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STOKES’ WAVE SOLUTION FOR GRAVITY WAVE AT THE INTERFACE OF TWO SUPERPOSED FLUID LAYERS OF FINITE THICKNESSES CONTENT''' 1.Introduction 2.Governing Equation 3.Method of Solution 4.Equations at the Order ε 5.Equation at the Order ε2 6.Equation at the Order ε3 7.Summary of the results and Conclusion 8.References

1.	INTRODUCTION : Stoke’s investigation (1847) on water waves is the starting point for non-linear theory of water waves. It was in this work that he obtained two very important results. First periodic waves are possible in nonlinear system though nont sinusoidal and second, the dispersion relation involves amplitude. His investigation are for infinite depth water and he found that the elevation ζ of the free surface for a weakly nonlinear surface gravity wave can be expanded in powers of amplitude of the first harmonic as follows :- ζ=a cos⁡(kx-Ωt)+1/2 ka^2 cos⁡2 (kx-Ωt)+ 3/8 k^2 a^3 cos⁡3 (kx-Ωt)+⋯,                        …(1.1) where the dispersion relation is Ω^2=gk (1+k^2 a^2+⋯)                                        …(1.2) This shows the dependence of dispersion relation on amplitude, unlike the linear dispersion relation ω^2=gk. This analysis of Stokes’ was latter extended by Whitham (1967) for finite depth water. In the present project Stokes wave solution is obtained for a gravity wave at the interface of two incompressible superposed fluid layers of different densities and of finite thicknesses. The solution obtained is correct upto order a^2, where ‘a’ is the amplitude of the first harmonic part of the wave. The technique developed by Montgomery and Tidman (1964) to solve Klien – Gordon equation with small nonlinear terms, is applied here to obtain the Stokes wave solution in the present investigation. In this paper Montgomery and Tidman also obtained amplitude dependent frequency shift for electromagnetic wave propagating in a cold plasma, the relativistic correction of which was later obtained; by Sluijter and Montgomery (1965), Das (1968, 1971) also used the method of Montgomery and Tidman (1964) to obtained nonlinear frequency shift for two transverse waves in a cold plasma and for an extra – ordinary wave in a cold magnetized plasma. The technique developed by Montgomery and Tidman (1964) is an extension of the perturbation method developed by Krylor and Bogoliubov (1947) to solve harmonic oscillator equations with small nonlinear terms. This was later altered slightly and mathematically justified by Bogoliubov and Mitropolsky (1961).

2.	GOVERNING EQUATIONS : Assuming the unperturbed interface of the two fluids as the z = 0 plane and the positive direction of z – axis pointing in the vertically upward direction the governing equations assume the following form :- (∂^2 " " )/(∂x^2 )+(∂^2 " " )/(∂z^2 )=0,  -h<z<ζ                                                        …(2.1) (∂/∂z)_(Z=-h)=0                                                                           …(2.2) ∂/∂z+∂ζ/∂t=∂/∂x  ∂ζ/∂x  ,at z= ζ                                                            …(2.3) (∂^2 ^')/(∂x^2 )+(∂^2 ^')/(∂z^2 )=0,  ζ<z<h^'                                                           …(2.4) ((∂^')/∂z)_(Z=h^' )=0                                                                           …(2.5) (∂^')/∂z+∂ζ/∂t=(∂^')/∂x  ∂ζ/∂x  ,at z= ζ                                                             …(2.6)  ∂/∂t-^' (∂^')/∂t-g("" -^' )ζ=/2 (∂/∂x)^2 +/2 (∂/∂z)^2-^'/2 ((∂^')/∂x)^2-^'/2 ((∂^')/∂z)^2 ,at z=ζ              …(2.7) where the equations (2.1) and (2.4) come from the equation of continuity, the equations (2.2) and (2.5) come from the condition that the velocity of fluid normal to the rigid boundary must be equal to zero, the equations (2.3) and (2.6) are consequences of Kinematic boundary condition at the interface of the two fluids and the equation (2.7) is a consequence of the continuity of pressure across the interface. The equation of the interface in the perturbed state has been taken as z=(x,t)                                                                           …(2.8) Here it has been assumed that the fluid flow is irrotational and also it is assumed that all the dependent variables are independent of y – coordinate. Therefore, the fluid velocities (u,0,w) and (u^',0,w^' ) in the lower and upper layers are given by u=-∂/∂x,w=-∂/∂z ;u^'=-(∂^')/∂x,w^'=-(∂^')/∂z                     …(2.9) In the equations (2.3), (2.6) and (2.7), φ and φ^'are be evaluated at z=ζ, i.e., φ=φ(x,ζ,t) and φ^'=φ^' (x,ζ,t) where z has been replaced by ζ. Therefore, Taylor expanding j and j^' appearing in these equations about z = 0 and keeping terms upto third degree, these three equations can be expressed as follows. (∂j/∂z)_0+∂ζ/∂t=b^((2))+b^((3))                                                                                 …(2.10) ((∂j^')/∂z)_0+∂ζ/∂t=c^((2) )+c^((3) )                                                                                …(2.11) (∂j/∂z)_0-^' ((∂j^')/∂t)_0-g("" -^' )ζ=f^((2))+f^((3))                                     …(2.12) where the terms on the right hand side are nonlinear terms given below and the notation (┤)_0 implies the value of the quantities inside bracket at z = 0. b^((2))=-ζ((∂^2 φ)/(∂z^2 ))_0+(∂j/∂x)  ∂ζ/∂x                                                             …(2.13) b^((3))=-1/2 ζ^2 ((∂^3 φ)/(∂z^3 ))_0+ζ((∂^2 " j" )/∂x∂z)_0  ∂ζ/∂x                                            …(2.14) c^((2))=-ζ((∂^2 φ^')/(∂z^2 ))_0+((∂j^')/∂x)_0  ∂ζ/∂x                                                       …(2.15) c^((3))=-1/2 ζ^2 ((∂^3 φ^')/(∂z^3 ))_0+ζ((∂^2 j^')/∂x∂z)_0  ∂ζ/∂x                                           …(2.16) f^((2))=- ζ((∂^2 " j" )/∂z∂t)_0+^' ζ((∂^2 j^')/∂z∂t)_0+1/2  (∂φ/∂x)_0^2 +1/2 (∂φ/∂z)_0^2-1/2 ^' ((∂φ^')/∂x)_0^2-1/2 ^' ((∂φ^')/∂z)_0^2                       …(2.17) f^((3))=-1/2  ζ^2 ((∂^3 " j" )/(∂z^2 ∂t))_0+1/2 ^' ζ^2 ((∂^3 j^')/(∂z^2 ∂t))_0+ ζ (∂φ/∂x)_0 ((∂^2 " j" )/∂x∂z)_0+ ζ(∂φ/∂z)_0 ((∂^2 φ)/(∂z^2 ))_0-^' ζ((∂φ^')/∂x)_0 ((∂^2 j^')/∂x∂z)_0 -^' ζ ((∂φ^')/∂z)_0 ((∂^2 φ^')/(∂z^2 ))_0                                                       …(2.18) where b^((2)),c^((2)),f^((2)) are quadric nonlinear terms and b^((3)),c^((3)),f^((3)) are cubic non-linear terms.

3.	METHOD OF SOLUTION : To get solution of equations (2.1), (2.2), (2.4), (2.5), (2.10), (2.11) and (2.12), which are the governing equations, we apply the perturbation method of Bogoliubov, Krylor and Mitropotsky developed by Montgomery and Tidman (1964). According to this method we expand φ,φ^' and ζ as follows :- φ=εE+ εφ^((1) ) (z,Ψ)+ε^2 φ^((2) ) (z,Ψ)+ε^3 φ^((3) ) (z,Ψ)+⋯ φ^'=εE^'+ εφ^'(1) (z,Ψ)+ε^2 φ^'(2)  (z,Ψ)+ε^3 φ^'(3)  (z,Ψ)+⋯ ζ=εζ^((1) ) (Ψ)+ε^2 {ζ^((2)) (Ψ)+ζ^((0)) }+ε^3 ζ^((3) ) (Ψ)+⋯                 …(3.1) where E,E^' and ζ^((0)) do not depend on z and Ψ but depends slowly on t, i.e., their derivative with respect to ‘t’ is of order ε. Here ε is a small parameter indicating the weakness of nonlinearity, which implies that the wave steepness (=wave amplitude / wave length) is very small; Ψ is the phase of the wave and for a travelling wave of constant wave length, ∂Ψ/∂x=k                                                                        …(3.2) and the appropriate expansion of ∂Ψ/∂t is ∂Ψ/∂t=-ω+ε^2 ω^((2))+⋯,                                      …(3.3) where ω is the frequency of the wave in absence of nonlinearity and –ω^((2)) is the amplitude dependent frequency shift. In the governing equations we replace x and t by the single variable Ψ by the use of (3.2) and (3.3) and then substitute the perturbation expansions (3.1) and (3.2) in these equations. Finally we equate coefficients of different powers of  on both sides of each equation. We thus get a sequence of equations of different orders. The governing equations written in terms of the single variable Ψ instead of the variables x, t become as follows which are respectively the equations (2.1), (2.2), (2.4), (2.5), (2.3), (2.6), (2.7) (∂^2 φ)/(∂z^2 )+k^2 (∂^2 φ)/(∂Ψ^2 )=0                                                                     …(3.4) (∂φ/∂z)_(z=-h)=0                                                                             …(3.5) (∂^2 φ^')/(∂z^2 )+k^2 (∂^2 φ^')/(∂Ψ^2 )=0                                                                   …(3.6) ((∂φ^')/∂z)_(z=h^' )=0                                                                             …(3.7) (∂φ/∂z)_0+^3  (∂ζ^((0)))/∂t+(-ω+^2 ω^((2)) )  ∂ζ/∂Ψ=b ̅^((2))+b ̅^((3))           …(3.8) ((∂φ^')/∂z)_0+^3  (∂ζ^((0)))/∂t+(-ω+^2 ω^((2)) )  ∂ζ/∂Ψ=c ̅^((2))+c ̅^((3))           …(3.9)  (-ω+^2 ω^((2)) ) (∂φ/∂Ψ)_0-^' (-ω+^2 ω^((2)) ) ((∂φ^')/∂Ψ)_0+^2   ∂E/∂t-^2 ^'  (∂E^')/∂t -g ("" -^' )ζ=f ̅^((2))+f ̅^((3))                                   …(3.10) where b ̅^((2)),c ̅^((2)),b ̅^((3)),c ̅^((3)),f ̅^((2)),f ̅^((3)) are same as b^((2)),c^((2)),b^((3)),c^((3)),f^((2)),f^((3)) respectively, where the variables x, t have been replaced by the single variable Ψ by the use of the relations (3.2) and (3.3). In these equations terms upto 0 (^3 ) have been retained.

4.	THE EQUATIONS AT THE ORDER  : At the order , i.e., at the lowest order, we get the following equations from the equations (3.4) – (3.10), which are the obtained after substituting the expansions (3.1) for φ,φ^',ζ and then equating coefficients of  on both sides of each equation, (∂^2 φ^((1)))/(∂z^2 )+k^2 (∂^2 φ^((1)))/(∂Ψ^2 )=0                                                                        …(4.1) ((∂φ^((1)))/∂z)_(z=-h)=0                                                                             …(4.2) (∂^2 φ^('(1)))/(∂z^2 )+k^2 (∂^2 φ^('(1)))/(∂Ψ^2 )=0                                                                      …(4.3) ((∂φ^('(1)))/∂z)_(z=h^' )=0                                                                             …(4.4) ((∂φ^((1)))/∂z)_0-ω (∂ζ^((1)))/∂Ψ=0                                                                        …(4.5) ((∂φ^('(1)))/∂z)_0-ω (∂ζ^((1)))/∂Ψ=0                                                                     …(4.6) -ω ((∂φ^((1) ))/∂Ψ)_0+ω^'  ((∂φ^'(1) )/∂Ψ)_0-g(""  -^' ) ζ^((1))=0            …(4.7) For a wave propagating along x – axis of wave number k and of linear frequency ω, we take ζ^((1))=a cos⁡Ψ                                                                               …(4.8) where the amplitude ‘a’ is a slowly varying function of t, i.e., it’s derivative with respect to t is of 0(^2 ). The expression (4.8) for ζ^((1)) suggests that we can take for φ^((1)) and φ^('(1)) the following expressions. φ^((1))=φ ̅^((1) ) (z) sin⁡Ψ,φ^('(1))=φ ̅^'(1)  (z)  sin⁡Ψ                            …(4.9) The choice of sin⁡Ψ is suggested by the equation (4.7). Therefore, from the equations (4.1) – (4.4) we get the following equations (d^2 φ ̅^((1)))/(dz^2 )-k^2 φ ̅^((1))=0,       ((dφ ̅^((1) ))/dz)_(z=-h)=0                          …(4.10) (d^2 φ ̅^('(1)))/(dz^2 )-k^2 φ ̅^('(1))=0,       ((dφ ̅^'(1) )/dz)_(z=h^' )=0                       …(4.11) We easily find that the solution of the system (4.10) and (4.11) are ├ ■(φ ̅^((1))=A^((1)) cos⁡h k (z+h),@@φ ̅^('(1))=B^((1))  cos⁡h k (z-h^' ) )} where A^((1)) and B^((1)) are two constants. Hence, the solutions for φ^((1)) and φ^('(1)) are φ^((1))=A^((1) ) cosh⁡k (z+h)  sin⁡Ψ,φ^('(1))=B^((1) )  cosh⁡k (z-h^' )  sin⁡Ψ         …(4.12) With the solutions for φ^((1)) and φ^('(1)) given by (4.12) and the expression for ζ^((1)) given by (4.8). We get the following equations from (4.5) to (4.7) – k A^((1)) sinh⁡kh+ωa=0                                        …(4.13) k B^((1)) sinh⁡〖kh^' 〗-ωa=0                                      …(4.14) -ω  A^((1)) cosh⁡kh+ω^'  B^((1))  cosh⁡〖kh^' 〗=g("" -^' )a             …(4.15) Therefore, from (4.13) and (4.14) we get A^((1))=-ωa/(k sinh⁡kh ) ,B^((1))=-ωa/(ksinh⁡kh^' )                           …(4.16) Consequently the solutions for φ^((1)) and φ^('(1)) as given by (4.12) becomes ├ ■(φ^((1))=-ωa/(k sinh⁡kh ) cosh⁡k (z+h)  sin⁡Ψ@@φ^('(1))=-ωa/(k sinh⁡〖kh^' 〗 )  cosh⁡k (z-h^' )  sin⁡Ψ )}                    …(4.17) Substituting the solutions for A^((1))  & B^((1)) in (4.15), we get – a [ω^2 ("" coth⁡〖kh+^'  coth⁡〖kh^' 〗 〗 )-gk("" -^' ) ]=0 Since for a wave to exist a≠0 we find that the expression inside the square bracket must be equal to zero. This gives ω^2=gk("" -^' )/("" coth⁡kh+^'  coth⁡〖kh^' 〗 ) or,     ω^2 (/+^'/^' )-gk("" -^' )=0                                 …(4.18) which is the linear dispersion relation for a surface – gravity wave at the interface of two superposed fluids of finite thicknesses. In (4.18), we use the notations, tanh⁡kh=    and    tanh⁡〖kh^' 〗=^'

5.	EQUATION S AT THE ORDER ^2 : The order ^2equations are obtained from the equations (3.4) – (3.10) after substituting the expansions (3.1) for φ,φ^' and ζ and then equating coefficients of ^2 on both sides of each equation. (∂^2 φ^((2)))/(∂z^2 )+k^2 (∂^2 φ^((2)))/(∂Ψ^2 )=0                                                  …(5.1) ((∂φ^((2) ))/∂z)_(z=-h)=0                                          …(5.2) (∂^2 φ^('(2)))/(∂z^2 )+k^2 (∂^2 φ^('(2)))/(∂Ψ^2 )=0                                               …(5.3) ((∂φ^'(2) )/∂z)_(z=-h^' )=0                                        …(5.4) (〖∂φ〗^((2) )/∂z)_0-ω (∂ζ^((2)))/∂Ψ=b ̅_2^((2) )                                         …(5.5) (〖∂φ〗^'(2) /∂z)_0-ω (∂ζ^((2)))/∂Ψ=c ̅_2^((2) )                                         …(5.6)  ∂E/∂t-^'  (∂E^')/∂t-ω (〖∂φ〗^((2) )/∂Ψ)_0+ω^'  (〖∂φ〗^'(2) /∂Ψ)_0+g("" -^' )(ζ^((2))+ζ^((0)) )=f ̅_2^((2))    …(5.7) We now first evaluate different nonlinear terms on the right hand side of equations (5.5), (5.6), (5.7). (I) Nonlinear terms of the right hand side of (5.5) b ̅_2^((2) )=-ζ^((1)) ((∂^2 φ^((1) ))/(∂z^2 ))_0+k^2 ((∂^2 φ^((1) ))/∂Ψ)_0 (∂ζ^((1)))/∂Ψ =-a cos⁡Ψ {-(ω a〖 k〗^2)/(k sinh⁡kh ) cosh⁡kh }  sin⁡Ψ+k^2 {-(ω a cosh⁡kh)/(k sinh⁡kh ) cos⁡Ψ }(-a sin⁡Ψ ) =(k ω a^2)/ sin⁡2Ψ Therefore, terms on the right hand side of (5.5) = (k ω a^2)/ sin⁡2Ψ. …(5.8) (II) Nonlinear terms of the right hand side of (5.6) c ̅_2^((2) )=-ζ^((1)) ((∂^2 φ^'(1) )/(∂z^2 ))_0+k^2 ((∂^2 φ^'(1) )/∂Ψ)_0 (∂ζ^((1)))/∂Ψ =-a cos⁡Ψ-(ω a〖 k〗^2 cosh⁡〖kh^' 〗)/(ksinh⁡kh^' )  sin⁡Ψ+k^2  (ω a cosh⁡〖kh^' 〗)/(ksinh⁡kh^' )  cos⁡Ψ (-a sin⁡Ψ ) =-(k ω a^2)/^'  sin⁡2Ψ Therefore, terms on the right hand side of (5.6) = -(k ω a^2)/^'  sin⁡2Ψ. …(5.9) (III) Nonlinear terms of the right hand side of (5.7) f ̅_2^((2))=ω  ζ^((1)) ((∂^2 φ^((1) ))/∂z∂Ψ)_0-ω ^' ζ^((1)) ((∂^2 φ^'(1) )/∂z∂Ψ)_0+1/2   k^2 ((∂φ^((1) ))/∂Ψ)_0^2 +1/2  ((∂φ^((1) ))/∂z)_0^2-1/2 ^' k^2 ((∂φ^'(1) )/∂Ψ)_0^2-1/2 ^'  ((∂φ^'(1) )/∂z)_0^2 Evaluation of different terms on the right hand side of (5.7) – (1) ω  ζ^((1)) ((∂^2 φ^((1) ))/∂z∂Ψ)_0=ω  a cos⁡Ψ {-(ω a k sinh⁡kh)/(k sinh⁡kh )  cos⁡Ψ } = -1/2 ω^2 a^2  - 1/2 ω^2 a^2  cos⁡2Ψ (2)-ω ^' ζ^((1)) ((∂^2 φ^'(1) )/∂z∂Ψ)_0=-ω ^'  a cos⁡Ψ {-(ω a k sinh⁡〖kh^' 〗)/(ksinh⁡kh^' )  cos⁡Ψ } = -1/2 ω^2 a^2 ^'+1/2 ω^2 a^2 ^' cos⁡2Ψ (3) 1/2   k^2  ((∂φ^((1) ))/∂Ψ)_0^2=1/2  k^2   (ω^2 a^2)/k^2   coth^2⁡kh  cos^2⁡Ψ =1/4  (ω^2 a^2 " " )/^2 +1/4  (ω^2 a^2 " " )/^2   cos⁡2Ψ (4) 1/2    ((∂φ^((1) ))/∂z)_0^2=1/2    (a^2 ω^2 k^2)/(k^2  sinh^2⁡kh )  sinh^2⁡kh  sin^2⁡Ψ =1/4 〖a^2 ω〗^2 -1/4 a^2 ω^2   cos⁡2Ψ (5)-1/2 ^'  k^2  ((∂φ^'(1) )/∂Ψ)_0^2=-1/2 ^'  k^2   (a^2 ω^2)/(k^2  sinh^2⁡〖kh^' 〗 )  cosh^2⁡〖kh^' 〗  cos⁡Ψ =-1/4  (〖a^2 ω〗^2 ^')/^'2 -1/4  (〖a^2 ω〗^2 ^')/^'2   cos⁡2Ψ (6)-1/2 ^'   ((∂φ^'(1) )/∂z)_0^2=-1/2 ^'   (ω^2 a^2)/(k^2  sinh^2⁡〖kh^' 〗 ) k^2  sinh⁡〖kh^' 〗  sin^2⁡Ψ =-1/4 ω^2 a^2 ^'+1/4 ω^2 a^2 ^'   cos⁡2Ψ Therefore, Right hand side of (5.7) = f ̅_2^((2)) = [-1/4 ω^2 a^2 ("" -^' )+1/4 ω^2 a^2 (/^2 -^'/^'2 ) ] +[-3/4 ω^2 a^2 ("" -^' )+1/4 ω^2 a^2  (/^2 -^'/^'2 ) ]  cos⁡2Ψ    …(5.10) The three equations (5.5), (5.6) and (5.7) therefore become as follows (〖∂φ〗^((2) )/∂z)_0-ω (∂ζ^((2)))/∂Ψ=(k 〖ω a〗^2  )/  sin⁡2Ψ                                     …(5.11) (〖∂φ〗^'(2) /∂z)_0-ω (∂ζ^((2)))/∂Ψ=-(k ω a^2)/  sin⁡2Ψ                                 …(5.12)  ∂E/∂t-^'  (∂E^')/∂t-ω ((∂φ^((2) ))/∂Ψ)_0+ω^' ((∂φ^'(2) )/∂Ψ)_0-g("" -^' )(ζ^((2))+ζ^((0)) ) =_0^((2))+_2^((2)) cos⁡2Ψ                                                      …(5.13) where  _0^((2) )=-1/4 〖 ω〗^2 a^2 ("r" -r^' )+1/4 ω^2 a^2    (/^2 -^'/^'2 ) _2^((2) )=-3/4 ω^2 a^2 ("r" -r^' )+1/4 ω^2 a^2 (r/s^2 -r^'/s^'2 )                                …(5.14) The expressions on the right hand side of equations (5.5), (5.6), (5.7) given respectively by (5.8), (5.9) and (5.10) suggests that we can take the following expression for φ^((2)),φ^('(2)) and ζ^((2)) ├ ■(φ^((2))=φ ̅^((2)) (z) sin⁡2Ψ@φ^('(2))=φ ̅^('(2)) (z)  sin⁡2Ψ@ζ^((2))=ζ ̅^((2))  cos⁡2Ψ  )}                                      …(5.15) Therefore, the equations (5.1) – (5.4), (5.11) – (5.13) give the following equations – (d^2 φ ̅^((2) ))/(dz^2 )-4k^2 φ ̅^((2))=0,((dφ ̅^((2) ))/dz)_(z=-h)=0                                             …(5.16) (d^2 φ ̅^'(2) )/(dz^2 )-4k^2 φ ̅^('(2))=0,((dφ ̅^'(2) )/dz)_(z=h^' )                                                    …(5.17) ((dφ ̅^((2) ))/dz)_0+2ωζ ̅^((2))=(kωa^2)/                                                  …(5.18) ((dφ ̅^'(2) )/dz)_0+2ω ζ ̅^((2))=-(kωa^2)/                                            …(5.19)  ∂E/∂t-^'  (∂E^')/∂t+g("" -^' ) ζ^((2))=_0^((2))                               …(5.20) -2ω (φ ̅^((2) ) )_0+2ω^' (φ ̅^'(2) )_0+g("" -^' ) ζ ̅^((2))=_2^((2))       …(5.21) where the last two equations are obtained from (5.13) by equating terms independent of Ψ and coefficient of cos⁡2Ψ on both sides. The solutions of the system (5.16) and (5.17) are φ ̅^((2))=A^((2) ) cosh⁡2k (z+h),                                 …(5.22) φ ̅^('(2))=B^((2) ) cosh⁡2k (z-h^' ),                              …(5.23) where A^((2)) and B^((2)) are two constants. substituting these solution in (5.18), (5.19) and (5.21), we get – A^((2)) k sinh⁡2kh+ω ζ ̅^((2))=(kωa^2)/2                                  …(5.24) B^((2)) k sinh⁡〖2kh^' 〗-ω ζ ̅^((2))=(kωa^2)/(2^' )                                …(5.25) -2ω A^((2)) cosh⁡2kh+2ω^'  B^((2))  cosh⁡〖2kh^'-g("r" -r^' )〗 ζ ̅^((2))=_2^((2))   …(5.26) From (5.24) and (5.25) we get A^((2))=-ω/k 1/sinh⁡2kh  ζ ̅^((2))+(ωa^2)/2  1/sinh⁡2kh                                …(5.27) B^((2))=ω/k 1/sinh⁡2kh^'  ζ ̅^((2))+(ωa^2)/(2^' sinh⁡2kh^' )                              …(5.28) Substituting these expression for A^((2)) and B^((2)) in (5.26) we get – ζ ̅^((2)) [4ω^2 (/_2 +^'/(_2^' ))-2gk("r" -r^' ) ] =2k_2^((2))+2kω^2 a^2 (r/("" _2 )-r^'/(^' _2^' )) or,ζ ̅^((2))=2k/f_2 [_2^((2))+ω^2 a^2 (r/(""  _2 )-r^'/(^' _2^' )) ]               …(5.29) where f_2=f(2ω,2k),                                                                     …(5.30) f(ω,k)≡ω^2 (r/-r^'/s^' )-gk("r" -r^' )=0                     …(5.31) being the linear dispersion relation. In the above expression _2=tanh⁡2kh,_2^'=tanh⁡〖2kh^' 〗. Therefore, from (5.24) and (5.25) we get A^((2))=-ω/k [2k/f_2 {_2^((2) )+ω^2 a^2 (r/(""  _2 )-r^'/(^' _2^' )) } ]  1/sinh⁡2kh +(ωa^2)/2  1/sinh⁡2kh   …(5.32) B^((2))=-ω/k [2k/f_2 {_2^((2) )+ω^2 a^2 (r/(""  _2 )-r^'/(^' _2^' )) } ]  1/sinh⁡〖2kh^' 〗 +(ωa^2)/(2^' )  1/sinh⁡〖2kh^' 〗   …(5.33) With these expression for A^((2)) and B^((2)) the expression for φ^((2)),φ^('(2)) as given by (5.15), (5.22), (5.23) become φ^((2))=[-2ω/f_2 Λ+(ωa^2)/2]  ( cosh⁡2k (z+h))/sinh⁡2kh   sin⁡2Ψ                        …(5.34) φ^('(2))=[2ω/f_2 Λ+(ωa^2)/(2^' )]  ( cosh⁡2k (z-h^'))/sinh⁡2kh^'   sin⁡2Ψ                        …(5.35) where Λ=_2^((2))+ω^2 a^2 (r/(""  _2 )-r^'/(^' _2^' ))                                                       …(5.35a) Also, the expression for ζ^((2)) as given by (5.29) becomes ζ^((2))=2k/f_2  Λ cos⁡2Ψ                                                 …(5.36)

6.	EQUATIONS AT THE ORDER ^3 : The order ^3 equations obtained from (3.4) – (3.10) following the same method of getting 0() and 0(^2) equation, are (∂^2 φ^((3)))/(∂z^2 )+k^2 (∂^2 φ^((3)))/(∂Ψ^2 )=0                                                     …(6.1) ((∂φ^((3) ))/∂z)_(z=-h)=0                                               …(6.2) (∂^2 φ^('(3)))/(∂z^2 )+k^2 (∂^2 φ^('(3)))/(∂Ψ^2 )=0                                                     …(6.3) ((∂φ^'(3) )/∂z)_(z=h^' )=0                                               …(6.4) (∂ζ^((0)))/∂t+((∂φ^((3) ))/∂z)_0-ω (∂ζ^((3)))/∂Ψ=-ω^((2)) (∂ζ^((1)))/∂Ψ+b ̅_3^((2))+b ̅_3^((3))                 …(6.5) (∂ζ^((0)))/∂t+((∂φ^'(3) )/∂z)_0-ω (∂ζ^((3)))/∂Ψ=-ω^((2)) (∂ζ^((1)))/∂Ψ+c ̅_3^((2))+c ̅_3^((3))                 …(6.6) -ω  ((∂φ^((3) ))/∂Ψ)_0+ω ^' ((∂φ^'(3) )/∂Ψ)_0-g("r" -r^' ) ζ^((3))=f ̅_3^((2))+f ̅_3^((3)) - ω^((2)) ((∂φ^((1)))/∂Ψ)_0+^'  ω^((2))  ((∂φ^('(1)))/∂Ψ)_0               …(6.7) Now we first evaluate different nonlinear terms on the right hand side of equations (6.5) – (6.7) (I) Nonlinear terms on the right hand side of (6.5) (i)  b ̅_3^((2))=-ζ^((0)) ((∂^2 φ^((1) ))/(∂z^2 ))_0-ζ^((1)) ((∂^2 φ^((2) ))/(∂z^2 ))_0-ζ^((2)) ((∂^2 φ^((1) ))/(∂z^2 ))_0 +k^2 ((∂φ^((1) ))/∂Ψ)_0 (∂ζ^((2)))/∂Ψ+k^2  (∂φ^((2)))/∂Ψ  (∂ζ^((1)))/∂Ψ Evaluation of different terms of b ̅_3^((2)) (1)-ζ^((0)) ((∂^2 φ^((1) ))/(∂z^2 ))_0=〖-ζ〗^((0)) {-ωak/ sin⁡Ψ } = ωak/ ζ^((0)) sin⁡Ψ (2)-ζ^((1)) ((∂^2 φ^((2) ))/(∂z^2 ))_0=-a cos⁡Ψ [-2ω/f_2 Λ+(ωa^2)/2]  (4k^2)/_2   sin⁡2Ψ =[(4ak^2 ω)/(f_2 _2 ) Λ-(a^3 ωk^2)/("" _2 )](sin⁡3Ψ+sin⁡Ψ ) (3)-ζ^((2)) ((∂^2 φ^((1) ))/(∂z^2 ))_0=-2k/f_2 Λ cos⁡2Ψ {-ωak/  sin⁡Ψ } =(ωak^2)/(f_2 " " ) Λ(sin⁡3Ψ-sin⁡Ψ ) (4) k^2 ((∂φ^((1) ))/∂Ψ)_0 (∂ζ^((2)))/∂Ψ=-(4k^3)/f_2  Λ sin⁡2Ψ {-ωa/k  cos⁡Ψ } =(2k^2 aω)/(f_2 " " ) Λ (sin⁡3Ψ+sin⁡Ψ ) (5) k^2 ((∂φ^((2) ))/∂Ψ)_0 (∂ζ^((1)))/∂Ψ=(-k^2 a sin⁡Ψ )(-2ω/f_2  Λ+(ωa^2)/2)   2cos⁡Ψ/_2 =(k^2 a)/_2 [2ω/f_2  Λ-(ωa^2)/2](sin⁡3Ψ-sin⁡Ψ ) Therefore, b ̅_3^((2))=_1 sin⁡Ψ+_3  sin⁡3Ψ                                               …(6.8) Where _1=(ωk^2 a)/f_2 (2/_2 +1/)Λ-(ωk^2 a^3)/(2" " _2 )+ωak/  ζ^((0))                       …(6.9) and _3=(3ωk^2 a)/f_2 (2/_2 +1/)Λ-(3ωk^2 a^3)/(2" " _2 )                                           …(6.10) (ii)  b ̅_3^((3))=-1/2 {ζ^((1) ) }^2 ((∂^3 φ^((1) ))/(∂z^3 ))_0+k^2 ζ^((1)) ((∂^2 φ^((1) ))/∂Ψ∂z)_0  (∂ζ^((1)))/∂Ψ =-1/2 a^2 cos^2⁡Ψ {-(ωak^3)/k  sin⁡Ψ }+k^2 a cos⁡Ψ  {–ωak/k  cos⁡Ψ }(-a sin⁡Ψ) =3/8 ωk^2 a^3 (sin⁡3Ψ+sin⁡Ψ )                                                          …(6.11) (iii)-ω^((2)) (∂ζ^((1)))/∂Ψ=aω^((2))  sin⁡Ψ                                                            …(6.11a) Therefore,right hand side of (6.5)=(_1+3/8 ωk^2 a^3+aω^((2) ) ) sin⁡Ψ +(_3+3/8 ωk^2 a^3 ) sin⁡3Ψ               …(6.12) (II) Nonlinear terms on the right hand side of (6.6) (i)  c ̅_3^((2))=-ζ^((0)) ((∂^2 φ^'(1) )/(∂z^2 ))_0-ζ^((1)) ((∂^2 φ^'(2) )/(∂z^2 ))_0-ζ^((2)) ((∂^2 φ^'(1) )/(∂z^2 ))_0 +k^2 ((∂φ^'(1) )/∂Ψ)_0 (∂ζ^((2)))/∂Ψ+k^2  (∂φ^('(2)))/∂Ψ  (∂ζ^((1)))/∂Ψ Evaluation of different terms of c ̅_3^((2)) (1)-ζ^((0)) ((∂^2 φ^'(1) )/(∂z^2 ))_0= -ωak/^' ζ^((0))  sin⁡Ψ (2)-ζ^((1)) ((∂^2 φ^'(2) )/(∂z^2 ))_0=[-(4ωk^2 a)/(f_2 _2^' ) Λ-〖ωk^2 a〗^3/(^' _2^' )](sin⁡3Ψ+sin⁡Ψ ) (3)-ζ^((2)) ((∂^2 φ^'(1) )/(∂z^2 ))_0=-(ωk^2 a)/(f_2 ^' ) Λ(sin⁡3Ψ-sin⁡Ψ ) (4) k^2 ((∂φ^'(1) )/∂Ψ)_0 (∂ζ^((2)))/∂Ψ=-(2〖ωk〗^2 a)/(f_2 ^' ) Λ (sin⁡3Ψ+sin⁡Ψ ) (5) k^2 ((∂φ^'(2) )/∂Ψ)_0 (∂ζ^((1)))/∂Ψ=[-(2ωk^2 a)/(f_2 _2^' ) Λ-(ωk^2 a^3)/(2^' _2^' )](sin⁡3Ψ-sin⁡Ψ ) Therefore, c ̅_3^((2))=_1 sin⁡Ψ+_3  sin⁡3Ψ                                               …(6.13) Where _1=-ωak/^' ζ^((0))-(2ωk^2 a)/(f_2 _2^' ) Λ-(ωk^2 a)/(f_2 ^' ) Λ-(ωk^2 a^3)/(2^' _2^' )                      …(6.14) and _3=-(6ωk^2 a)/(f_2 _2^' ) Λ-(3ωk^2 a)/(f_2 ^' ) Λ-3/2 (ωk^2 a^3)/^'                                     …(6.15) (ii)  c ̅_3^((3))=-1/2 {ζ^((1) ) }^2 ((∂^3 φ^'(1) )/(∂z^3 ))_0+k^2 ζ^((1)) ((∂^2 φ^'(1) )/∂Ψ∂z)_0  (∂ζ^((1)))/∂Ψ =-1/2 a^2 cos^2⁡Ψ (-ωak^2  sin⁡Ψ )+k^2 a cos⁡Ψ  (–ωa cos⁡Ψ )(-a sin⁡Ψ ) =3/8 ωk^2 a^3 (sin⁡3Ψ+sin⁡Ψ )                                                           …(6.16) (iii)-ω^((2)) (∂ζ^((1)))/∂Ψ=aω^((2))  sin⁡Ψ                                                            …(6.16a) Therefore,right hand side of (6.6)=(_1+3/8 ωk^2 a^3+aω^((2) ) ) sin⁡Ψ +(_3+3/8 ωk^2 a^3 ) sin⁡3Ψ               …(6.17)

III. Nonlinear terms on the right hand side of (6.7) (i) f ̅_3^((2))=ω ζ^((1))  ((∂^2 φ^((2) ))/∂z∂Ψ)_0+ω ζ^((2))  ((∂^2 φ^((1) ))/∂z∂Ψ)_0+ω ζ^((0))  ((∂^2 φ^((1) ))/∂z∂Ψ)_0 -ω^' ζ^((1))  ((∂^2 φ^'(2) )/∂z∂Ψ)_0-ω^'  ζ^((2))  ((∂^2 φ^'(1) )/∂z∂Ψ)_0-ω^'  ζ^((0))  ((∂^2 φ^((3) ))/∂z∂Ψ)_0 + k^2  ((∂φ^((1) ))/∂Ψ)_0 ((∂φ^((2) ))/∂Ψ)_0-k^2 ^' ((∂φ^'(1) )/∂Ψ)_0 ((∂φ^'(2) )/∂Ψ)_0 +   ((∂φ^((1) ))/∂z)_0 ((∂φ^((2) ))/∂z)_0-^'  ((∂φ^'(1) )/∂z)_0 ((∂φ^'(2) )/∂z)_0 (1) ω  ζ^((1))  ((∂^2 φ^((2) ))/∂z∂Ψ)_0=[- (4ω^2 ka" " )/f_2  Λ+(ω^2 〖ka〗^3 " " )/](cos⁡3Ψ+cos⁡Ψ ) (2) ω  ζ^((2))  ((∂^2 φ^((1) ))/∂z∂Ψ)_0=-  (ω^2 ka" " )/f_2  Λ(cos⁡3Ψ+cos⁡Ψ ) (3) ω  ζ^((0))  ((∂^2 φ^((1) ))/∂z∂Ψ)_0=- ω^2  a ζ^((0))  cos⁡Ψ (4)-ω ^' ζ^((1))  ((∂^2 φ^'(2) )/∂z∂Ψ)_0=[ (4ω^2 ka^')/f_2  Λ+(ω^2 〖ka〗^3 ^')/^' ](cos⁡3Ψ+cos⁡Ψ ) (5) ω ^'  ζ^((2))  ((∂^2 φ^'(1) )/∂z∂Ψ)_0=  (ω^2 ka^')/f_2  Λ(cos⁡3Ψ+cos⁡Ψ ) (6)-ω ^' ζ^((0))  ((∂^2 φ^'(1) )/∂z∂Ψ)_0=ω^2  a^'  ζ^((0))  cos⁡Ψ (7) k^2  ((∂φ^((1) ))/∂Ψ)_0 ((∂φ^((2) ))/∂Ψ)_0= [ (2ω^2 ka" " )/(f_2 " " _2 ) Λ-(ω^2 〖ka〗^3 " " )/(2" " 2_2 )](cos⁡3Ψ+cos⁡Ψ ) (8) 〖-k〗^2 ^' ((∂φ^'(1) )/∂Ψ)_0 ((∂φ^'(2) )/∂Ψ)_0= [-(2ω^2 ka^')/(f_2 ^' _2^' ) Λ-(ω^2 〖ka〗^3 ^')/(2^'2 _2^' )](cos⁡3Ψ+cos⁡Ψ ) (9) ((∂φ^((1) ))/∂z)_0 ((∂φ^((2) ))/∂z)_0=(2ω^2 ka" " )/f_2 Λ-(ω^2 ka^3 " " )/2 (cos⁡Ψ-cos⁡3Ψ ) (10)-^' ((∂φ^'(1) )/∂z)_0 ((∂φ^'(2) )/∂z)_0=[-(2ω^2 ka^')/f_2  Λ-(ω^2 ka^3 ^')/(2^' )](cos⁡Ψ-cos⁡3Ψ ) Therefore,f ̅_3^((2))=_1 cos⁡Ψ+_3  cos⁡3Ψ                                        …(6.18) where _1=-(3ω^2 ka)/f_2 Λ("" -^' )+(ω^2 ka^3)/2 (/+^'/^' ) +(2ω^2 ka)/f_2 Λ(/(""  _2 )-^'/(^' _2^' ))-(ω^2 ka^3)/2 (/(^2 _2 )+^'/(^'2 _2^' )) -ω^2 a ("r" -r^' ) ζ^((0))                                                            …(6.19) and _3=(-7ω^2 ka)/f_2 Λ("r" -r^' )+(3ω^2 ka^3)/2 (/+^'/^' ) +(2ω^2 ka)/f_2 Λ(/(""  _2 )-^'/(^' _2^' ))-(ω^2 ka^3)/2 (/(^2 _2 )+^'/(^' _2^' ))  …(6.20) (ii) f ̅_3^((3))=ω/2  {ζ^((1) ) }^2 ((∂^3 φ^((1) ))/(∂z^2 ∂Ψ))_0-ω/2 ^' {ζ^((1) ) }^2 ((∂^3 φ^'(1) )/(∂z^2 ∂Ψ))_0 +k^2  ζ^((1)) ((∂φ^((1) ))/∂Ψ)_0 ((∂^2 φ^((1) ))/∂Ψ∂z)_0 + ζ^((1)) ((∂φ^((1) ))/∂z)_0 ((∂^2 φ^((1) ))/(∂z^2 ))_0-k^2 ^'  ζ^((1))  ((∂φ^'(1) )/∂Ψ)_0 ((∂^2 φ^'(1) )/∂Ψ∂z)_0 - ^' ζ^((1))  ((∂φ^'(1) )/∂z)_0 ((∂^2 φ^'(1) )/(∂z^2 ))_0 =1/2 ωa^2  cos^2⁡Ψ {-ωak/  cos⁡Ψ }-1/2 ω^' a^2  cos^2⁡Ψ  {ωak/^'   cos⁡Ψ } + k^2  a cos⁡Ψ {-ωa/k cos⁡Ψ }{-ωa cos⁡Ψ } -^' a cos⁡Ψ {-ωa sin⁡Ψ } {ωak/^'   sin⁡Ψ } =1/2 ω^2 ka^3 (/+^'/^' )[3/2  cos⁡Ψ-1/4 (cos⁡3Ψ+cos⁡Ψ ) ] or f ̅_3^((3))=1/8 ω^2 ka^3 (r/+r^'/^' )[5 cos⁡Ψ-cos⁡3Ψ ]                                          …(6.21) (iii)- ω^((2)) ((∂φ^((1) ))/∂Ψ)_0+^' ω^((2)) ((∂φ^'(1) )/∂Ψ)_0 =- ω^((2)) (-ωa/k cos⁡Ψ )+^'  ω^((2)) (-ωa/(k^' )  cos⁡Ψ ) =(ω^((2)) ωa)/k (r/s+r^'/s^' ) cos⁡Ψ                                                                 …(6.21a) Therefore, right hand sides of (6.7) = [_1+5/8 ω^2 ka^3 (r/s+r^'/s^' )+(w^((2)) wa)/k (r/s+r^'/s^' ) ] cos⁡Ψ +[_3-1/8 ω^2 ka^3 (r/s+r^'/s^' ) ] cos3⁡Ψ                              …(6.22) Therefore, the three equations (6.5) – (6.7) assume the following forms, (∂ζ^((0)))/∂t+((∂φ^((3) ))/∂z)_0-ω (∂ζ^((3)))/∂Ψ=(_1+3/8 ωk^2 a^3+ω^((2)) a)  sin⁡Ψ +(_3+3/8 ωk^2 a^3 ) sin⁡3Ψ                                 …(6.23) (∂ζ^((0)))/∂t+((∂φ^'(3) )/∂z)_0-ω (∂ζ^((3)))/∂Ψ=(_1+3/8 ωk^2 a^3+ω^((2)) a)  sin⁡Ψ +(_1+3/8 ωk^2 a^3 ) sin⁡3Ψ                                 …(6.24) -ω  ((∂φ^((3) ))/∂Ψ)_0+ω ^' ((∂φ^'(3) )/∂Ψ)_0-g("r" -r^' ) ζ^((3)) =[_1+(5/8 ω^2 ka^3+(ω^((2)) ωa)/k)(r/s+r^'/s^' ) ] cos⁡Ψ +[_3-1/8 ω^2 ka^3 (r/s+r^'/s^' ) ] cos⁡3Ψ                              …(6.25) The right hand side of equations (6.23) – (6.25) and also those of equations and boundary conditions (6.1) – (6.4) suggest that we can assume the following forms of φ^((3)),φ^('(3)) and ζ^((3)). ├ ■(φ^((3))=φ ̅_1^((3) ) (z) sin⁡Ψ+φ ̅_3^((3) ) (z)  sin⁡3Ψ@φ^('(3))=φ ̅_1^'(3)  (z)  sin⁡Ψ+φ ̅_3^'(3)  (z)  sin⁡3Ψ@ζ^((3))=ζ ̅_1^((3) )  cos⁡Ψ+ζ ̅_3^((3) )  cos⁡3Ψ )}                       …(6.26) Substituting the above expression for φ^((3)),φ^('(3)),ζ^((3)) in the equations (6.1) – (6.4), (6.23) – (6.25) and then equating coefficients of sin⁡Ψ and sin⁡3Ψ on both sides of equations (6.1) – (6.4), (6.23) – (6.24) and coefficients of  cos⁡Ψ and  cos⁡3Ψ on both sides of (6.25), we get two sets of equations. Also equating terms independent of Ψ on both sides of equations (6.23) and (6.24) we get another set of equations. (A) Equations obtained by equating terms independent of Ψ In this case we get contributions only from the equations (6.23) and (6.24). Both the equations produce the same equation, which is the following (∂ζ^((0)))/∂t=0                                                                         …(6.27) which gives ζ^((0))=0                                                                             …(6.28) (B) Equations obtained by equating coefficients of sin⁡〖Ψ or〗 cos⁡Ψ In this case the equations obtained from equations (6.1) – (6.4) after equating coefficients of sin⁡Ψ are the following (d^2 φ ̅_1^((3)))/(dz^2 )-k^2 φ ̅_1^((3))=0                                                    …(6.28a) (dφ ̅_1^((3) ))/dz=0  when  z=-h                                              …(6.29) (d^2 φ ̅_1^'(3) )/(dz^2 )-k^2 φ ̅_1^('(3))=0                                                    …(6.30) (dφ ̅_1^'(3) )/dz=0 when z=h^'                                                  …(6.31) Similarly the equations (6.23) and (6.24) produce the following two equations after equating coefficients of sin⁡Ψ ((dφ ̅_1^((3) ))/dz)_0+ω ζ ̅^((3))=_1+3/8 ωk^2 a^3+ω^((2)) a                  …(6.32) ((dφ ̅_1^'(3) )/dz)_0+ω ζ ̅_1^((3) )=_1+3/8 ωk^2 a^3+ω^((2)) a                  …(6.33) While the equation (6.25) give the following equation after equating coefficients of cos⁡Ψ on both sides of the equation -ω  (φ ̅_1^((3) ) )_0+ω ^' (φ ̅_1^'(3) )_0-g("" -^' ) ζ ̅_1^((3) ) =_1+(5/8 ω^2 ka^3+(ω^((2)) ωa)/k)(r/s+r^'/s^' )                             …(6.34) The solutions of equations (6.28) and (6.30) satisfying the boundary conditions (6.29) and (6.30) are ├ ■(φ ̅_1^((3))=A_1^((3) ) cosh⁡k (z+h)@@φ ̅_1^('(3))=B_1^((3) )  cosh⁡k (z-h^'))}                             …(6.35) where A_1^((3) ) and B_1^((3) ) are two constants. With these solutions for φ ̅_1^((3)) and φ ̅_1^('(3))  we get the following from equations (6.32) – (6.34) A_1^((3) ) k sinh⁡kh+ω ζ ̅_1^((3) )=_1+3/8 ωk^2 a^3+ω^((2)) a                …(6.36) -B_1^((3) ) k sinh⁡〖kh^' 〗+ω ζ ̅_1^((3) )=_1+3/8 ωk^2 a^3+ω^((2)) a           …(6.37) -ω  A_1^((3) ) cosh⁡kh+ω ^'  B_1^((3) )  cosh⁡〖kh^' 〗-g("" -^' ) ζ ̅^((3)) =_1+ (5/8 ω^2 ka^3+(ω^((2)) ωa)/k)(r/s+r^'/s^' )                   …(6.38) From (6.36) and (6.37) we get ├ ■(A_1^((3) )=-ω/(k sinh⁡kh ) ζ ̅_1^((3) )+1/(k sinh⁡kh ) (_1+3/8 ωk^2 a^3+ω^((2)) a)@@B_1^((3) )=-ω/(k sinh⁡〖kh^' 〗 ) ζ ̅_1^((3) )-1/(ksinh⁡kh^' ) (_1+3/8 ωk^2 a^3+ω^((2)) a) )}     …(6.39) Substituting these expressions for A_1^((3) ) and B_1^((3) ) in (6.38) we have – -ω  cosh⁡kh [-ω/(k sinh⁡kh ) ζ ̅_1^((3) )+1/(k sinh⁡kh ) (_1+3/8 ωk^2 a^3+ω^((2)) a) ] +ω ^' cosh⁡〖kh^' 〗 [ω/(k sinh⁡〖kh^' 〗 ) ζ ̅_1^((3) )-1/(k sinh⁡〖kh^' 〗 ) (_1+3/8 ωk^2 a^3+ω^((2)) a) ] -g("" -^' ) ζ ̅_1^((3) ) = _1+(5/8 ω^2 ka^3+(ω^((2)) ωa)/k)(r/s+r^'/s^' ) or,ζ ̅_1^((3) ) [ω^2 (/+^'/^' )-gk("" -^')]=k_1+ω(("" _1)/+(^' _1)/^' ) +ω^2 k^2 a^3 (/+^'/^' )+2ω^((2)) aω(/+^'/^' )                …(6.40) Since the expression within the square bracket on the left hand side of this equation vanishes due to the linear dispersion relation (4.18), ζ ̅_1^((3) ) remains undetermined and consequently the equation (6.40) gives the following expression for w^((2)), the negative of which is the nonlinear amplitude dependent frequency shift of the interfacial wave, ω^((2))=-(k_1)/2aω (/+^'/^' )^(-1)-1/2a ((_1 " " )/+(_1 ^')/^' ) (/+^'/^' )^(-1)-1/2 ωk^2 a^2   …(6.41) Now substituting the expressions for _1,_1 and _1 given respectively by (6.19), (6.9) and (6.14), we get (1)-(k_1)/2aω=-k/2aω [-(3ω^2 ka)/f_2 Λ ("" -^' )+(ω^2 ka^3)/2 (r/s+r^'/s^' ) ┤ +(2ω^2 ka)/f_2 Λ(r/("s" _2 )+r^'/(s^' _2^' ))-├ (ω^2 ka^3)/2 (r/(s^2 _2 )+r^'/(s^'2 _2^' )) ] =(3ωk^2)/(2f_2 ) Λ("" -^' )-(ωk^2 a^2)/4 (r/s+r^'/s^' ) -(ωk^2)/f_2 Λ(r/("s" _2 )+r^'/(s^' _2^' ))+(ωk^2 a^2)/4 (r/(s^2 _2 )+r^'/(s^'2 _2^' )) and (2)-1/2a ((_1 " " )/+(_1 ^')/^' )=-1/2a [(2ωk^2 a" " )/(f_2 _2 " " ) Λ+(ωk^2 a" " )/(f_2 ^2 ) Λ┤ ├ -(ωk^2 a^3 " " )/(2^2 _2 )-(2ωk^2 a^')/(f_2 _2^' ^' ) Λ-(ωk^2 a^')/(f_2 ^'2 ) Λ-(ωk^2 a^3 ^')/(2^'2 _2 )] =-(ωk^2)/f_2 Λ(r/(_2 " s" )-r^'/(_2^' s^' ))-(ωk^2)/(2f_2 ) Λ(r/s^2 -r^'/s^'2 )+(ωk^2 a^2)/4 (r/(s^2 _2 )+r^'/(s^'2 _2^' )) Therefore,- (k_1)/2aω-1/2a ((_1 " " )/+(_1 ^')/^' ) =(3ωk^2)/(2f_2 ) Λ("" -^' )-(2ω k^2)/f_2 Λ(r/("s" _2 )-r^'/(s^' _2^' )) -(ωk^2)/(2f_2 ) Λ(r/s^2 -r^'/s^'2 )-(ωk^2 a^2)/4 (r/+r^'/^' )+(ωk^2 a^2)/2 (r/(s^2 _2 )+r^'/(s^'2 _2^' )) and hence from (6.41) we get ω^((2))=[(3ωk^2)/(2f_2 ) Λ("r" -r^' )-(2ωk^2)/f_2 Λ(r/("" _2 )-r^'/(s^' _2^' )) ┤ -(ωk^2)/(2f_2 ) Λ(r/s^2 -r^'/s^'2 )+(ωk^2 a^2)/2 (r/(s^2 _2 )-r^'/(s^2 _2^' )) (r/+r^'/^' )^(-1)-3/4 ωk^2 a^2  …(6.42) The expression for w^((2)) in absence of the upper layer, i.e., for ^'=0. For ^'=0 we get the following expression for f_2 and Λ. f_2=f(2ω,2k)= ((4ω^2)/_2 -2gk) =((4ω^2)/_2 -(2ω^2)/)=2ω^2  (2/_2 -1/),  since ω^2=gk =2ω^2  {2(1+^2 )/2-1/} Since,_2=tanh⁡2kh=2/(1+^2 ) Therefore, f_2=2ω^2 . Λ=_2^((2))+ω^2 a^2 /("" _2 )=-3/4 ω^2 a^2 +1/4 ω^2 a^2  /^2 +ω^2 a^2  /∙(1+^2)/2 =1/4 ω^2 a^2  (3/^2 -1) Now, for ^'=0, we have the following expressions for the different terms of ω^((2)) having the factor (r/+r^'/^' ). (1)   (3ωk^2)/(2f_2 ) Λ /=(3ωk^2)/2  1/(2ω^2 " " )  1/4  ω^2 a^2  (3/^2 -1) = 3/16 ωk^2 a^2 (3/^2 -1) (2) - (2ωk^2)/f_2  Λ /("" _2 )∙/=-(2ωk^2)/(2ω^2 " " )  1/4  ω^2 a^2  (3/s^2 -1)  (1+^2)/2 = 1/8 ωk^2 a^2 (1-2/^2 -3/^4 ) (3) - (ωk^2)/〖2f〗_2  Λ(/^2 )  /=-(ωk^2)/2  1/(2ω^2 " " )   1/4 ω^2 a^2  (3/s^2 -1)  1/ = 1/16.k ωk^2 a^2 (1-2/^2 -3/^4 ) (4) ωk^2 a^2  /(^2 _2 )∙/=1/2 ωk^2 a^2  1/   (1+^2)/2 = 1/4 ωk^2 a^2 (1+1/^2 ) Therefore, for ^'=0, the expression for ω^((2)) given by (6.42) becomes ω^((2))=1/16 ωk^2 a^2 ├ [9/^2 ┤-3+2-4/^2 -6/^4 +1/^2 -3/^4 +4+4/^2 -12] or,ω^((2))=-(ωk^2 a^2)/(16^4 ) (9^4-10^2+9) Therefore, the frequency shift =-ω^((2))=∆ω becomes ∆ω=(ωk^2 a^2)/(16^4 ) (9^4-10^2+9)                                       …(6.43) which is same as that obtained by Whitham (1967, 1974). For infinite depth water, i.e. for →1 the above expression for frequency shift becomes ∆ω=1/2 ωk^2 a^2                                                                       …(6.44) This is the well-known Stores’ result.

7.	SUMMARY OF THE RESULTS AND CONCLUSION : The main aim of the present project is to get Stokes’ wave solution for the perturbed elevation ζ of the interface of two superposed fluid layers of finite thicknesses due to the propagation of a gravity wave in the following form like that obtained by Stokes (1847) given in the section – 1 of this project, which is for a surface gravity wave in deep water ζ= ϵ  cos⁡(kx-Ωt)+ϵ^2 ζ ̅^((2)) cos⁡2(kx-Ωt),                   …(7.1) where       Ω=ω-ϵ^2 ω^((2))                                                                               …(7.2) These are obtained correct to 0(ϵ^2 ) terms. ζ ̅^((2)) is obtained from order ϵ^2 equations, but to get ω^((2)) we are to go to the 0(ϵ^3 ) equations. The expressions for ζ ̅^((2)) and w^((2)) as given by equations (5.29) and (6.41) respectively assume the following forms after simplification. ζ ̅^((2))=(ka^2)/4 ["" (3/^2 -1)-^' (3/^'2 -1) ] ("" +^' ^' )^(-1),          …(7.3) ω^((2) )=1/8 ωk^2 a^2 [{"" (3/s^2 -1)-^' (3/s^'2 -1) } ┤ {""  (1/2-3/(2^2 ))-^'  (1/2-3/(2^'2 )) } ("" ^'+^' ^' ) (/+^'/^' )^(-1)                                     +{2/ (1/^2 +1)-(2^')/^'  (1/^'2 +1) } (/+^'/^' )^(-1) ├ ■(@-6@)]               …(7.4) The technique developed by Montgomery and Tidman (1964) has been used to obtain the above results. The Stokes’ wave solution for surface gravity wave has recently found importance in the investigation of the formation of extra-ordinarily large amplitude localized water surface excitation known as rogue waves. This is produced due to Benjamin – Feir (1967) instability to two Stokes wave [Onorato at al. (2006)].

8.	REFERENCES : Bengamin, T. B. and Feir, J. E. (1967), J. Fluid. Mech. 27, 417. Bogoliubov, N. and MItroplosky, Y. A. (1961). Asymptotic Methods in the Theory of Nonlinear Oscillations, New York : Gordon and Breach. Das, K. P. (1968), Phys. Fluids 11, 2055. (1971), Phys. Fluids 14, 2235. Krylov, N. and Bogoliubov, N. (1947). Introduction to Nonlinear Mechanics, Priceton University Press. Montgomery, D. and Tidman, D. A. (1964) Physc. Fluids, 7, 242. Ontarato, M. et al, (2006) Phys. Rev. Lett. 96, 014503. Sluijter, F. W. and Montgomery, D. (1965) Phys. Fluids, 8, 551. Stokes, G. G. (1847) Camb. Trans., 8, 441. Whitham, G. B. (1974). Linear and Nonlinear Waves ; John Wiley & Sons. (1967) J. Fluid. Mech., 27, 399.