0 k 3 ( d A 2 d ξ + d 3 A 2 d ξ 3 ) = ω 1 cos ξ + 3 2 c 0 k sin ξ cos ξ . (17)

It can be seen from the above formula that the non-duration condition here is

ω 1 = 0. (18)

Then Equation (17) is reduced to

d A 2 d ξ + d 3 A 2 d ξ 3 = 3 2 β 0 k 2 sin ξ cos ξ . (19)

Its special solution is

A 2 = − 1 4 β 0 k 2 sin 2 ξ . (20)

Taking Equations (15), (16), (18) and (20) into third-order approximation Equation (14), we can obtain

c 0 β 0 k 3 ( d A 3 d ξ + d 3 A 3 d ξ 3 ) = ω 2 cos ξ + ( − 9 8 c 0 β 0 k + 9 4 c 0 k α 2 α 1 2 ) sin 2 ξ cos ξ . (21)

It can be seen from the above formula that the non-duration condition here is

ω 2 = − 9 ( 2 k 2 α 2 β 0 − α 1 2 ) c 0 32 k β 0 α 1 2 . (22)

And Equation (21) can be simplified as

d A 3 d ξ + d 3 A 3 d ξ 3 = − 9 ( 2 k 2 α 2 β 0 − α 1 2 ) 32 α 1 2 β 0 2 k 4 cos ξ + 9 ( 2 α 2 β 0 k 2 − α 1 2 ) 8 α 1 2 β 0 2 k 4 sin 2 ξ cos ξ . (23)

Its special solution is

A 3 = − 3 ( 2 k 2 α 2 β 0 − α 1 2 ) 64 α 1 2 β 0 2 k 4 sin 3 ξ . (24)

The perturbation solution of Equation (1) is obtained by using the PLK method

A = ε sin ξ − ε 2 1 4 β 0 k 2 sin 2 ξ − ε 3 3 ( 2 k 2 α 2 β 0 − α 1 2 ) 64 α 1 2 β 0 2 k 4 sin 3 ξ + o ( ε 4 ) . (25)

The circular frequency is

ω = c 0 k − c 0 β 0 k 3 − 9 ( 2 k 2 α 2 β 0 − α 1 2 ) c 0 32 k β 0 α 1 2 ε 2 + o ( ε 3 ) .

Noting that ε = h 2 − h 1 h 1 h 2 η 0 and α 1 = 3 2 h 2 − h 1 h 1 h 2 , the above formula can be adapted as

ζ = η 0 sin ξ − α 1 6 β 0 k 2 η 0 2 sin 2 ξ − 2 k 2 α 2 β 0 − α 1 2 48 β 0 2 k 4 η 0 3 sin 3 ξ + o ( ε 4 ) . (26)

And the circular frequency is

ω = c 0 k − c 0 β 0 k 3 − 2 k 2 α 2 β 0 − α 1 2 8 k β 0 c 0 η 0 2 + o ( ε 3 ) . (27)

Formula (27) is the dispersion relation of nonlinear internal solitary wave EKdV equation. By truncating formula (27), we can get the truncated expression of dispersion relation

ω = c 0 k − c 0 β 0 k 3 − 2 k 2 α 2 β 0 − α 1 2 8 k β 0 c 0 η 0 2 . (28)

Any wave equation has its specific dispersion relation, so we can determine the wave parameters of wave according to the dispersion relation. We select a set of data of Andaman Sea area to discuss the influence of hydrological parameters on dispersion relation. The water depth of upper layer h 1 = 230 m , the water depth

of lower layer h 2 = 863 m , density difference ratio Δ ρ ρ = 0.003 , amplitude

η 0 = 60 m [

The expressions of phase velocity and group velocity can be calculated respectively based on the dispersion relation (28)

C p = c 0 − c 0 β 0 k 2 − 2 k 2 α 2 β 0 − α 1 2 8 k 2 β 0 c 0 η 0 2 , (29)

C g = c 0 − 3 c 0 β 0 k 2 − 2 k 2 α 2 β 0 + α 1 2 8 k 2 β 0 c 0 η 0 2 . (30)

Combining the expression of phase velocity [

C p = c 0 ( 1 + η 0 ( h 2 − h 1 ) 2 h 1 h 2 ) , (31)

the value of wave number k can be deduced by formula (29) and (31) using the measured data.

As seen from

As seen from

In this paper, the dispersion relation with the perturbation solution of EKdV equation is obtained by using PLK method. The dispersion relation derived in this paper is related to water number and amplitude. The expressions of phase

velocity and group velocity are obtained by using the dispersion relation, which can be used to study the propagation characteristics and energy transmission of ocean internal waves. Under the actual hydrological data, the influence of water depth and density difference ratio on the descending internal solitary waves is discussed. The value of ω decreases with the increase of the upper and lower water depth, but it increases with the increase of density difference ratio. We hope to provide a better theoretical basis for solving internal wave parameters by using dispersion relation.

This work has been supported by the project of Beijing Information Science and Technology University No. Z2018057.

The authors declare no conflicts of interest regarding the publication of this paper.

Feng, P.P. and Meng, X.H. (2021) The Dispersion Relation of Internal Wave Extended-Korteweg-de Vries Equation in a Two-Layer Fluid. Journal of Applied Mathematics and Physics, 9, 1056-1064. https://doi.org/10.4236/jamp.2021.95072