Modulational instability in crossing sea states over finite depth water Sumana Kundu, S. Debsarma, and K. P. Das

Sumana Kundu, S. Debsarma, and K. P. Das

Citation: Physics of Fluids (1994-present) 25, 066605 (2013); doi: 10.1063/1.4811695 View online: http://dx.doi.org/10.1063/1.4811695

View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/25/6?ver=pdfcov Published by the AIP Publishing

Modulational instability in crossing sea states over finite depth water

Sumana Kundu,^1,a)S. Debsarma,^2,b)and K. P. Das^2,b)

1Salkia Mrigendra Dutta Smriti Balika Vidyapith (High), Salkia, Howrah 711106, India

2Department of Applied Mathematics, University of Calcutta, 92 A.P.C. Road, Kolkata 700009, India

(Received 29 August 2012; accepted 30 May 2013; published online 28 June 2013)

Nonlinear evolution equations are derived in a situation of crossing sea states charac- terized by water waves having two different spectral peaks. The nonlinear evolution equations derived here are valid for any water depth except for shallow water depth case. These evolution equations are then employed to study the instability prop- erties of two Stokes wave trains considering both unidirectional and bidirectional perturbations. Figures have been plotted showing the growth rate of instability for various depths of water and for different values of the angle of interaction of the two wave systems. All the figures serve as an evidence to the fact that freak waves can be formed as a result of modulational instability in crossing sea states over finite depth water. It is observed that the growth rate of instability in crossing sea states situation over finite depth water is much higher than that for infinite depth case and it increases with the decrease of the depth of water.^C 2013 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4811695]

I. INTRODUCTION

Freak waves are surprisingly large amplitude waves, localized both in space and time.

Devastating power of freak waves even leading to ship accidents (Toffoli et al.¹ and Cavaleri et al.²) and causing damage to offshore engineering structures have led many scientists to carry out research in order to have adequate knowledge about the probability of occurrence of freak waves, the mechanism of generation of such waves, their dynamics of evolution and other characteristics.

So, freak waves have been focused in many research works^3–15in recent times. Study of White and Fornberg³shows formation of freak waves at random locations when ocean swell traverses an area of random current. Some investigations have been made showing the generation of freak waves as a result of modulational instability or Benjamin-Feir instability (see Benjamin and Feir¹⁶) of two obliquely interacting wave systems. Studies of Shuklaet al.⁶and Onoratoet al.⁷are in this direction.

Starting from Zakharov equation [see Zakharov¹⁷ and Krasitskii¹⁸], Onorato et al.⁷ derived two coupled nonlinear Schr¨odinger equations that describe evolution of two wave packets propagating in two dtfferent directions in deep water and carried out stability analysis for unidirectional perturbation.

Shuklaet al.⁶extended the results obtained by Onoratoet al.⁷by performing stability analysis for two-dimensional perturbations. Dyachenko and Zakharov⁵have shown formation of freak waves as a result of modulational instability by performing numerical experiments in which they started with the Stokes wave train and perturbed it by a long wave with 20 times less amplitude. Onoratoet al.,⁸based on two independent experiments performed in two wave basins of different geometries, reported that the probability of the formation of extreme waves strongly depends on the directional properties of the waves. Hjelmervik and Trulsen⁹ derived non-potential current modified nonlinear Schr¨odinger equation and also performed Monte Carlo simulations to investigate the effect of nonlinearity on the

a)E-mail:sumanakundu.mail@gmail.com

b)Electronic addresses:sdappmath@caluniv.ac.inandkalipada_das@yahoo.com

1070-6631/2013/25(6)/066605/13/$30.00 25, 066605-1 ^C 2013 AIP Publishing LLC

significant wave height and the occurrence of freak waves, for waves propagating on inhomogeneous currents. Laine-Pearson¹⁰ studied the interaction of two weakly nonlinear wave systems having different carrier frequencies and propagating in two different directions, similar in nature to crossing sea states. His investigation shows that the leading order growth rates due to nonlinear interactions for such a more general two-wave systems can be larger than those for short-crested waves. Gramstad and Trulsen¹¹derived modified nonlinear Sch¨odinger equation describing evolution of short waves influenced by a swell. Using this swell-modified evolution equation they studied the effect of a swell on irregular and directional short waves. They found that for short crested waves, a swell can increase the probability of freak waves by 5%–20% compared with a corresponding sea without swell. They also observed that for orthogonal swell this modification of the number of freak waves is much less than that for non-orthogonal swell. Onoratoet al.¹⁵ have shown that an initially stable wave train may become unstable when it enters a region of an opposing current flow and such instability can trigger the formation of rogue waves.

In this paper, we have derived a set of nonlinear evolution equations consisting of three coupled equations for two wave systems having the same carrier frequency and interacting obliquely over the surface of finite depth water. This is an extension to finite depth case of the papers by Shuklaet al.⁶ and Onoratoet al.⁷ for infinite depth water. The nonlinear evolution equation for a single surface gravity wave packet was first obtained by Hasimoto and Ono¹⁹ for infinite depth water. Later on, Davey and Stewartson²⁰obtained two coupled nonlinear evolution equations for finite depth water and for a three dimensional single wave packet. Benney and Roskes²¹also studied the same problem for finite depth water but for a single two dimensional wave packet. The coupled evolution equations derived here can be considered as an extension of Davey-Stewartson equations to the situation of crossing seas. As in the case of Davey-Stewartson equations the nonlinear evolution equations derived in the present paper remain valid for any water depth except for shallow water depth case.

These equations also reduce to the equations of Hasimoto and Ono¹⁹as depth of water approaches to infinity. Using these evolution equations we have studied stability of two Stokes wave trains which are uniform solutions of the evolution equations. Considering bidirectional perturbations we have plotted figures showing the growth rate of instability in perturbed wave number plane for some different depths of water and also for different values ofθ,θbeing the half of the angle between the two wave directions. It is observed that the growth rate of instability in crossing sea states situation over finite depth water is much higher than that for the infinite depth case. We have also plotted growth rate of instability curves for unidirectional perturbations. Figures confirm the fact that the growth rate of instability in crossing sea states is much higher than in the case of modulation of a single wave packet, even in the case of finite depth water. So, freak waves can be formed as a result of modulational instability over finite depth water.

For nonlinear evolution of a single wave system in shallow water, Benney and Roskes²¹ and later on, Davey and Stewartson²⁰ showed that there is modulational instability if k_chexceeds the critical value 0.380. Herek_cis a characteristic wavenumber andhis the mean depth of water. But in a situation of crossing sea states, this critical value depends upon the angle of interaction between the two wave systems and the depth of water. It is also observed that the growth rate of instability increases with the decrease of the depth of water.

II. BASIC EQUATIONS

We choose a Cartesian system of co-ordinates oxyz, with the oxy plane coinciding with the undisturbed free surface of the water and oz axis pointing vertically upwards. The sea bed is defined by the planez= −h. We consider that the two gravity wave packets move in thexyplane with carrier wave numberska =(k,l) andkb=(k,−l),respectively. Letz=ζ(x,y,t) be the equation to the free surface at any time t in the perturbed state, andφ(x,y,z,t) be the perturbed velocity potential satisfying the following Laplace equation:

∇²φ=0, −h<z< ζ. (1)

The kinematic boundary condition to be satisfied at the free surface is

∂φ

∂z −∂ζ

∂t =∂φ

∂x ·∂ζ

∂x +∂φ

∂y ·∂ζ

∂y on z=ζ. (2)

The condition of continuity of pressure at the free surface gives

∂φ

∂t +gζ = −1 2

∇φ2

on z=ζ. (3)

Also, at the bottomφsatisfies the following condition:

∂φ

∂z =0 on z= −h. (4)

We consider weakly nonlinear interaction of two wave packets which are assumed to be narrow- banded. For crossing sea states we take solution of the above equations in the following form:

G=G00+ ∞

m=0

∞

n=0

Gmnexpi(mψ1+nψ2)+G^∗_mnexp−i(mψ1+nψ2)

, (5)

where

ψ₁=kx+l y−ωt, ψ₂=kx−l y−ωt

andGstands forφandζ. Hereφ₀₀,φ_mn, andφ^∗_mn are slowly varying functions of z,x₁ =ǫx,y₁

=ǫy,t₁ =ǫt;ζ₀₀,ζ_mn,ζ_mn^∗ are function ofx₁,y₁,t₁. Here (*) denotes complex conjugate andǫis a slow ordering parameter measuring the weakness of nonlinearity. The linear dispersion relation obtained from Eqs.(1)–(4)after linearization is

ω=(gkcσ)¹², (6)

wherekc=√

k²+l²,σ =tanh (kch) andgis the gravitational acceleration. The group velocity of either wave packet is given by

Cg = g 2ω

σ+kch

1−σ²

. (7)

III. DERIVATION OF THE EVOLUTION EQUATIONS A. Equations forφmnandζmnwhen (m, n)=(0, 0)

Substituting the expansion(5)in(1)and(4)and then equating the coefficients of expi(mψ1

+nψ₂) for (m,n)=[(1,0), (0,1) (2,0), (0,2), (1,1), (1,−1)], we obtain the following equations:

d²φmn

d z² −²_mnφmn=0, −h<z< ζ, (8) dφmn

d z =0 on z= −h, (9)

where the operatormnis given by mn ≡

(m+n)k−iǫ ∂

∂x₁ 2

+

(m−n)l−iǫ ∂

∂y₁ 21/2

. The solution of Eq.(8)satisfying bottom boundary condition(9)is given by

φ_mn =cosh [(z+h)_mn]A_mn, (10)

whereA_mn’s are functions ofx₁,y₁, andt₁. Substituting the expansion(5)in the Taylor expanded forms of Eqs.(2)and(3) about z =0 and then equating coefficients of expi(mψ1 +nψ2) for

(m,n)=[(1,0), (0,1), (2,0), (0,2), (1,1), (1,−1)] on both sides, we obtain six sets of equations of the following type:

mnsinh (hmn)Amn+i Wmnζmn=amn, (11)

−i Wmncosh (hmn)Amn+gζmn =bmn, (12) whereWmnis the operator

Wmn ≡(m+n)ω+iǫ ∂

∂t1

anda_mn,b_mn’s are contributions from nonlinear terms. EliminatingA10[A01] from the two equations corresponding to the set (m,n)=(1,0)[(0,1)], we get the following two equations:

−W₁₀² cosh (h10)+g₁₀sinh (h10)

ζ₁₀=i W₁₀cosh (h10)a₁₀+₁₀sinh (h₁₀)b₁₀, (13)

−W₀₁² cosh (h01)+g01sinh (h01)

ζ01 =i W01cosh (h01)a01+01sinh (h01)b01. (14) We now introduce the following perturbation expansion:

G_mn=

p

ǫ^pG^(p)_mn, (15)

where the index p starts from p=1 for (m,n)=(1,0), (0,1), (0,0) and starts from p=2 for (m,n)

=(2,0), (0,2), (1,1), and (1,−1). Using expansion(15)in Eqs.(11)and(12)for (m,n)=(1,0) and (0,1) we obtain solutions forA⁽¹⁾₁₀,A⁽²⁾₁₀,A⁽¹⁾₀₁,A⁽²⁾₀₁ in terms ofζ10andζ01. Next, using expansion(15) in Eqs.(11)and(12)for (m,n)=(2,0), (0,2), (1,1), (1,-1) we obtain solutions forA⁽²⁾₂₀,ζ₂₀⁽²⁾,A⁽²⁾₀₂,ζ₀₂⁽²⁾, A⁽²⁾₁₁,ζ₁₁⁽²⁾, A⁽²⁾₁₋₁,ζ₁⁽²⁾₋₁in terms ofζ10andζ01. All these solutions are given in AppendixA. Using these solutions we find nonlinear expressions fora₁₀,b₁₀,a₀₁,b₀₁in terms ofζ₁₀,ζ₀₁, andφ₀₀.

B. Equations forφ00andζ00

We now substitute the expansion(5)in Eq.(1), in Taylor expanded forms of Eqs.(2)and(3) and also in Eq.(4). Then equating coefficients of the terms independent ofψ₁andψ₂on both sides, we get the following equations:

ǫ² ∂²

∂x₁² +ǫ² ∂²

∂y₁² + ∂²

∂z²

φ00=0, −h<z<0, (16)

∂φ00

∂z −ǫ∂ζ00

∂t1 =a₀₀ on z=0, (17)

ǫ∂φ₀₀

∂t₁ +gζ00=b00 on z=0, (18)

∂φ00

∂z =0 on z= −h. (19)

When the perturbation expansion(15)is substituted in Eq.(16)it follows from the equation at 0(ǫ) thatφ⁽¹⁾₀₀ is independent of z. Also, from the equation at 0(ǫ²) it follows thatφ₀₀⁽²⁾is independent of z. At order 0(ǫ³), we get the following equation:

∂²φ₀₀⁽³⁾

∂z² + ∂²

∂x₁² + ∂²

∂y₁²

φ⁽¹⁾₀₀ =0, −h<z<0. (20)

Integrating equation(20)with respect tozbetween the limits−hto 0 and making use of the bottom boundary condition(19), we get

∂φ₀₀⁽³⁾

∂z = −(z+h) ∂²

∂x₁² + ∂²

∂y₁²

φ⁽¹⁾₀₀, −h ≤z≤0. (21) Eliminatingζ00between(17)and(18), we get

ǫ²∂²φ₀₀

∂t₁² +g∂φ₀₀

∂z =ǫ∂b₀₀

∂t₁ +ga00 on z=0. (22) Asφ₀₀⁽¹⁾andφ₀₀⁽²⁾are independent of z, Eq.(22)at 0(ǫ³) gives

ǫ³∂²φ₀₀⁽¹⁾

∂t₁² +ǫ³g∂φ₀₀⁽³⁾

∂z =ǫ∂b00

∂t₁ +ga00 on z=0. (23) Now substituting^∂φ

(3) 00

∂z on z=0 as given by Eqs.(21)in(23), we get ǫ³

∂²

∂t1 −gh ∂²

∂x₁² + ∂²

∂y₁²

φ₀₀⁽¹⁾=ǫ∂b00

∂t1 +ga00 on z=0. (24)

C. Set of evolution equations

Next, we substitute the expressions fora₁₀,b₁₀,a₀₁,b₀₁,a₀₀,b₀₀in Eqs.(13),(14), and(24)and then simplify the operators appearing in Eqs.(13)and(14)so as to write these equations correct up to 0(ǫ³). These three equations are then made dimensionless using the following change of variables:

X =k_cx₁, Y =k_cy₁, T =ωt₁, h^′=k_ch, k^′=k/k_c, l^′=l/k_c,

(25) ζ₁₀^′ =kcζ10, ζ₀₁^′ =kcζ01, E =φ₀₀⁽¹⁾/

ω k_c²

.

Thus we obtain the following three equations that describe the nonlinear evolution at the lowest order under crossing sea states assumption over finite depth water:

2i∂ζ10

∂T +iβ₀

k ∂

∂X +l ∂

∂Y

ζ₁₀+β₁

k ∂

∂X +l ∂

∂Y 2

ζ₁₀+β₂

l ∂

∂X −k ∂

∂Y 2

ζ₁₀

=λ1ζ₁₀²ζ₁₀^∗ +λ2ζ10ζ01ζ₀₁^∗ +ζ10L₊E, (26)

2i∂ζ01

∂T +iβ₀

k ∂

∂X −l ∂

∂Y

ζ₀₁+β₁

k ∂

∂X −l ∂

∂Y 2

ζ₁₀+β₂

l ∂

∂X +k ∂

∂Y 2

ζ₀₁

=λ1ζ₀₁²ζ₀₁^∗ +λ2ζ01ζ10ζ₁₀^∗ +ζ01L₋E, (27)

σ² ∂²

∂T² −hσ( ∂²

∂X² + ∂²

∂Y²)

E =L₊(ζ10ζ₁₀^∗)+L₋(ζ01ζ₀₁^∗). (28) In Eqs.(26)–(28)the primes onζ₁₀^′ ,ζ₀₁^′ ,h^′,k^′,l^′have been dropped out. Thusk≡cosθ,l≡sinθ, whereθis the angle made by the direction of propagation of either wave packet with the positive direction of x axis. The coefficientsβ_i,λ_i’s are given in AppendixBand the operatorsL_±appearing in Eqs.(26)–(28)are given below:

L_±≡2

k ∂

∂X ±l ∂

∂Y

−(1−σ²) ∂

∂T.

The two wave systems travel with their own group velocities and their evolutions are governed by Eqs.(26)–(28). The terms containing group velocity, i.e., the terms containingβ0in Eqs.(26)and(27) cause inconvenience in performing stability analysis. In order to avoid this difficulty we now rewrite Eqs.(26)–(28)in a reference frame that moves along positive x-direction with a velocity equal to the group velocity component of either wave packed along that direction. This is implemented by the following transformation:

ξ =X−kcgT, η=Y, τ =ǫT, (29) wherecgis the dimensionless group velocity given bycg=Cg/

ω k_c

=β0/2 . Equations(26)–(28) then assume the following forms:

2i∂ζ₁₀

∂τ +iβ0l∂ζ₁₀

∂η +β1

k ∂

∂ξ +l ∂

∂η 2

ζ10+β2

l ∂

∂ξ −k ∂

∂η 2

ζ10

=λ1ζ₁₀²ζ₁₀^∗ +λ2ζ10ζ01ζ₀₁^∗ +ζ10L₊E, (30)

2i∂ζ01

∂τ −iβ0l∂ζ01

∂η +β1

k ∂

∂ξ −l ∂

∂η 2

ζ01+β2

l ∂

∂ξ +k ∂

∂η 2

ζ01

=λ1ζ₀₁²ζ₀₁^∗ +λ2ζ01ζ10ζ₁₀^∗ +ζ01L₋E, (31)

μ1

∂²E

∂ξ² +μ2

∂²E

∂η² =L₊(ζ10ζ₁₀^∗)+L₋(ζ01ζ₀₁^∗), (32) whereL_±now stands for

L_±≡

2+(1−σ²)cg k ∂

∂ξ ±2l ∂

∂η.

The coefficientsμ1,μ2appearing in Eq.(32)are given in AppendixB. Equations(30)–(32)reduce to Davey-Stewartson²⁰ equations in the absence of the second wave packet. Moreover, in the limit h → ∞, the coefficientsβi’s andλi’s are in agreement with the corresponding coefficients of the equations derived in Onoratoet al.⁷and also with Shuklaet al.⁶For very smallh, the coefficientsβ0, β1,β2,λ1,λ2,μ1, andμ2are of orderO(1),O(h²),O(1),O(h⁻⁴),O(h⁻²),O(h²),O(h²), respectively.

So, Eqs.(30)–(32)are appropriate for describing crossing sea states for any depth of water except for shallow water depth case for which the depth of water is much less than the wavelength, i.e., when the dimensionless depth of water k_ch ≪1. Crossing sea states in shallow water for small crossing angles can be described by KP equation (Kadomtsev and Petviashvili²²) that has recently been reported by Ablowitz and Baldwin.²³

IV. STABILITY ANALYSIS

Equations(30)–(32)possess the following solution which is uniform in space:

ζ₁₀= A₀e^{−iτ ω1}, ζ₀₁=B₀e^{−iτ ω2}, E =E₀, (33) whereA0,B0, andE0are three real constants and the nonlinear frequency shiftsω1andω2are given by

ω1=1 2

λ1A²₀+λ2B₀²

, ω2= 1 2

λ1B₀²+λ2A²₀

. (34)

To study the stability of the uniform solution(33), we introduce the following infinitesimal pertur- bation:

ζ10= A0e^{−iτ ω1} 1+A^′

, ζ01=B0e^{−iτ ω2} 1+B^′

, E =E0(1+E^′). (35) Substituting(35)in Eqs.(30)–(32), then linearizing with respect to perturbed quantities and assuming A^′=A^′_r+i A^′_i, B^′=B_r^′+i B_i^′, E^′=E_r^′+i E_i^′, we separate real and imaginary parts. Thus we

obtain six equations in A^′_r, A^′_i, B_r^′, B_i^′, E_r^′, and E_i^′. Next, we take Fourier transform of these equations with respect toξ,ηdefined by

fˆ= 1 2π

_∞

−∞

f (ξ, η)e^{−i(Kξ+Lη)}dξdη, (36) wherefstands forA_r^′,A^′_i,B_r^′,B_i^′,E_r^′, andE_i^′. Assuming time dependence of ˆA^′_r, ˆA^′_i, ˆB_r^′, ˆB_i^′, ˆE_r^′, and Eˆ_i^′to be of the form exp (−iτ), we finally arrive at the following set of six equations:

−[β1(k K +l L)²+β2(l K−k L)²+2λ1A²₀] ˆA_r^′ +i(2−β0l L) ˆA^′_i

−2λ2B₀²Bˆ_r^′−i[

2+(1−σ²)cg

k K +2l L]E0Eˆ_r^′ =0, (37)

−i(2−β0l L) ˆA^′_r−[β1(k K+l L)²+β2(l K−k L)²] ˆA^′_i−i[

2+(1−σ²)cg

k K+2l L]E0Eˆ_i^′=0, (38)

−2λ2A²₀Aˆ^′_r−[β1(k K −l L)²+β2(l K+k L)²+2λ1B₀²] ˆB_r^′ +i(2+β0l L) ˆB_i^′−i[

2+(1−σ²)cg

k K −2l L]E0Eˆ_r^′ =0, (39)

−i(2+β0l L) ˆB_r^′−[β1(k K−l L)²+β2(l K+k L)²] ˆB_i^′−i[

2+(1−σ²)cg

k K−2l L]E0Eˆ_i^′=0, (40) 2i[

2+(1−σ²)cg

k K+2l L]A²₀Aˆ^′_r+2i[

2+(1−σ²)cg

k K −2l L]B₀²Bˆ_r^′

+[μ1K²+μ2L²]E0Eˆ_r^′ =0, (41)

[μ1K²+μ2L²]E0Eˆ_i^′=0. (42) Since the factor (μ1K²+μ2L²) is non-vanishing forh=0, (K,L)=(0, 0), from Eq.(42), it follows that ˆE^′_i=0. Moreover, using Eq. (41)we can eliminate ˆE_r^′ from the other Eqs.(38)–(40). Thus, the system of Eqs.(37)–(42)reduces to a system of four equations in four unknowns ˆA^′_r, ˆA^′_i, ˆB_r^′, Bˆ_i^′. Using the condition for existence of a non-trivial solution of this reduced system, we get the following nonlinear dispersion relation:

(2−β0l L)²−P₊{P₊+2(λ1+R Q²₊)A²₀}

(2+β0l L)²−P₋{P₋+2(λ1+R Q²₋)B₀²}

=4P₊P₋(λ2+R Q₊Q₋)²A²₀B₀². (43) The expressions forP_±,Q_±, and R are as follows:

P_±=β₁(k K ±l L)²+β₂(l K ∓k L)², Q_±=

2+(1−σ²)cg

k K ±2l L,

(44) R=[μ1K²+μ2L²]⁻¹.

To solve Eq.(43)numerically we construct a mesh in the perturbed (K,L) wavenumber plane by drawing two families of parrel straight lines: K=Ki =is, L= Lj =js, i,j=1(1)N, sbeing the step size. At each mesh point (Ki,L_j) Eq.(43)is a biquadratic equation in. Each such equation is solved usingMATLABtakings=0.0045,N=445 and keepinghandθ as constants. Then we have plotted growth rate of instability (the imaginary part of) in (K,L) plane for different values of water depth and takingθ =π/8, π/4, and 3π/8. These are shown in Figs.1–5. In Figs.1–3,

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

K

L

θ = π/8

−1 −0.5 0 0.5 1

K θ = π/4

−1 −0.5 0 0.5 1

K θ = 3π/8

0 0.005 0.01 0.015 0.02 0.025

FIG. 1. Growth rate of instability in perturbed wavenumber plane as obtained from(43),A0=B0=0.1,h=1.

A0 =B0 =0.1 while in Figs.4–5,A0 =0.12,B0 =0.07. For largeh, figures are in qualitative agreement with the corresponding figures for infinite depth case.

For unidirectional perturbations along x-direction, Eq.(43)can be solved foras follows:

²= 1 4P K²

P K²+(λ1+S)(A²₀+B₀²)±

{(λ1+S)(A²₀−B₀²)}²+4(λ2+S)²A²₀B₀²

, (45) wherePandSare given below:

P=(β1k²+β2l²), S= {2+(1−σ²)cg}²k²/μ1. (46) In the limith→ ∞,given by(45)is in agreement withgiven by Eq. (11) of Onoratoet al.⁷ Equation(45)shows that for unidirectional perturbations along x-direction, instability occurs only when the following inequality holds:

P

(λ1+S)(A²₀+B₀²)±

{(λ1+S)(A²₀−B₀²)}²+4(λ2+S)²A²₀B₀²

<0. (47)

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

K

L

θ = π/8

−1 −0.5 0 0.5 1

K θ = π/4

−1 −0.5 0 0.5 1

K θ = 3π/8

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

FIG. 2. Growth rate of instability in perturbed wavenumber plane as obtained from(43),A0=B0=0.1,h=2.

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

K

L

θ = π/8

−1 −0.5 0 0.5 1

K θ = π/4

−1 −0.5 0 0.5 1

K θ = 3π/8

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

FIG. 3. Growth rate of instability in perturbed wavenumber plane as obtained from(43),A0=B0=0.1,h=5.

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

K

L

θ = π/8

−1 −0.5 0 0.5 1

K θ = π/4

−1 −0.5 0 0.5 1

K θ = 3π/8

0 0.005 0.01 0.015 0.02

FIG. 4. Growth rate of instability in perturbed wavenumber plane as obtained from(43),A0=0.12,B0=0.07,h=2.

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1

K

L

θ = π/8

−1 −0.5 0 0.5 1

K θ = π/4

−1 −0.5 0 0.5 1

K θ = 3π/8

0 0.005 0.01 0.015 0.02 0.025

FIG. 5. Growth rate of instability in perturbed wavenumber plane as obtained from(43),A0=0.12,B0=0.07,h=5.

0 1 2 3 4 5 6 7 8 0

10 20 30 40 50 60 70 80 90

h θ^o

β1k²+β

2l²=0 II

I I I

I

S S

S

FIG. 6. Stability and instability regions inh−θplane for unidirectional perturbations: “S” denotes stability and “I” denotes instability.

The instability criterion (47) is diagrammatically presented in Fig. 6. This figure demonstrates whether there will be modulational instability or not for a given value ofθ and for a given dimen- sionless depth of waterh. The termP=(β1k²+β₂l²) which is, in fact, the coefficient of ^∂_∂X^2ζ102 in Eq.(26)and the coefficient of^∂_∂X^2ζ012 in Eq.(27), changes sign across the curve:θ=ar ctan(√

−β1/β2) in h − θ plane. It is clearly observed in Fig. 6 that for large h the curve is asymptotic to θ =ar ctan(1/√

2), which is the critical value ofθ for infinite depth water (Shuklaet al.⁶ and Onoratoet al.⁷). For largeh, instability is observed again in the range 68.02⁰< θ≤90⁰, which is in agrement with Fig.2of Onoratoet al.⁷In Figs.7–9, we have plotted growth rate of instabilityGr, imaginary part ofas obtained from Eq.(45)for unidirectional perturbations. Figure7shows that the growth rate of instability of one wave packet increases with the increase of the amplitude of the second wave packet. From Fig.8, it becomes clear that the growth rate of instabilityG_rdecreases

0 0.1 0.2 0.3 0.4 0.5 0.6

0 0.01 0.02 0.03

K Gr

h=1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0

0.01 0.02 0.03 0.04

K Gr

h=2.5

FIG. 7. Growth rate of instabilityGras obtained from(45)forθ=15^o, ———A0=B0=0.1, - - - - -A0=0.1,B0=0.05, . . . . .A0=0.1,B0=0.

0 0.1 0.2 0.3 0.4 0

0.004 0.008 0.012 0.016

K Gr

θ = 35^o

0 0.1 0.2 0.3

0 0.003 0.006 0.009

K G_r

θ = 75^o

FIG. 8. Growth rate of instabilityGras obtained from(45):A0=B0=0.1, ———-h=2, - - - -h=3, - . - . - . - .h=4 , . . . . .h=6.

0 0.1 0.2 0.3 0.4 0.5 0.6

0 0.01 0.02 0.03

K Gr

h = 1.5

0 0.2 0.4 0.6 0.8

0 0.01 0.02 0.03 0.04

K Gr

h = 2.5

FIG. 9. Growth rate of instabilityGras obtained from(45):A0=B0=0.1, ———-θ=4^o, - - - -θ=10^o, - . - . - . - .θ=17^o, . . . .θ=23^o.

with the increase of water-depth. Figure9shows the variation inG_rwith respect to the change in values ofθwhen the dimensionless depth of water is kept constant.

V. CONCLUSIONS

We have derived nonlinear evolution equations for two wave systems having the same central frequency and meeting obliquely over the surface of water. Both the wave systems are assumed to be narrow banded. The evolution equations derived here are valid for any depth of water except for shallow water. Using the evolution equations we have carried out stability analysis of two Stokes wave trains. Results are all shown graphically. Figures showing the growth rate of instability establish the possibility of generation of freak waves over finite depth water. Moreover, in a situation of crossing sea states over finite depth water, growth rate of instability is much higher than that for the similar type of interaction over infinite depth water. It is also found that the growth rate of instability becomes higher as the depth of water becomes lesser.

ACKNOWLEDGMENTS

The authors are grateful to the referees for their valuable suggestions for improvement of the earlier version of this paper.

APPENDIX A:Amn,ζmnAT THE LOWEST ORDER

A10 = − ig ωcosh(kch)

1+iǫh

kc

tanh(kch)(k ∂

∂x1 +l ∂

∂y1

)−iǫ ω

∂

∂t1

ζ10,

A01 = − ig ωcosh(kch)

1+iǫh

kc

tanh(kch)(k ∂

∂x1 −l ∂

∂y1

)−iǫ ω

∂

∂t1

ζ01,

A20 =F1ζ₁₀², ζ20 =F2ζ₁₀², A02 =F1ζ₀₁², ζ02=F2ζ₀₁², A₁₁ =F₃ζ₁₀ζ₀₁, ζ₁₁=F₄ζ₁₀ζ₀₁,

A₁₋₁ =0, ζ₁₋₁=

k_ctanh(kch)−(k²−l²)g² ω²

ζ₁₀ζ₀₁^∗,

F1 = igk_c 2ωcosh(2kch)

2 sinh²(kch)−1

sinh(2kch) −2gkc

f {3+2 sinh²(kch)}

,

F2 = 2gk_c² f

3+2 sinh²(kch) ,

F₃ = i

2ωcosh(2kh) 2g

f1{−4gk²cosh(2kh)+3gkkcsinh(2kh) tanh(kch)

−k(k²−l²)g²

ω² sinh(2kh)} +3gkctanh(kch)−(k²−l²)g² ω²

,

F4 = 1 f1

8gk²cosh(2kh)−2ksinh(2kh){3gkctanh(kch)−(k²−l²)g² ω² }

,

f =4ω²cosh(2kch)−2gkcsinh(2kch), f1=4ω²cosh(2kh)−2gksinh(2kh).

APPENDIX B: COEFFICIENTS OF THE EVOLUTION EQUATIONS

β0 =1−h(σ−σ⁻¹), β1= −1 4−1

2h(σ−σ⁻¹)+1

4h²(3σ²−σ⁻²−2), β2 = 1

2−1

2h(σ−σ⁻¹), λ1=6−σ²−σ⁻²−3−σ⁻²

1+σ² +(3−σ²)(3−σ⁴) 2σ⁴(1+σ²) , λ₂ =2+8k²−4l²−σ²−σ⁻²−[(k²−l²)σ⁻¹−σ]²−[ktanh(2kh)−2k²σ⁻¹]

×[(k²−l²)σ⁻¹−3σ]−[4k²−ktanh(2kh){3σ−(k²−l²)σ⁻¹}]

×[ktanh(2kh)−2k²σ⁻¹−(k²−l²)σ⁻¹+σ]/[2σ−ktanh(2kh)], μ1 =k²c²_gσ²−hσ, μ2 = −hσ.

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