Interfacial waves with free-surface boundary conditions: an approach via a model equation

Physica D 150 (2001) 278–300

Interfacial waves with free-surface boundary conditions:an approach via a model equation

F. Diasa,∗, A. Il’ichev b

a Centre de Mathématiques et de Leurs Applications, Ecole Normale Supérieure de Cachan,61 avenue du Président Wilson, 94235 Cachan cedex, France

b Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, 117966 Moscow, Russia

Received 18 September 2000; accepted 3 January 2001Communicated by C.K.R.T. Jones

Abstract

In a two-fluid system where the lower fluid is bounded below by a rigid bottom and the upper fluid is bounded above bya free surface, two kinds of solitary waves can propagate along the interface and the free surface: classical solitary wavescharacterized by a solitary pulse or generalized solitary waves with nondecaying oscillations in their tails in addition to thesolitary pulse. The classical solitary waves move faster than the generalized solitary waves. The origin of the nonlocal solitarywaves can be understood from a physical point of view. The dispersion relation for the above system shows that short wavescan propagate at the same speed as a “slow” solitary wave. The interaction between the solitary wave and the short wavescreates a nonlocal solitary wave. In this paper, the interfacial-wave problem is reduced to a system of ordinary differentialequations by using a classical perturbation method, which takes into consideration the possible resonance between short wavesand “slow” solitary waves. In the past, classical Korteweg–de Vries type models have been derived but cannot deal with theresonance. All solutions of the new system of model equations, including classical as well as generalized solitary waves, areconstructed. The domain of validity of the model is discussed as well. It is also shown that fronts connecting two conjugatestates cannot occur for “fast” waves. For “slow” waves, fronts exist but they have ripples in their tails. © 2001 Elsevier ScienceB.V. All rights reserved.

Keywords:Interfacial waves; Dynamical systems; Solitary wave; Two-fluid system

1. Introduction

When one considers dispersive waves, i.e. waves for which the phase velocityc depends on the wavenumberk, oneexpects solitary waves in the limitk → 0. But it might happen that the linear phase velocity of a periodic wave withwavenumberk 6= 0 be equal toc(0). In that case, a resonance can occur between the solitary wave and the periodicwave. The result is a generalized solitary wave, characterized by ripples in the far field in addition to the solitarypulse. This situation occurs for example in the context of capillary-gravity surface waves. Various mathematicaland numerical results were obtained in the framework of the full water-wave problem in [1–6]. The classicalKorteweg–de Vries (KdV) equation, which is a model for long water waves, does not exhibit generalized solitary

∗ Corresponding author. Fax:+33-14740-5901.E-mail address:[email protected] (F. Dias).

0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0167-2789(01)00149-X

F. Dias, A. Il’ichev / Physica D 150 (2001) 278–300 279

waves. However, the fifth-order KdV equation, which is valid when the Bond number is close to13, admits solutions

in the form of generalized solitary waves and a number of papers have been written on it (see for example [7]).Generalized solitary waves can also occur in the context of continuously stratified fluids, even in the absence of

surface tension. Akylas and Grimshaw [8] computed generalized solitary waves by using a singular-perturbationprocedure. Vanden-Broeck and Turner [9] studied numerically long waves propagating in a channel bounded aboveand below by horizontal walls, the fluid consisting of two layers of constant densities separated by a region in whichthe density varies continuously.

The present paper is devoted to two-layer systems in which the upper, lighter fluid is bounded above by a freesurface. Peters and Stoker [10] used a classical perturbation method to derive a KdV equation and studied itssolitary wave solutions. A similar approach was followed in [11]. In all these papers, the asymptotic expansionswhich are used do not lead to generalized solitary waves. To our knowledge, Sun and Shen [12] were the firstto give a theoretical proof of the existence of generalized solitary waves in a two-fluid system in the absence ofcapillarity effects at the interface. The main purpose of the present paper is to provide a system of model equationswith the same properties as the full equations. This system is roughly a Korteweg–de Vries type equation coupledwith an oscillator. Numerical results on the full equations have been given in [13] for classical solitary waves, in[14] for generalized solitary waves and in [15] for periodic waves. Experiments have been performed in [16] andin [17].

At a critical thickness ratio, the coefficient of the quadratic nonlinearity in the model vanishes and the cubicnonlinearity becomes dominant [11]. We will show that our model with cubic nonlinearity admits solutions in theform of generalized fronts, i.e. fronts with ripples in their tails.

The outline of the paper is as follows. In Section 2, the problem is formulated and the model is derived. In Section3, travelling wave solutions of the model are studied. In Section 4, travelling wave solutions of the modified modelincluding cubic nonlinearity are studied. Section 5 provides a discussion on the results. In Appendix A, conjugateflows are considered. It is shown that fronts connecting two conjugate states cannot occur for “fast” waves. InAppendix B, a new integral relation for “fast” solitary waves is derived.

2. Formulation of the problem

The geometry of the problem is as follows: a layer of a lighter fluid rests on top of a layer of a heavier fluid. Theheavy fluid is bounded below by a rigid horizontal bottom. The lighter fluid is bounded above by a free surface. All

Table 1Physical parameters and their dimension

Symbol Physical quantity Dimension

c Wave velocity [L][T]−1

g Acceleration due to gravity [L][T]−2

hj Depth of the layerj ; j = 1: bottom layer,j = 2: top layer [L]ρj Density of the fluid in the layerj [M][L] −3

k Wavenumber [L]−1

ω Wave frequency [T]−1

(x∗, y∗) Physical coordinates [L](u∗j , v

∗j ) Velocity components in layerj [L][T] −1

ϕ∗j (x, y) Velocity potential:(u∗

j , v∗j ) = ∇ϕ∗

j [L] 2[T]−1

η∗1(x

∗, t∗) Profile of the interface [L]η∗

2(x∗, t∗) Profile of the free surface [L]

λ Characteristic wave length [L]a Characteristic wave amplitude [L]

280 F. Dias, A. Il’ichev / Physica D 150 (2001) 278–300

Table 2Dimensionless quantities

Symbol Definition Dimensionless quantity

K kh1 Dimensionless wavenumberKres (kh1)res Dimensionless resonant wavenumber

Ωωh1

cDimensionless wave frequency

Hh2

h1Depth ratio

Hjhj

h1 + h2Relative depth of layerj

Rρ2

ρ1Density ratio

C2 c2

gh1Square of Froude number based on bottom layer depth

xx∗

λDimensionless horizontal coordinate

ξ(x∗ − ct∗)(1 − R)−1/2

h1Dimensionless horizontal coordinate in moving frame after (2.12)

yy∗

h1Dimensionless vertical coordinate

tt∗(gh1)

1/2

λDimensionless time

η1(x, t)η∗

1

aDimensionless profile of the interface before (2.12)

η1(x, t)η∗

1

h1Dimensionless profile of the interface after (2.12)

η2(x, t)η∗

2

aDimensionless profile of the free surface before (2.12)

η2(x, t)η∗

2

h1Dimensionless profile of the free surface after (2.12)

uj u∗j (gh1)

−1/2 Dimensionless horizontal velocity in layerj after (2.12)

ϕjϕ∗j (gh1)

−1/2

gaλDimensionless potential

quantities related to the upper fluid layer have the index 2, while those related to the lower layer are indexed with1. All the physical parameters as well as the dimensionless numbers are provided in Tables 1 and 2. Thex∗-axiscoincides with the interface at rest. They∗-axis is the vertical axis.

The governing equations are

0=(∂2

∂x∗2+ ∂2

∂y∗2

)ϕ∗i , i = 1,2, −h1 < y∗ < h2 + η∗

2,

0= ∂η∗2

∂t∗+ ∂η∗

2

∂x∗∂ϕ∗

2

∂x∗ − ∂ϕ∗2

∂y∗ , y∗ = h2 + η∗2,

0= ∂ϕ∗2

∂t∗+ 1

2

[(∂ϕ∗

2

∂x∗

)2

+(∂ϕ∗

2

∂y∗

)2]

+ gη∗2, y∗ = h2 + η∗

2,

0= ∂η∗1

∂t∗+ ∂η∗

1

∂x∗∂ϕ∗

i

∂x∗ − ∂ϕ∗i

∂y∗ , i = 1,2, y∗ = η∗1,


0= ρ1∂ϕ∗

1

∂t∗− ρ2

∂ϕ∗2

∂t∗+ 1

2ρ1

[(∂ϕ∗

1

∂x∗

)2

+(∂ϕ∗

1

∂y∗

)2]

− 1

2ρ2

[(∂ϕ∗

2

∂x∗

)2

+(∂ϕ∗

2

∂y∗

)2]

+ (ρ1 − ρ2)gη∗1,

y∗ = η∗1,

0= ∂ϕ∗1

∂y∗ , y∗ = −h1. (2.1)

The dispersion relation for periodic waves,

(1 + R tanhkh1 tanhkh2)c4 − g

k(tanhkh1 + tanhkh2)c

2 + g2

k2(1 − R) tanhkh1 tanhkh2 = 0, (2.2)

consists of two branches, corresponding to the so-called surface and interfacial modes. Note that in the limitR → 0,the dispersion relation (2.2) simply is(

c2 − g

ktanhkh1

) (c2 − g

ktanhkh2

)= 0.

In dimensionless form, the relation (2.2) becomes

(1 + R tanhK tanhKH)C4 − 1

K(tanhK + tanhKH)C2 + 1

K2(1 − R) tanhK tanhKH = 0. (2.3)

The limitK → 0 gives the relation satisfied by the solitary wave speedsC+ (“fast”) andC− (“slow”):

C4± − (1 +H)C2

± +H(1 − R) = 0. (2.4)

The larger root of (2.4),C+, satisfiesC2+ > 1 andC2+ > H whileC− satisfiesC2− < 1 − R andC2− < H(1 − R).The interfacial mode corresponding to the speedC− resonates with a surface mode of wavenumberKresbelongingto the branch originating atC+. So one expects the appearance of oscillating waves in the tail of the solitary wave.In [14], various configurations were explored to check that this resonance can be satisfied in physical conditions.In the limit R → 0, the two speeds areC2± = (1, H). In the limitR → 1, the two speeds areC2± = (0,1 + H).In oceanic conditions(R → 1), the resonant wavenumberKres is rather large but the corresponding wave lengthremains physical (of the order of 1 m). A discussion is provided in Section 5.

The full dispersion relation as well as the dispersion curve obtained from our model (see next section) are shownin Fig. 1 for two sets of parameter values:H = 0.4,R = 0.8 andH = 1,R = 0.1. A density ratio of 0.8 is typicalin laboratory experiments (see [16,17]). The validity of the model is discussed in Section 5. We insist here on thefact that our “model” dispersion relation is different from the order 2 dispersion relation that would be obtained byexpanding for smallK the dispersion relation (2.3):

(1 + RK2H)C4 − [1 +H − 13K

2(1 +H 3)]C2 + (1 − R)H [1 − 13K

2(1 +H 2)] = 0. (2.5)

As a matter of fact, asK → +∞ in (2.5), the speeds satisfy

RHC4 + 13(1 +H 3)C2 − 1

3(1 − R)H(1 +H 2) = 0.

It can be shown that the speedC(K → ∞) of the upper branch of (2.5) is above the speedC− for slow solitarywaves. SinceC decreases monotonically withK for both branches of (2.5), (2.5) cannot lead to any resonancebetween the slow solitary waves and periodic waves.1

1 We were not able to prove this result mathematically, but for all parameter values we tested numerically, it turned out to be true.


Fig. 1. Dispersion curves for the full problem (dotted line) and for the model (solid line) for two sets of parameter values. ForR = 0.8,H = 0.4,the resonant wavenumber isK = 16.8 for the full problem andK = 3.1 for the model. ForR = 0.1,H = 1,Kfull = 1.4 andKmodel = 1.7.

Eqs. (2.1) written in dimensionless variables become

0=(µ∂2

∂x2+ ∂2

∂y2

)ϕi, i = 1,2, −1< y < H + εη2,

0= ∂η2

∂t+ ε

∂η2

∂x

∂ϕ2

∂x− 1

µ

∂ϕ2

∂y, y = H + εη2,

0= ∂ϕ2

∂t+ ε

2

[(∂ϕ2

∂x

)2

+ 1

µ

(∂ϕ2

∂y

)2]

+ η2, y = H + εη2,

0= ∂η1

∂t+ ε

∂η1

∂x

∂ϕi

∂x− 1

µ

∂ϕi

∂y, i = 1,2, y = εη1,

0= ∂ϕ1

∂t− R

∂ϕ2

∂t+ ε

2

[(∂ϕ1

∂x

)2

+ 1

µ

(∂ϕ1

∂y

)2]

− εR

2

[(∂ϕ2

∂x

)2

+ 1

µ

(∂ϕ2

∂y

)2]

+(1 − R)η1, y = εη1,

0= ∂ϕ1

∂y, y = −1, (2.6)

where

µ =(h1

λ

)2

, ε = a

h1,

are the usual small parameters in the modeling of long waves. These parameters control dispersive and nonlineareffects, respectively. It is assumed that

ε, µ 1.

Of course, these assumptions are essential in the analysis. Their validity will be discussed in Section 5. In particular,we will argue that although the wavenumber(kh1)res of the short wave in resonance with the long wave is of order1 or higher (see Fig. 1), the model still makes sense, at least qualitatively.


To obtain the model equations for long waves of small amplitude we use the expansion at the pointy = 0:

ϕi = ϕ0i + yϕ0

i,y + 12y

2ϕ0i,yy + 1

6y3ϕ0i,yyy + 1

24y4ϕ0i,yyyy+ 1

120y5ϕ0i,yyyyy+ · · · . (2.7)

Here the subscripty denotes differentiation with respect to this variable. From the first equations in (2.6) we obtain

ϕ0i,yy = −µϕ0

i,xx, ϕ0i,yyy = −µ(ϕ0

i,y)xx, ϕ0i,yyyy = µ2ϕ0

i,xxxx, ϕ0i,yyyyy= µ2(ϕ0

i,y)xxxx, i = 1,2.

Using the last equation in (2.6) (boundary condition at the bottom), (2.7), and applying the expansion seriesmethod for smallµ up to (but not including) terms of orderµ3, we obtain the following expression forϕ0

1,y :

ϕ01,y = −µϕ0

1,xx − 13µ

2ϕ01,xxxx. (2.8)

Then using (2.7), the second equation in (2.6) and applying the same procedure, we obtain forϕ02,y , keeping terms

in µ, µε andµ2:

ϕ02,y = µη2,t +Hµϕ0

2,xx + µε[η2ϕ02,x ]x + 1

2(µ2H 2)η2,txx + 1

3(µ2H 3)ϕ0

2,xxxx. (2.9)

Next, substituting relations (2.8) and (2.9) into the third, fourth (fori = 1,2) and fifth equations in (2.6) andneglecting terms of order o(µ, ε) yields (dropping the superscript0)

0= η1,t + εη1,xϕ1,x + εη1ϕ1,xx + ϕ1,xx + 13µϕ1,xxxx,

0= η2,t − η1,t + εη2,xϕ2,x + εη2ϕ2,xx +Hϕ2,xx − εη1,xϕ2,x − εη1ϕ2,xx + µ12H

2η2,txx + µ13H

3ϕ2,xxxx,

0= ϕ2,t + µHη2,tt + µ12H

2ϕ2,txx + 12εϕ

22,x + η2,

0=Rϕ2,t − ϕ1,t + 12εR(ϕ2,x)

2 − 12ε(ϕ1,x)

2 − (1 − R)η1. (2.10)

The system (2.10) is ill-posed. Of course, this is an important issue when studying the stability of waves. However,there are tricks to transform the system into a well-posed one. In this paper, since we are only interested in travellingwave solutions, we keep (2.10) as it is. Making the change of variablesξ = x − Ct, integrating the first and secondequations in (2.10) and eliminatingηi from the last ones yields at order O(µ, ε2)

µu1 = 3

(C2

1 − R− 1

)u1 − 3C2R

1 − Ru2 − 9

2(1 − R)εCu2

1 + 3

2(1 − R)εCRu2

2 + 3R

1 − RεCu1u2

+ 3ε2

2(1 − R)u3

1 − 3ε2R

2(1 − R)u1u

22,

µu2 = 1

κ

− C2

1 − Ru1 +

(C2

1 − R−H

)u2 + 1

2(1 − R)εCu2

1 − 3

2(1 − R)εCu2

2 + 1

1 − RεCu1u2

+ ε2

2(1 − R)u3

2 − ε2

2(1 − R)u2

1u2

, (2.11)

where the dots denote differentiation with respect toξ , ui = ϕi are the corresponding velocities andκ = H(C4 −HC2 + 1

3H2) > 0. The profiles of the interface and the free surface, at the required order inµ andε, are given by

η1 = C

1 − Ru1 − RC

1 − Ru2 − ε

2(1 − R)u2

1 + εR

2(1 − R)u2

2, η2 = Cu2 − µCH(C2 − 12H)u2 − 1

2εu22.

(2.12)

Making in (2.11) the further change of variables

ξ =√(1 − R)µξnew, ui new = εui, ηi new = εηi, i = 1,2,


and dropping the subscript “new”, we obtain finally

u1 = −α1(C)u1 + α2(C)u2 − 9

2Cu2

1 + 3

2CRu2

2 + 3CRu1u2 + 3

2u3

1 − 3

2Ru1u

22,

u2 = γ1(C)u1 + γ2(C)u2 + C

2κu2

1 − 3

2κCu2

2 + C

κu1u2 + 1

2κu3

2 − 1

2κu2

1u2,

α1(C) = −3(C2 − 1 + R), α2(C) = −3RC2, γ1(C) = −C2

κ, γ2(C) = C2 −H(1 − R)

κ. (2.13)

Below we will use the notationsαi(C±) = αi , γi(C±) = γi , i = 1,2. Note thatα1(C−) > 0, γ1(C−) < 0,α2(C−) < 0, γ2(C−) < 0, whileα1(C+) < 0, γ1(C+) < 0,α2(C+) < 0, γ2(C+) > 0.

Recall that the equivalent of (2.13) in physical variables is obtained by the change of variables

u∗i =

√gh1ui, x∗ − ct∗ = h1

√1 − Rξ, η∗

i = h1ηi. (2.14)

3. Travelling wave solutions

We consider the travelling wave solutions of (2.13) forC = C± + ν, whereν is a small positive number. Thesystem of equations (2.13) may be written in the form of a spatial “dynamical” system

w = Aw + F(ν,w), (3.1)

wherew = u1, u2, u1, u2T, F denotes a nonlinear function of its arguments:F(0,0) = 0, ∂wF(0,0) = 0, and

A =

0 0 1 00 0 0 1

−α1 α2 0 0γ1 γ2 0 0

.

The system (3.1) is reversible, i.e.AR = −RA, F(ν,Rw) = −RF(ν,w), whereR = diag−1,−1,1,1. Thereversibility means that the set of solutions of (3.1) should contain even functionsu1 andu2.

The eigenvaluesσ of A satisfy

σ 4 + (α1 − γ2)σ2 − γ1α2 − γ2α1 = 0. (3.2)

This dispersion relation is different from the one that one would obtain from the system of equations (2.10). Thisdifference comes from the fact that our model equation was obtained from (2.10) by eliminatingη1 andη2. Ofcourse, this elimination is made asymptotically, i.e. in the long wave limit. Therefore, the linear part of (2.10)is disturbed further. The only part which coincides between (2.10) and our model is the “long-wave part” of thedispersion relation (up to terms in the square of the wavenumber). The dispersion relation (2.3) has been plotted inFig. 1 and can be compared with the full dispersion relation (2.3). Of course the agreement is good for small valuesof σ = iK (see discussion in Section 5). Settingγ1α2+γ2α1 equal to zero recovers Eq. (2.4) forC±. The coefficientof σ 2, α1 − γ2, is positive whenC = C− and negative whenC = C+. Consequently, whenC = C+ Eq. (3.2) hastwo roots (double zero) lying on the imaginary axis, while whenC = C− it has four roots: double zero and±iq,q > 0. It is easy to verify that forC = C+ +ν with ν > 0, there are two real eigenvalues coming from zero atν = 0and the other two eigenvalues are also real. ForC = C− + ν with ν > 0, there are two real eigenvalues comingfrom zero atν = 0 and the other pair is imaginary. Therefore, one should expect pure solitary wave solutions of(2.13) in a vicinity ofC+, and solitary waves with ripples in a vicinity ofC−. Solitary waves and solitary waves with


ripples may exist forν > 0. The properties of solitary wave-type solutions of (2.13) being different forC = C− +νandC = C+ + ν, these two cases are considered separately. In this section we shall consider the case where thethird-order nonlinearity is dominated by the quadratic one, and thus neglect the cubic terms in (2.13). As is wellknown in the case of capillary–gravity surface waves, there are two types of generalized solitary waves: solitarywaves with exponentially small ripples, and solitary waves with ripples of algebraic size. Below we will distinguishbetween these two cases by calling solitary waves with exponentially small ripples generalized solitary waves andsolitary waves with ripples all the others.

3.1. Generalized solitary waves

Throughout this subsection the wave speed is slightly aboveC−: C = C− + ν, with ν > 0. In (3.1) we replacew by

w = a0φ0 + a1φ1 + a+φ+ + a−φ−, a+ = a−, φ+ = φ−,

where the bar denotes complex conjugation, andAφ0 = 0, Aφ1 = φ0, Aφ+ = iqφ+. The eigenvectors andgeneralized eigenvectors are given by

φ0 =

−γ2

γ1

1

0

0

, φ1 =

0

0

−γ2

γ1

1

, φ+ =

−α1

γ1

1

− iqα1

γ1iq

.

The eigenvectors and generalized eigenvectors of the adjointA∗ are given by

ψ1 = 1

q2

0

0

γ1

α1

, ψ0 = 1

q2

γ1

α1

0

0

, ψ+ = − 1

2q3

γ1q

γ2q

iγ1

iγ2

.

Recall thatq2 = α1 − γ2. The velocities are given by the expressions

u1 = −γ2

γ1a0 − α1

γ1(a+ + a−), u2 = a0 + a+ + a−, (3.3)

and at leading order

η1 = − C

1 − R

(R + γ2

γ1

)a0, η2 = Ca0.

The nonlinearityF can be decomposed as

F = Fs(a0)+ Fz(a0, a+, a−),


with

Fs =

0

0(α2(C)+ γ2

γ1α1(C)

)a0 + C−

(3

2R − 9

2

γ 22

γ 21

− 3Rγ2

γ1

)a2

0

(γ2(C)− γ2

γ1γ1(C)

)a0 + C−

κ

(−3

2− γ2

γ1+ 1

2

γ 22

γ 21

)a2

0

andFz(a0,0,0) = 0.In the new dependent variablesa0, a1, a±, the Eqs. (2.13) read

a0 = a1, a1 = fs1 + 〈Fz, ψ1〉, a+ = iqa+ + ifs2 + 〈Fz, ψ+〉,a− = −iqa− − ifs2 + 〈Fz, ψ−〉, (3.4)

where〈·, ·〉 denotes the scalar product inC4, and the real polynomialsfs1 = 〈Fs , ψ1〉, fs2 = −i〈Fs , ψ+〉 are givenby

fs1 = νda0 −Λa20, fs2 = − γ2

2q3

[−r1(C)+ α1

α2r2(C)

]a0 +

[−s1 + α1

α2s2

]a2

0

,

where

d = 6(1 − R)

C3−

C2−(1 +H)− 2H(1 − R)

C2−(1 +H 2 + 3RH)−H(1 +H), Λ = 9(1 − R)

2C−C4− + C2−(1 − 2H)+H 2 − 1

C2−(1 +H 2 + 3RH)−H(1 +H),

(3.5)

r1(C)=(γ2(C)− γ2γ1(C)

γ1

), s1 = C−

κ

(−3

2+ γ 2

2

2γ 21

− γ2

γ1

), r2(C) = α2(C)+ γ2

γ1α1(C),

s2 =C−

(3

2R − 9

2

γ 22

γ 21

− 3Rγ2

γ1

).

It can be shown that the coefficientd(R,H) is always positive. The presence of the(1−R) factor in the coefficientsd andΛ comes from the change of variables introduced at the end of Section 2. Using the usual scaling in the system(3.4) (see for example [3]), one can deduce thata0 = O(ν), a± = O(ν2) and, therefore,Fz is of order higher thanν2.

The velocities of the solitary waves are given by

u1 = −3γ2

2γ1

νd

Λsech2

√νd

2ξ + O(ν2), u2 = 3

2

νd

Λsech2

√νd

2ξ + O(ν2). (3.6)

The profiles of the solitary waves are given by

η1 = −3

2

νd

Λ

(H − C2

C

)sech2

√νd

2ξ + O(ν2), η2 = 3

2

νd

ΛC sech2

√νd

2ξ + O(ν2). (3.7)

These velocities and profiles agree with the results in [10]. Expressions in physical variables can be obtained from(2.14). The interface is of depression forΛ > 0 and of elevation forΛ < 0. For the free surface, it is the opposite.


From (3.5) we deduce that for small 1− R

Λ ∼ 9√

1 +H(1 −H)

2H√H

√1 − R.

Small values of 1−R are typical for stratified fluids in oceans and lakes. ForH < 1, orh2 < h1, the surface waveis of elevation and the internal wave is of depression, and vice versa forH > 1, orh2 > h1. As follows from (3.7)the internal solitary wave is dominant for small values of 1− R, namelyη1 = (1 − R)−1O(η2).

The expressions in (3.6) give the principal part of the solitary wave solution of (3.4). Higher order terms in (3.4)lead to nondecreasing periodic oscillations at infinity on top of the solitary wave (3.6). Sinceα1 > 0 andγ1 < 0 forslow solitary waves, (3.3) shows that the oscillations in the horizontal velocity along the interface and along the freesurface are in phase at infinity. The oscillating component of the solitary waves would disappear if the coefficientsof a0 anda2

0 in f2s in (3.4) were equal to zero. In this casea+ = 0 and one would have a pure solitary wave. Yet, ournumerical computations show that the coefficient ofa2

0 in fs2 is always different from zero forR < 1 andH > 0.Let us consider the local structure of the solitary wave solution (3.6) in the wavenumber domain. Dropping the

higher order terms, we rewrite the last two equations in (3.4) in the form:

La+ = if2s(a∗0), La− = −if2s(a

∗0), L = ∂

∂ξ− iq, (3.8)

wherea∗0 coincides with the lowest order term in the expression foru2 in (3.6). Making use of the Fourier transform

a± = 1

2π

∫ ∞

−∞a± exp(−ikξ)dξ,

one gets from (3.8)

a+ = − f2s

k + q, a− = f2s

k − q, (3.9)

where

f2s = (b1k + b2k3)csch

(πk√νd

). (3.10)

The precise values of the constantsb1 = O(1) andb2 = O(1) can be obtained from the expression forf2s (see forexample [18]), but they are not important for our analysis. From (3.3) and (3.9), it follows that at lowest order inν

u1 = − A

q2 − k2

α1

γ1exp

(− πq√

νd

), u2 = A

q2 − k2exp

(− πq√

νd

)as k → ±q. (3.11)

The explicit expression forA can only be computed from the full system (3.4), because the higher order terms inν

also contribute to the magnitude ofA. In other words, instead of the polynomial ink in the expression (3.10) forf2s , the infinite series ink should be used. But since we are only interested in the qualitative behavior ofu1,2 atinfinity, we omit this analysis here. We are looking for even solutionsu1, u2 of (3.1) which are consistent with thereversibility property. Namely, the generalized solitary wave under analysis has its principal part given by (3.6) andthe same asymptotics at both infinities. We have

u1,2 = 1

2

∫Γ=Γ1∪Γ2

u1,2 exp(ikξ)dk, (3.12)


where the contourΓ1 lies above the poles ofu1,2 on the real axis and the contourΓ2 below. The contoursΓ1,2

contribute to the asymptotics at minus and plus infinities correspondingly. From (3.11) and (3.12), it follows that

u1 → ±π Aq

α1

γ1exp

(− πq√

νd

)sin(qξ), u2 → ∓π A

qexp

(− πq√

νd

)sin(qξ) as ξ → ±∞. (3.13)

The asymptotics (3.13) gives an exponentially small oscillatory tail on top of the solitary wave pulse (3.6) at infinity.

3.2. Classical solitary waves

ForC equal toC+, Eq. (3.2) has only a double zero root lying on the imaginary axis. The matricesA andA∗

have two generalized eigenvectorsφ0, φ1 andψ0, ψ1 as given above. LetC = C+ + ν with ν > 0. According tothe center-manifold reduction theorem [19], one has

w = a0(ξ)φ0 + a1(ξ)φ1 + h(ν, a0, a1),

whereh(ν, a0, a1) is a nonlinear function of its arguments. The unknownsa0 anda1 satisfy

a0 = a1, a1 = νda0 −Λa20 + o(|νa| + |a|2), (3.14)

wherea = a0, a1T.In this case the expressions for d andΛ in (3.14) are also given by (3.5), except that one should replaceC− by

C+. Keeping this in mind one deduces that for small 1− R

Λ ∼ 9

2(1 +H)2√

1 +H(1 − R).

As in the previous case, the profiles of the solitary waves are given by (3.7), but now, both the surface and internalwaves are waves of elevation (C2+ > H , see also [20]). In the limith2 → 0, the amplitude of the internal wavetends to the amplitude of the surface wave. The solution (3.6) now persists for the full system (3.14) (for a generaltheorem, see for example Section 3.1 in [3]). In other words, it gives a pure solitary wave. The terms of higher orderin ν also decay exponentially at both infinities.

3.3. Solitary waves with ripples, periodic and quasi-periodic waves

We now go back to values of the speed slightly aboveC−, as in Section 3.1;C = C− + ν, with ν > 0. Thedifference between this subsection and Section 3.1 is the following. We are now looking for solutions of the coupledsystema0, a1, a+, a− with a+ of orderν. The scaling will become evident below. In Section 3.1, we were lookingfor solutions witha+ of orderν2. To make the analysis more complete we put the system (3.4) into normal form.The normal form of (3.4) in the space of polynomials in(ν,a), wherea is the vector functiona = a0, a1, a+, a−T,is constructed using a characterization of Elphick et al. [21]. We use here a slightly modified version valid forparameter dependent systems, which has been given in [19, Theorem I.20].

For the flow of the type (3.4) the normal form is given by

∂ξa0 = a1, ∂ξ a+ = iqa+ + ia+Ψ (µ, a0, |a+|2), ∂ξ a1 = Φ(µ, a0, |a+|2),∂ξ a− = −iqa− − ia−Ψ (µ, a0, |a+|2), (3.15)

whereΦ andΨ are real polynomials to the given ordern

Φ =n∑i=1

ci(ν)ai0 − d2(ν)|a+|2 + · · · , Ψ = g0(ν)+ g1(ν)a0 + · · · .


The new coordinates are still denoted bya0, a1, a+ anda−. The system (3.15) has two integralsK0 = a+a− andH0 = a2

1 − Φ, ∂a0Φ = 2Φ. The coefficientsc1, c2 andd2 are found to be

c1(ν)= νd + O(ν2), c2(ν) = −Λ+ O(ν),

d2(ν)= − C−γ 2

1 κq2[α3

1 − α21γ1(2 + 9κ)+ 3γ 3

1 κR − 3α1γ21 (1 + 2κR)] + O(ν). (3.16)

Settinga+ = r exp(iθ), one obtains from (3.15) and (3.16) via the scalinga0 = νa0(ζ ), K0 = ν2k0, θ =ϑ(ζ )/

√d|ν|, ζ = √

d|ν|ξ , h0 = ν2dH0

∂2ζ a0 = sgnν

(a0 − Λ

da2

0 − d2

dk0

)+ O(ν), ∂ζ r = 0, ∂ζ ϑ = q + Ψ (ν, a0, k

20) = q + O(ν). (3.17)

There are exactly two equilibria. At lowest order inν they are given by

a∗0 = d

2Λ− sgnν

d

2Λ

√1 − 4d2d−2Λk0, a∗∗

0 = d

2Λ+ sgnν

d

2Λ

√1 − 4d2d−2Λk0,

wherea∗0 stands for the saddle point anda∗∗

0 for the center. From (3.17) one gets

12(∂ζ a0)

2 = sgnνf (a0, h0, k0),

where

f (a0, h0, k0) = 1

2a2

0 − Λ

3da3

0 − d2

dk0a0 + h0.

For the existence of solutionsa0 we need sgnνf (a0, h0, k0) > 0. Hence, all the solutions live in the regionG ofthe(k0, h0) plane bounded by the curvesΓ1 = f (a∗

0) = 0,Γ2 = f (a∗∗0 ) = 0 and the axisk0 = 0 sincek0 must be

positive.The analysis is restricted toν > 0 since our interest lies in solitary waves. Then there are two cases to consider:

Λ andd2 of the same sign (Case 1), andΛ andd2 of opposite signs (Case 2). Both cases are possible in our model.The domainG in the(k0, h0) plane for Case 1 withΛ > 0, d2 > 0 is shown in Fig. 2.

As follows from Fig. 2, at the origin O of the(k0, h0) plane, one has the homoclinic connection to zero, whichcorresponds to the generalized solitary wave, described in Section 3.1. Moving along the curveΓ1 we get solitarywaves with ripples (see for example Section 3.2 in [3]). These waves result from the superposition of a solitarywave and a periodic wave with resonant wavenumberq (short wave). The amplitude of the periodic componentis measured byk0 and increases ask0 grows; vice versa, the amplitude of the solitary wave pulse decreases ask0

increases. At the edge point O1 = (14d

2d−12 Λ−1, 1

24d2Λ−2), both equilibria points coincide, and only the short

periodic wave with nonzero mean-value remains. Moving from the point O1 along the curveΓ2, one has shortperiodic waves with increasing mean-value.

On the intervalk0 = 0,h0 ∈ (−16d

2Λ2,0) and inside the domainG, one finds waves which are the superpositionof long and short periodic waves. These waves can be periodic waves of the second kind [3] (i.e. periodic waveson a torus, if the ratio of periods of short-period and long-period waves is a rational number) or quasi-periodicwaves, which tend to the generalized solitary wave as(k0, h0) → 0. The form of periodic and quasi-periodicwaves at lowest order inν is given by the linear combination of the square of an elliptic cosine with ordinarycosine.

As said above, when one moves along the curveΓ1 from left to right, the amplitude of the ripples of the general-ized solitary wave grows and at the point O1, the solitary wave with ripples becomes a periodic wave. Profiles of the


Fig. 2. Closed domainG in which bounded solutions exist in the(k0, h0) plane (Case 1,Λ > 0, d2 > 0). The behavior of the polynomialf (a0, h0, k0) is shown at various parts of the boundary and inside the domainG. Solitary waves with ripples exist along the curveΓ1. Thecoordinates of O1 are( 1

4d2d−1

2 Λ−1, 124d

2Λ−2). The coordinates of O2 are(0,− 16d

2Λ2).

interface and the free surface are given by

η1 = − C−1 − R

[(γ2

γ1+ R

)a0 +

(α1

γ1+ R

)(a+ + a−)

],

η2 =C−(a0 + a+ + a−)+ C−H1 − R

(C2 − 1

2H)q2(a+ + a−),

where

a0 = νa∗0 + 3

2ν

(d

Λ− 2a∗

0

)sech2

[1

2

√(d − 2Λa∗

0)νξ

], a+ + a− = 2ν

√k0 cosqξ.

Recall thatq = √α1 − γ2. The parameterk0 varies between 0 and14d

2d−12 Λ−1. Several profiles are shown in Fig. 3

for some values ofR andH . A similar behavior for solutions of the full equations was described in [14]. As saidearlier, the internal wave is much larger than the free-surface wave when the density ratio is close to 1.

The domainG in the (k0, h0) plane for Case 2 (Λ < 0, d2 > 0) is shown in Fig. 4. NowG is unbounded andtherefore, no transition from solitary waves with ripples to periodic waves takes place. In Case 1 withΛ < 0,d2 < 0, respectively, in Case 2 withΛ > 0, d2 < 0, the domainG is exactly the same as the one given in Fig. 2,respectively, Fig. 4. The difference is in the behavior of the polynomialf (a0, k0, h0); it has to be reflected aboutthe vertical axis.

4. Solutions near the critical depth ratio

In this section we consider the situation where the coefficientΛ is close to zero, which is possible only for thelowest branch of the dispersion relation, originating atC−. In Appendix A, we show how one can find the values ofR andH for whichΛ = 0, without deriving any differential equation.


Fig. 3. Four examples of profiles of the free surface and the interface for physically acceptable values of the parameters. (The parameters valuesare indicated on top.) The plots withk0 = 0 provide the principal part of generalized solitary waves. Whenk0 6= 0, the waves are solitary waveswith ripples. The ripples are barely visible on the free surface. For these four cases, bothΛ andd2 are positive.

The unfolding of the singularityΛ = 0 requires to take into consideration cubic terms in (2.13). The expressionfor fs1 in (3.4) now reads

fs1 = νda0 −Λa20 + Γ a3

0,

where

Γ = α1

2κq2

[1 − 2(C2− −H(1 − R))2

C4−+ (C2− −H(1 − R))4

C8−R

].

The expression forΓ was compared with the expression obtained in [11]. We were not able to reduce both expressionsto the same expression, but we also find thatΓ is positive for all values of the parametersH > 0 andR < 1. Next


Fig. 4. Unbounded domainG in which bounded solutions exist in the(k0, h0) plane for Case 2 (Λ < 0, d2 > 0). Again solitary waves withripples exist alongΓ1.

we assume that the coefficientΛ is of orderν1/2 and introduce the new change of variables

a0 =√νd

ΓY (ζ ), ζ =

√νdξ.

Eqs. (3.4) at lowest order inν then read

Y ′′ − Y + QY2 − Y 3 = 0, a+ = iqa+ + ifs2(Y ), a− = −iqa− − ifs2(Y ), (4.1)

where

Q = Λ√νdΓ

and prime denotes differentiation with respect toζ . WhenQ = ±3/√

2 the first equation in (4.1) admits solutionsin the form of fronts, which connect two conjugate states (see Appendix A for a discussion of conjugate flows):

Y ∗ = ± 1√2

(1 + tanh

1

2ζ

). (4.2)

The corresponding profiles for the interfacial and free-surface fronts are given at leading order by

η1 = −√νd

2Γ

H 2 − C

C

(1 + tanh

√νd

2ξ

), η2 =

√νd

2ΓC

(1 + tanh

√νd

2ξ

). (4.3)

Again, in oceanic conditions, the density ratioR is close to 1 and the interfacial front is much larger than thefree-surface front.

Eq. (4.2) only provides the principal part of the solution of (4.1). Higher order terms in (4.1) lead to nondecreasingperiodic oscillations at infinity on top of the front (4.2). The functionfs2 is now of orderν and, as the front (4.2)tends to a nonzero constant asζ → +∞, one has

fs2(Y∗) → νP 6= 0 as ζ → +∞,

where the constantP = O(1) can be computed from (3.5).


The operatorL in (3.8) is no longer invertible, because the functionfs2(Y ∗) does not tend to zero at infinity. Inorder to makeL invertible let us differentiate the last two equations in (4.1) with respect toζ :

z = iqz+ igs2, ˙z = −iqz− igs2, z = a+, z = a−,

where

gs2 = fs2 = − γ2

2q3

±ν d

2√

2Γ

[−r1(C)+ α1

α2r2(C)

]sech2

ζ

2

+ν3/2d√d

2Γ

[−s1 + α1

α2s2

](sech2

ζ

2+ sinh

ζ

2sech3

ζ

2

).

The expressions forri andsi , i = 1,2, are given in (3.5). Proceeding as in Section 3.1, we get at lowest order

z = − g2s

k + q, ˆz = g2s

k − q, g2s = (

√νl1k + il2k

2)cschπk√νd,

wherel1 andl2 are some constants of order 1 inν, depending on the parametersR andH . Finally one gets

z+ z = ∓π [√νl1 sin(qξ)+ l2q cos(qξ)]exp

(− πq√

νd

)as ξ → ±∞,

a+ + a− = −2νP

q− π

[− l1

√ν

qcos(qξ)+ l2 sin(qξ)

]exp

(− πq√

νd

)as ξ → +∞,

a+ + a− = π

[− l1

√ν

qcos(qξ)+ l2 sin(qξ)

]exp

(− πq√

νd

)as ξ → −∞

or at lowest in order inν

ui =Ai− exp

(− πq√

νd

)sin(qξ) as ξ → −∞,

ui =√νBi + Ai+ exp

(− πq√

νd

)sin(qξ) as ξ → +∞, (4.4)

whereAi−,Bi ,Ai+, i = 1,2 are some real constants of order O(1) in ν. The asymptotics (4.4) gives an exponentiallysmall oscillatory tail on top of the front (4.2) at infinity. In addition to generalized fronts, one could compute as inSection 3.3 fronts with ripples.

Before concluding, it is important to make the following remark at this stage. In Appendix A, it is conjectured thatfor the full problem with fixed values of the density ratioR and the thickness ratioH , solitary waves with ripplescannot have a speedC larger thanCmax. AsC increases towardsCmax, the solitary waves broaden and eventuallybecome fronts. The same occurs for modified KdV-type systems like (4.1) and one can compute a maximum valueνmax for the bifurcation parameter. But since only solitary waves with ripples exist, what can one say about theamplitude of the ripples at infinity? For a given value of the Froude numberC, we have shown in Section 3 thatthe amplitude of the ripples can be exponentially small. What about the full equations? The numerical results ofMichallet and Dias [14] indicate that asC increases towardsCmax the minimum amplitude of the ripples increasesand can become of the same order of magnitude as the solitary pulse. In that case, there is less and less “room” leftfor nonperiodic solutions and they may well disappear before the maximum speed is reached. Only a numericalstudy of the full equations will provide an answer.


Fig. 5. Plot of the wavenumber at resonanceKresversus the thickness ratio, for various values of the density ratioR. From bottom to top,R = 0,0.1, 0.8.

5. Conclusion and discussion

A simple model for describing waves in a two-layer configuration with free-surface boundary conditions has beenderived. This model has some drawbacks and ought to be used to obtain qualitative rather than quantitative results.The key feature for waves having speeds slightly aboveC− is the presence of a long/short wave resonance. Themodel reproduces this resonance, which was not the case for earlier models. However, the value of the wavenumberin resonanceKres may be different from the real one as was shown in Fig. 1.

Moreover, the analysis is based on the assumption that the wavenumbers are much smaller than 1. This assumptionis satisfied for example when deriving the fifth-order KdV equation in the problem of capillary–gravity waves.Fig. 5, which gives the value ofKres for interfacial waves as a function ofH for various values ofR, clearlyshows thatKres is not small in most cases. For small values ofR, there is a minimum value ofKres(H). Buteven forR = 0.05, this minimum, which is obtained atH = 1.5, is 1.08, a value which is not much smallerthan 1!

To summarize, one can say that the present model gives quite accurately the principal part of the solitary wavesolutions and qualitatively the oscillations superimposed to the solitary wave. Only an analysis performed on thefull Euler equations, as was done for capillary–gravity waves in [3], would provide a more accurate model. To ourknowledge, such a tedious analysis has not been performed yet. The normal form will be similar to the one obtainedin Section 3, but the coefficients will be different.

Finally, an important conclusion from a physical point of view is that in oceanic conditions (R close to 1) thedeformations of the free surface are much smaller than the deformations of the interface. Solitary waves with ripplesseem to have been observed in laboratory experiments by Walker [16]. The ripples can only be observed alongthe internal wave. If one replaces the top fluid (air) by a heavier fluid, computations show that solitary waves withripples may be more easily reproduced in the laboratory [22].

Acknowledgements

This work was performed in the framework of the INTAS-RFBR project No. 95-0435.


Appendix A. Conjugate flows

We consider the occurrence of fronts at the interface between two finite layers of fluid in the presence of a freesurface. We suppose that there is a uniform flow upstream and a uniform flow downstream, possibly with differentproperties. The properties of the flow downstream are denoted with primes.

The interest here is not in the shape of the interface, just in the upstream and downstream states. Upstream, thevelocity c is the same in the lower and upper layers. The layers have depthsh1 andh2. The bottom is located aty = 0.

We suppose that all quantities are known upstream. We ask ourselves whether there are in addition to the trivialsolution(c′1, h

′1, c

′2, h

′2) = (c, h1, c, h2) other uniform solutions downstream. Note that only the special case where

both fluids have the same velocity upstream is considered.The conservation of mass in both layers gives

c′ih′i = chi , i = 1,2. (A.1)

Next we write down the conservation of momentum. The pressure is zero along the free surface. Bernoulli’sequation gives

1

2c2 + gy+ p2

ρ2= 1

2c2 + g(h1 + h2), (A.2)

1

2c2 + gy+ p1

ρ1= 1

2c2 + gh1 + ρ2

ρ1gh2. (A.3)

It follows that the momentum upstream is equal to

(ρ1h1 + ρ2h2)c2 + 1

2ρ1gh21 + 1

2ρ2gh22 + ρ2gh1h2. (A.4)

Downstream, Bernoulli’s equation gives

1

2c′2

2 + gy+ p′2

ρ2= 1

2c2 + g(h1 + h2), (A.5)

1

2c′1

2 + gy+ p′1

ρ1= 1

2c2 + gh1 + ρ2

ρ1gh2. (A.6)

It follows that the momentum downstream is equal to

ρ1h′1c

′21 + ρ2h

′2c

′22 + ρ1

∫ h′1

0

[1

2(c2 − c′21 )+ g(h1 − y)

]dy

+ρ2gh2h′1 + ρ2

∫ h′1+h′

2

h′1

[1

2(c2 − c′22 )+ g(h1 + h2 − y)

]dy.

After computing the integrals and setting equal the momentum on the left and the momentum on the right, oneobtains

(ρ1h1 + ρ2h2)c2 = 1

2ρ1h′1(c

2 + c′21 )+ 12ρ2h

′2(c

2 + c′22 )− 12ρ1g(h1 − h′

1)2 − 1

2ρ2g(h2 − h′2)

2

−ρ2g(h1 − h′1)(h2 − h′

2). (A.7)

Finally, we write down the conservation of energy. The pressure is required to be continuous across the interface.It follows that downstream one has

12ρ1(c

2 − c′21 )+ g(h1 − h′1)(ρ1 − ρ2) = 1

2ρ2(c2 − c′22 ). (A.8)


Let us now look for nontrivial solutions of Eqs. (A.1), (A.7) and (A.8). There are four unknowns(c′1, h′1, c

′2, h

′2)

for four equations. However, there is also the constraint that the pressure downstream is equal to zero on the freesurface:

12c

′22 + g(h′

1 + h′2) = 1

2c2 + g(h1 + h2). (A.9)

Note that (A.8) and (A.9) providec′21 andc′22 :

12c

′21 = 1

2c2 + g(h1 − h′

1)+ gR(h2 − h′2), (A.10)

12c

′22 = 1

2c2 + g(h1 − h′

1)+ g(h2 − h′2). (A.11)

Let us multiply (A.10) byh′21 and (A.11) byh′2

2 . Sincec′1h′1 = ch1 andc′2h

′2 = ch2, one has

gRh′21 (h2 − h′2) = (h1 − h′

1)[12c

2(h1 + h′1)− gh′2

1 ], (A.12)

gh′22 (h1 − h′

1) = (h2 − h′2)[

12c

2(h2 + h′2)− gh′2

2 ]. (A.13)

On the other hand, replacingc′21 by (A.10) andc′22 by (A.11) in (A.7) leads to

(h1 − h′1)[c

2 − g(h′1 + Rh′

2)+ 12g(h1 − h′

1)+ gR(h2 − h′2)]

= R(h2 − h′2)[−c2 + g(h′

1 + h′2)− 1

2g(h2 − h′2)]. (A.14)

Therefore in general we expect the downstream state to be the same as the upstream state, unless some degeneracycondition is satisfied. Let us now find this degeneracy condition. The three equations for the two unknownsh′

1 andh′

2 are rewritten in dimensionless form. Eqs. (A.12)–(A.14) become

R(H2 −H ′2) = (H1 −H ′

1)

[1

2F 2H1 +H ′

1

H ′21

− 1

], (A.15)

H1 −H ′1 = (H2 −H ′

2)

[1

2F 2H2 +H ′

2

H ′22

− 1

], (A.16)

(H1 −H ′1)[F

2 − (H ′1 + RH′

2)+ 12(H1 −H ′

1)+ R(H2 −H ′2)]

= R(H2 −H ′2)[−F 2 +H ′

1 +H ′2 − 1

2(H2 −H ′2)], (A.17)

where we have set

F 2 = c2

g(h1 + h2)= C2

1 +H, Hi = hi

h1 + h2, H ′

i = h′i

h1 + h2.

Eq. (A.15) givesH ′2. This expression forH ′

2 can then be plugged into the two remaining equations (A.16) and(A.17). One is inH ′5

1 while the other is inH ′81 . Requiring both equations to have a common root leads to the

vanishing of the resultant of both polynomials. One finds through symbolic calculation that

P1(H1)× P2(H1) = 0, (A.18)

where

P1(H1) = F 4 − F 2 + (1 − R)H1(1 −H1), (A.19)


Fig. 6. Speed of conjugate flow (solid line) as well as the critical speedsF± = C±/(1+H)1/2 for two values of the density ratioR. The regionbetween the bottom curve and the middle curve is the conjectured region of existence of solitary waves with ripples. The bottom curve and themiddle curve touch at the point(H ∗

1 , F∗) which satisfies (A.21) and (A.22).

and

P2(H1)= 16(1 − R)5H 81 − 96F 2(1 + 2R)(1 − R)4H 7

1 + a6H61 + a5H

51 + a4H

41

+a3H31 + a2H

21 + a1H1 + a0. (A.20)

The expressions of the coefficientsai are rather long and we do not reproduce them here.Therefore, eitherP1 = 0, which leads to the equation satisfied by the critical speeds (2.4), orP2 = 0. In fact, the

equationP2 = 0 is a relation betweenH1, F andR. It can be shown numerically that for all values ofR between0 and 1 and for all values ofH1 between 0 and 1, there is always a unique solution forF . The curveF(H1) isplotted in Fig. 6 for various values ofR. The curves giving the critical speedsF± = C±/(1 + H)1/2 are plottedas well. WhenP2 = 0, (A.16) and (A.17) have a common root, which is given by a messy expression that we donot reproduce here. The corresponding value forH ′

2 is given by (A.15). The value forH ′1 is equal toH1 when the

curves forP1 = 0 andP2 = 0 touch each other. This corresponds to the critical heightH ∗1 . Requiring thatP1 = 0

andP2 = 0 have a common root leads to the vanishing of the resultant of both polynomialsP1 andP2. The curvegiving the critical Froude numberF ∗ versusR is given by

(8 + R)F 6 − 12F 4 + 2F 2(3 − R)− (1 − R) = 0, (A.21)

and is plotted in Fig. 7. The curve giving the critical height versusR is given by

1 − 6H1 + 3RH1 +H1R2 + 12H 2

1 − 10RH21 − 2H 2

1R2 − 8H 3

1 + 7H 31R + R2H 3

1 = 0, (A.22)

and is plotted as well in Fig. 7. It does not vary much as a function ofR. Note that this curve is identical to the curvecorresponding to the vanishing of the quadratic termΛ (3.5) in the KdV-type equation (3.17).

From Fig. 6, one sees that fronts are possible only in the case of generalized solitary waves. Therefore, one doesnot expect classical fronts in the two-layer configuration with free-surface boundary conditions, but only fronts withripples. One may conjecture that generalized solitary waves exist in the region between the curve for fronts and thecurve forC−. Fig. 8 shows the value ofνmax = (Ffront − F−)/(H1)

1/2 as a function ofH1. The implications ofthe presence of this maximum value forν are discussed in Section 4. For the parameter values used in Fig. 3, therespective values ofνmax are 0.03 (R = 0.8, H = 0.4) and 0.04 (R = 0.99, H = 0.01), while the respective valuesof ν are 0.01 and 0.001. Therefore, it is justified not to use the modified model of Section 4, valid near the criticaldepth ratioH ∗

1 .


Fig. 7. Critical values of the Froude numberF ∗ (solid line) and of the critical heightH ∗1 (dashed dotted line) as a function of the density ratioR.

Finally, let us mention a detailed analysis of conjugate flows by Lamb [23] in a three-layer configuration. Ourconfiguration is a special case.

Appendix B. Integral properties

In this appendix, all the equations are written in a frame moving with the wave. The horizontal coordinate isdenoted byx.

The conservation of horizontal momentum (in an integral form) implies that∫ h1+η1

0(p1 + ρ1u

21)dy +

∫ h1+h2+η2

h1+η1

(p2 + ρ2u22)dy (B.1)

Fig. 8. Maximum value of the bifurcation parameterν for solitary waves with ripples for two values of the density ratio:R = 0.1 (solid line),R = 0.8 (dashed dotted line).


is independent ofx. Therefore it is equal to its value atx = −∞∫ h1

0[ρ1c

2 + g(ρ1h1 + ρ2h2)− ρ1gy] dy +∫ h1+h2

h1

[ρ2c2 + gρ2(h1 + h2)− ρ2gy] dy. (B.2)

This equality can be rewritten as∫ h1+η1

0[p1 + ρ1u

21 + ρ1g(y − h1)− ρ1c

2 − ρ2gh2] dy +∫ h1+h2+η2

h1+η1

[p2 + ρ2u22 + ρ2g(y − h2)

−ρ2c2 − ρ2gh1] dy = 1

2(ρ1 − ρ2)gη

21 − (ρ1 − ρ2)c

2η1 + 1

2ρ2gη

22 − ρ2c

2η2. (B.3)

The conservation of vertical momentum (in local form) can be written as

ui∂vi

∂x+ vi

∂vi

∂y= − 1

ρi

∂pi

∂y− g. (B.4)

Introducing the notationUi = (ui, vi)T, it follows that

ρiv2i + pi + ρig(y − h1 − h2) = div(ρiyviUi)+ ∂

∂y[ypi + ρigy(y − h1 − h2)]. (B.5)

Considering a control volume bounded by the bottom, two vertical planes atx = x−, x = x+ and the free surface,Eq. (B.5) leads to

∫ x+

x−

∫ h1+η1

0[ρ1v

21+p1+ρ1g(y − h1 − h2)] dy dx +

∫ x+

x−

∫ h1+h2+η2

h1+η1

[ρ2v22 + p2 + ρ2g(y − h1 − h2)] dy dx

=∫ x+

x−[(ρ1 − ρ2)g(h1 + η1)(η1 − h2)+ ρ2g(h1 + h2 + η2)η2] dx +Λ(x+)−Λ(x−), (B.6)

where

Λ =∫ h1+η1

0ρ1yv1u1 dy +

∫ h1+h2+η2

h1+η1

ρ2yv2u2 dy.

Finally, adding the integral with respect tox of (B.3) and (B.6) leads to (after using Bernoulli’s equation)

∫ x+

x−

[(ρ1 − ρ2)η1

(3

2gη1 + gh1 − c2

)+ ρ2η2

(3

2gη2 + gh1 + gh2 − c2

)]dx +Λ(x+)−Λ(x−) = 0.

(B.7)

Now, takex+ = +∞ andx− = −∞. Moreover, use the dimensionless variables introduced above in (2.14).Eq. (B.7) becomes

3

2(1 − R)

∫ +∞

−∞η2

1 dξ + (1 − R)(1 − C2)

∫ +∞

−∞η1 dξ + 3

2R

∫ +∞

−∞η2

2 dξ + R(1 +H − C2)

∫ +∞

−∞η2 dξ = 0.

(B.8)

Of course, all this makes sense for the classical solitary waves (otherwise there are oscillations at infinity).Eq. (B.8) was almost obtained by Pennell [24]. He introduced the pressure integralP (difference between actualand hydrostatic pressure at the interface), the potential energyV , the mass of the free-surface waveMf and the mass


of the interfacial waveMi :

P =∫ ∞

−∞pi − ρ2g(h2 − η1)dx, V =

∫ ∞

−∞

∫ η1

−h1

(ρ1 − ρ2)gydy dx +∫ ∞

−∞

∫ h2+η2

η1

ρ2g(y − h2)dy dx,

Mf =∫ ∞

−∞ρ2η2 dx, Mi =

∫ ∞

−∞(ρ1 − ρ2)η1 dx.

Pennell showed that

3V = (c2 − gh1)Mi + (c2 − gh2)Mf − h1P.

But it is easy to show that in fact

P = gMf .

Therefore

3V = (c2 − gh1)Mi + [c2 − g(h1 + h2)]Mf ,

which is precisely Eq. (B.8).

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Interfacial waves with free-surface boundary conditions: an approach via a model equation