- Email: [email protected]
Instability of capillary-gravity waves in water of arbitrary uniform depth
WAVE MOTION 9 (1987) 483-492 NORTH-HOLLAND
483
INSTABILITY OF CAPILLARY-GRAVITY WAVES IN WATER OF ARBITRARY UNIFORM DEPTH* C.V. E A S W A R A N and S.R. M A J U M D A R Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada T2N IN4 Received 17 July 1986, Revised 9 February 1987
The Benjamin-Feir instability of periodic capillary-gravity waves on a liquid layer of arbitrary uniform depth is investigated. When surface tension is present, there is always instability for some wavenumber and liquid depth and bounds on the sideband frequencies for unbounded amplification are derived. The results are compared with the slow modulation theory using an averaged Lagrangian.
1. Introduction This p a p e r investigates the effect o f surface tension on the B e n j a m i n - F e i r instability o f small amplitude surface waves on a liquid layer o f arbitrary u n i f o r m depth h. The p r o b l e m o f u n b o u n d e d amplification o f sidebands (the " n o i s e " ) propagating along with a primary wave has first been considered by Benjamin and Feir [1] for a liquid o f infinite depth. Subsequently Benjamin [2] extended the analysis for a liquid layer o f uniform depth h. His result s h o w e d that u n b o u n d e d amplification o f sidebands occurs if the w a v e n u m b e r k o f the main. wave satisfies kh > 1.363 . . . . This particular limit has been f o u n d in a different context by W h i t h a m [3]. The slow m o d u l a t i o n equations for water waves changes their character from hyperbolic to elliptic for kh = 1.363 . . . . The effect o f surface tension on B e n j a m i n - F e i r instability for an infinitely deep liquid has been investigated by Barakat [4]. He finds that capillary-gravity waves are unstable to sideband m o d u l a t i o n s in a deep liquid and gives b o u n d s on the sideband frequencies for the growth o f instability. O u r aim here is to extend the analysis to a liquid layer o f uniform depth h. The analysis closely follows [2] and we retain m a n y o f its notations for easy reference. The main difficulty is the excessive algebra involved. Even the process o f deriving Stokes-wave solutions with surface tension up to s e c o n d - o r d e r terms in ka (where a is the amplitude, small but finite) is formidable. We have been able to use the capabilities o f the algebraic m a n i p u l a t i o n system M A C S Y M A to handle the algebra t h r o u g h o u t this paper. We first briefly recall the B e n j a m i n - F e i r analysis. The primary wavetrain is assumed to have amplitude a, w a v e n u m b e r k and frequency to. Two perturbing wavetrains with wavenumbers k~ and k2, frequencies to~ and to2 respectively are introduced and allowed to interact nonlinearly with the primary wavetrain, It is assumed that k, = k( l + k'),
tot = to(1 + to'),
k2 = k ( 1 - k'),
to2 = to(1 - to'),
(1)
* This paper was presented at the Tenth U.S. National Congress of Applied Mechanics held in Austin, Texas, June 16-20, 1986. 0165-2125/87/$3.50 O 1987 Elsevier Science Publishers B.V. (North-Holland)
484
C. V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves
where k', to' are small compared to 1. The amplitudes of these sidebands, e~ and e2, are taken to be small compared to a, and slowly varying functions of time. The arguments of the fundamental harmonics of the sidebands are ( i = 1,2),
~=k,x-to,t-3,,
(2)
where Yi (i = 1, 2 ) a r e unknown slowly varying functions of time. As the nonlinear interaction between the main waves and the sidebands evolves, the product of sideband terms and the second harmonics of the main wavetrain give rise to terms with arguments 2~--ffl,2-----~2,1+("/1+"/2). NOW if the condition 0 = y l + 3,2~ constant ( S 0 , rr) is satisfied, these two modes become resonant with each other and the amplitudes of the sidebands can grow exponentially in time with the resultant distortion of the main wavetrain. The main task of the Fourier-mode analysis is to show that the condition yl + 3'2 -~ constant is possible.
2. Capillary-gravity Stokes waves
In this section we derive the Stokes-wave solution for periodic capillary-gravity waves in a liquid of arbitrary uniform depth h. We take the x-axis horizontally on the liquid surface, the y-axis vertically up, with the orgin at the undisturbed liquid surface. We denote the surface by y = ~7(x, t).
(3)
Considering an irrotational inviscid liquid, the velocity potential q~(x, y, t) satisfies (suffix denoting partial derivative) q0xx+ q0yy= 0,
t > 0,
throughout the liquid,
(4)
subject to the no-slip condition at the bottom qby = 0,
(5)
y=-h
On the free surface we have the two boundary conditions rh + r/xqbx- q0y = 0
(6)
o n y = ~7(x, t),
( ~ t ..~- ~(q~x+qb~)-rr/xx(l+~72)-3/2+g~7 1 2
=0
o n y = r/(x, t).
(7)
In (7), ~" denotes surface tension per unit density. We look for small amplitude periodic wavetrain solutions (Stokes waves) for (3)-(7). If a denotes the amplitude and k the wavenumber, we require that ka is small. To facilitate the analysis we introduce the nondimensional groups p = zk2/g,
(8)
H = kh.
Then by successive approximation the following Stokes-wave solution is found up to O(k2a 2) terms: rl = ka2A + a cos ~'+ k a 2 P cos 2~',
q0 _ toa cosh(ky + H ) sin ~+ toa2Q c o s ( 2 k y + 2 H ) k sinh H
(9) sin 2if,
(10)
C. V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves
485
where ~"= k x - tot,
(11)
A = -½(1 +p) cosech (2H),
(12)
p =
(13)
1 - 2 sinh 2 H ) sinh(2H) + sinh(4H) tO2o' 4 sinh 2 H{2togcosh(2H) - gk(1 + 4 0 ) sinh(2H)}
( 1 - 2 sinh 2 H)tog+ g(1 + 4 p ) sinh(2H) Q = 4 sinh 2 H{2tog cosh(2H) - gk(1 + 40) sinh(2H)}'
(14)
to 2= gk(1 + p ) tanh H.
(15)
The dispersion relation up to O(k2a 2) is found by suppressing secular terms at the third order: to 2 = tO2o(1+
k2a2D2),
(16)
where D 2=
2Q cosh 2 H + P cosech(2H) + 2A cosech(2H) + 1.
(17)
The perturbation solution breaks down when the denominators in (13) and (14) vanish; this occurs when (1 + 4 p ) sinh(2H) = 2(1 + p ) t a n h ( H ) cosh(2H).
(18)
This breakdown is associated with second harmonic resonance and solutions valid at and near the critical wavenumbers can be constructed using modified scales similar to the PLK method [5]. In this paper we confine ourselves to nonsingular wavenumbers where the solutions (9)-(10) are valid.
3. Perturbation analysis As is usual in similar situations, the perturbation scheme proceeds by introducing a small disturbance and investigating its asymptotic behaviour in time. Thus we set
n =n,+,~,
(19)
(~ ~ (~1 + ~,
(20)
where ~1, ~1 are the main waves given by (9) and (10) and ~, ~ are small perturbations whose nature is to be determined. To derive the governing equations for ~ and ~ we substitute (19) and (20) into (6) and (7) and reduce the resulting equations evaluated at y = Th to equations evaluated at y = 0 using a Taylor series about y = 0. After substituting the expressions (9) and (10) for ~ and ~, into these equations, we obtain the following two linear equations governing the perturbations ~ and ~: ~ , - ¢~yly=0+ a[-k{C~x + ~ coth(H)} sin ~ + {-t~yy + ~x coth(H)} cos ~']y=o
+ a2[{kto~x(Q sinh(3H) cosech(H) - Q+½) -¼~yy - Pk~yy} cos(2~') + { k 2 w ~ ( 2 Q - 2 Q sinh(3H) c o s e c h ( n ) - 1 ) - ½ k ~ y - 2 P k Z ~ x } sin(2~) + ½kzo~x- ~''~ryy- kAq~yy]y=O= 0,
(21)
486
C. V. Easwaran, S.R. Majumdar / Instability of capiUary-gravity waves
~t[y=O-'l'~xx + g~ + a[to~y sin ¢ + {__to2~ .at_to(~x coth(H) + ~ty} cos
~']y=O
+ a2[{kto~y(½Q sinh(4H) cosech2(H) - 2Q coth(H) +½ coth(H)) +½tot~yy} sin(2¢) + {QkzoZ~(7coth(n) - sinh(4H) cosech2(H)) +/aoqbx(½Q cosh(4H) cosech2(H) +½Q cosech2(H)(1-2 cosh(H)) +~)1..t_~tot~xyl ~ coth(H) +a~i)tyy + °1
kP~ty } cos(2~')
+½kzo2~ coth(H)+½kzo~ "[-~£O~xy ' " coth(H) "~-4(~lyy ' " "~ kAtib,y]y=o " = 0.
(22)
The next step in the analysis is to introduce the right form of the perturbing wavetrains ~. Following Benjamin [2] we take this as ~ = ~1 + ~2 with ~, = e, cos ~, + ka{A, cos(~"+ ~',) + B, cos(~" - ~',)}+ O(k2a2e,)
(i = 1, 2),
(23)
where
~, = k,x - tod - % (i = 1, 2).
(24)
As (23) shows, the perturbations consist of a pair of sideband modes together with the product of their nonlinear interaction with the primary wavetrain. As shown in [1], the neglected terms with arguments 2ff + ~'~ are nonresonant and need not be considered while those with arguments 2 ~ - ~ may be merged with those having argument ~. It is further assumed that ei, y, are slowly varying functions of time, with (25)
"Yi= O(tok2a2),
di = O(tokEa2ei),
to', k'= O(ka).
(26)
I[ will also be necessary to assume that the unknown coefficients Ai, B~ are O(1). The perturbing velocity potentials q~, are such that V2q~, = 0, (q~,)y = 0,
(27)
y=-h.
(28)
These are then assumed to be of the form ~ _ cosh ki(y + h) {e,(to~L, + "~,M~) sin ~ + i,N~ cos ~,} ki sinh(k/h) coshl(k + ki)(y + h)l
+ toae, Ci
sinhl(k + k,)h I
c o s h l ( k - k,)(y + h)l sin(~+~',)+D,
sinhl(k_k~)h I
I sin(~-~',).j
(i=1,2). (29)
Here, too, the coefficients L~, M~, Ni, C~, D,, whose natures are unknown, will be taken to be O(1). We now proceed to determine the equations governing the coefficients in ~ and t~. Towards this end, equations (23) and (29) for ~ , ~i are substituted in (21) and (22) and all terms are reduced to simple harmonic components. The process is quite laborious and the excellent abilities of MACSYMA were amply used. Once the simple harmonic components are separated, we require that each of the components satisfy the equations independently. Thus separating the components with arguments ~ - ~ and ~"+ ~ we obtain the following 8 linear equations for A~, B~, C~ and Di (i -- 1, 2):
k( to + toi)A~ - to( k + k,)C~ = ¼(k + k,) cosech(H) cosech(k~h){(to + to~)(sinh((k + k~)h) + sinh((k - k~)h))}
(i = 1, 2),
(30)
487
C. V. Easwaran, S.R. Majumdar / Instability of capillary.gravity waves
k[g + r ( k + k~)2]A, - to(to + to,) coth((k + k J h ) C, = ~ cosech(H) cosech(k,h){(to2 + to2) cosh((k + ks)h) - (to + to,)2 cosh((k - ki)h)}
( i = 1,2), (31)
k(to - to,)B,- to( k - k,) D, = ¼cosech(H) cosech(k,h){(to + to,) sinh((k + k,)h) - (to - to,) sinh((k - k,)h)}
(i = 1, 2),
(32) k[g + z k ( k - kJZ]B, - to(to - to,) coth((k - k J h )D, = ¼cosech(H) cosech(kih){(to -to,)2 cosh((k + k,)h) - (to2+ to2) cosh((k - k,) h)}
( i = 1,2). (33)
Before proceeding it is necessary to consider the ratio of wavenumber and frequency perturbations, to' and k' respectively. For this purpose we note that to a first approximation the sidebands may be expected to satisfy the linear dispersion relation at wavenumber k. Noting that k', to' are assumed to be much smaller than k and to, we have, to a first approximation, toto' d kk' = group velocity at wavenumber k =
[ ( g k + rk 3) tanh(kh)],
or say, (34)
t o ' / k ' = ½{(1+ 3p)/(1 + p ) + 2 H cosech(2H)} = A. The expression (34) has been obtained by omitting O(kZa 2) terms. The approximate solutions of (30)-(33) are now readily obtained, in the limit to, ~ to, k, -~ k:
(35)
A, = 2P, c o t h ( n ) +½H cosech2(n) ~ l* H coth(n)-A 2 J' L
(36)
B, = - f
C,=2Qsinh(2H)
D1 •1 = +H{ ½)t
(37)
( i = 1,2),
cosech2(H) + coth2(H) ~ j.
(38)
Next we separate components at wavenumbers k, from (21) and (22) with (23) and (29) substituted in. Taking approximations to O(tok2a%J and O(to2ka2eJ terms, we obtain the pair of equations (39) and (42): e,[to,(1 - L,) + 3~,(1- M,)] sin ¢, + g,(1 - N~) cos t', = ~tok2a2[e,R sin ~', + (6,1 e2 + 6,2el)S sin(C, + 0)]
(i = 1, 2),
(39)
where R = 3 + (2A + A + B) c o t h ( H ) + 2 C coth(2H) - I D I / H ,
(40)
S = 3+ ( P + B) c o t h ( H ) + 2Q c o t h ( 2 H ) - I D I / H ,
(41)
and A. B. C, D are given by (35)-(38) and 0 = Yl + Y2. e,[to.l(gk, + rk~) tanh(k,h) - to,L,- 3)i(1 + M,)] cos ~', + d,(l + N J sin(¢J = ltok2a2[e,U cos ¢/+ (6/le2 + 6,2et) V cos({,+ 0)],
(42)
C. V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves
488
where U = --~+ (2/t + A + B) t a n h ( H ) - 2C cosech(2H) + IDI/14, V = -¼+ ( P + B) t a n h ( H ) - 2 Q +
(43)
IDI/H.
(44)
One observes that the forms of these equations are similar to those of Benjamin [2] (except for a change of sign in front of IDI which is probably a misprint in [2]) into which they reduce if surface tension is made zero. Once again we require that the separate simple harmonic components satisfy (39) and (42) independently. Hence upon equating coefficients of cos ~ and sin ~i in these equations we obtain ~,(1 - N~) = ½wk2a2{(8,1e2 + 6,2e~)S sin 0},
(45)
e~(1 + Ni) = -½¢ok2a2{(6~le2+ 6~2e,) V sin 0}.
(46)
Adding these two equations, we have
ei = (½cok2 a2X sin 0)(t~i,ezq- t~i2el) ,
(47)
x = ½ ( s - v ) = 1 + ( p + B) cosech(2H) + 2 Q coshE(H) - I D I / H .
(48)
where
To obtain equations governing 0 we separate sin ~ components from (39) and cos ~ components from (42) and get, respectively, ei[~oi(1 - L,) + 3~i(1- M~)] = ½~ok2a2[eiR + (8ile2+ 6~2el)S cos 0]
( i = 1,2),
(49)
e i [ o ~ ( g k i + zk 3) tanh(k~h) - w~L~- ~i(1 + M~)] =½ojk2aZ[e~U+(6~2e2+8~zeOVcos 0]
( i = 1,2).
(50)
Subtracting (50) from (49) results in
~/~=~¢071[(gk~+,k3)tanh(kih) - w~]+~wk2 ~ 2a 2[~(R1 _ U)4 6ile2+ Bi2el X cos 0. 6iael + 6i2e2
(51)
Adding the two equations contained in (47) gives 0 = ½[w~-'{(gk, + zk 3) tanh(k~h) - w~z} + wz~{(gk2 + rk 3) tanh(kzh) - ¢o2}] +½~ok2a2[ (R-U)+e~+e2Xe,e2 cos 0].
(52)
If we denote
f ( k,) = [ (gk, + Tk 3) tanh( k,h ) ] ~/2,
(53)
then it is easy to see that, [2], to a second approximation, fZ(k,) - ro~z = f2(k) - toz - w2~o,2 Y, where
Y = 1-½[f2(k)]"/[f'(k)] 2, where the primes in (54) denote k-derivatives.
(54)
C. V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves
489
Using this expression and noting that ½(R - U) - / ) 2 = X,
(55)
we get from (52) O = tok2 a2 X ( l + e 2 + e2 } 2ele: cos 0 -toto'2Y.
(56)
This expression is crucial to the whole stability issue. If only dispersive effects were present, we will have, if cg is the group velocity at wavenumber k,
~0+ cg~ = (k'k)2ff(k) = to'2ff(k)ff(k)/[f'(k)] 2, Ot
8x
(57)
where, here, f(k) is the disperson relation to = f ( k ) . Thus the property 0 ~ constant cannot be realized unless there are effects counteracting the dispersion. This is precisely the first term on the right of (56) which represents the nonlinear effects balancing the dispersive effects given by the second term.
4. Instability criteria Integrating (47) we find
e,(t)=e,(O)cosh{½tok2a2X f~sinOdt}+[Si~e2(O)+ei2el(O)]sinh{½tok2a2 Io'sinOdt},
(58)
so that if O = constant ( # 0, rr), as t-~ ~ , ei(t)-~ oo, so that the sideband instability can be achieved. To obtain precise criteria for instability we need to uncouple the differential equations for el and e 2. This is achieved as follows: Defining T = tok2a2t, we have from (56), ~,2 y )
d(cos 0) - e l e 2 - d- T -
X - - - k - ~ a2 ele2sin 0 + ~1( e l2+ e2)Xsin 2 0 cos 0,
(59)
and from (47),
d~ d~ dT
(60)
- - -- X e l e 2 sin 0. dT
Using (60) in (59), we get cos
--0,
(61)
or, say, e 18 2 COS O + a e 2 = constant = tr,
(62)
oe= 1 - to,2y/(k2aEX)"
(63)
where
C. V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves
490
From (60), we have el2 - e 2 = constant = 2ira (1 - g),
/~ being a constant.
(64)
From (60), upon using (62) and (64), we get ~--~/ = X2{(1 - ot2) e 4 + 2cra/xe 2 - or2}.
(65)
The stability characteristics of the sidebands are determined from the quadratic in e 2 on the right of (65). We refer to Benjamin and Feir [1] for a detailed discussion of this aspect. For the present purpose it is sufficient to distinguish the following three cases: (i) - 1 < a < l: instability, (ii) a = - 1 : marginal instability, (iii) a < - 1 : stability. Using the definition of a given by (63), case (i) gives (66)
2k2a2X > 00'2 Y
as the criterion for instability. Therefore it is seen that when X ~ Y < 0 there is no instability and when X ~ Y > 0 the sidebands grow unbounded so long as the perturbation frequency satisfies co ,2 <~2k2 a2 X / y..
(67)
5. Relation to slow modulation approach
As pointed out in the introduction the wavenumber limit kh = 1.363... for gravity waves also follows from Whitham's slow modulation approach using an averaged Lagrangian. We have shown that an average Lagrangian for capillary-gravity Stokes waves is given by, [6],
ttg~rK
k2E 2
/
-3cr 2
Tk2+rgv
)Kio
-V
1} 3-rg
2to / 4r o j.
(68)
Here the notation is that of Whitham [7, Section 16.6], with E = l(g + ~.k2)a 2 (the energy density of capillary-gravity waves) To = tanh(kho),
U=
(g + zk2)(3 - T°2) 4To( T2(g + rk 2) - 3 ~'k2)"
In the above expressions, ho is the undisturbed water depth, h is the surface h(x, t) and 3',/3 are related to the mean velocity and height of the surface. Using the variational principle L:~ ---0 (see [7, Chapter 16] one can deduce the dispersion relation
(O = O)O-t-
k2E CO
2Co0ooD2 2Co-½Co ~--~ ~-~-5] g + rk 2
kho(gho - C 2)
1 kho
+O(E2) '
(69)
C.V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves
491
where co, Co are the phase and group velocities respectively:
Co=[(g+zk2)k-lTo] '/2,
C -'c rg+3rk2 2kho ] o - ~ oL g + r k 2 sinh(2kho)J"
Then the characteristic velocities are ( k ~ @ ) Co± tog(k)/22
1/2
,
(70)
where/22 is the coefficient of k2E/co in (69). Thus the waves are unstable to slow modulations if tog/22 < 0 and stable for tog/22>O. In the next section we indicate how this approach and the Benjamin-Feir instability compare on a stability diagram.
6. Discussion of results
Fig. 1 is a stability diagram based on the Fourier-mode perturbation analysis and is valid when ka can be considered small. The significant effect of surface tension is quite evident from the diagram. One observes that the neutral stability curve starting at kh = 1.363... for p = 0 tends to lower values of kh as p increases. The branch (a) corresponds to Y = 0 and branch (b) corresponds to equation (18). In both situations the perturbation analysis is invalid and modified scales need to be introduced. 5.0
4.0
3.0
i~ i!i!i!i!i!¸iiili
ii i i i i i i i i i i i i i iiiiiiiiiiiiiiiiiiii i i i i i i i i i i i i iiiiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii i i i i i i i i i iiiiiiiiiiiiiiiiiiiiii !!i iiiiiiiiiiii
kh 2.0
1.0
0.O
o.o
~.o
2.0
-rk2/g
3.0
4.0
~.o
Fig. 1. Stability diagram based on Fourier-mode analysis. Shaded areas indicate instability.
Fig. 2 is a stability diagram based on the slow modulation approach of Section 5. This is very similar to Fig. 1 except for the appearance of two additional branches (c) and (d). (c) corresponds to gho = C~; this is where the group velocity Co coincides with the phase velocity of long waves, gv~o. In this situation 1"22becomes singular. The branch (d) corresponds to 122 = 0. Once again near (c) the perturbation analysis is invalid. The branches (c) and (d) are missing in Fig. 1 because as shown by equations (56) and (54) the dispersive effect balancing nonlinearity is taken only up to the first order in the Fourier mode analysis. As seen from (69) the branches (c) and (d) in Fig. 2 arise from the second order effects of dispersion.
492
C, V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves 5.0
4.0
iiiiiiii!ii!!iiiiiiiiii,ii i iiii iiii!iiiiiiiiiiiiiiiiiiii ( d ) iiiiiiiiii!ili!iiiiiii!iiii
kh 2.0 :::::::::
::::::::::
:::
1.0
O.O
o.o
,~.o
2.o
3.o
4.o
5.o
-r k2/g Fig. 2. Stability diagram based o n the averaged Lagrangian. Shaded areas indicate instability.
This instability diagram is identical to that of Kawahara [8] who uses a multiple scale formalism to derive a nonlinear Sehr6dinger equation governing the self-modulation of capillary-gravity waves. It is interesting to note that if surface tension is not zero there is always instability at some value of kh. Apart from surface tension, viscosity is expected to have a significant effect on the Benjamin-Feir instability, particularly at small wavelengths. Inclusion of viscosity significantly modifies the Stokes-wave expansion and the resulting analysis of instability poses extreme algebraic difficulties. We hope to use MACSYMA to overcome this difficulty in future work.
Acknowledgment This work was done with the aid of MACSYMA, a large symbolic manipulation program developed at the MIT Laboratory for Computer Science and supported from 1975 to 1983 by the National Aeronautics and Space Administration under grant NSG 1323, by the Office of Naval Research under grant N00014-77C-0641, by the U.S. Department of Energy under grant ET-78-C-02-4687, by the U.S. Airforce under grant F49620-79-C-020, and since 1982 by Symbolics Inc. of Cambridge, MA. This research was supported by grant A5294 from the Natural Sciences and Engineering Research Council of Canada.
References [ 1] [2] [3] [4] [5]
T.B. Benjamin and J. Feir, "The disintegration of wavetrains on deep water, Part 1: Theory," J. Fluid Mech. 27, 417-430 (1967). T.B. Benjamin, "Instability of periodic wavetrains in nonlinear dispersive systems", Proc. Roy. Soc. London A 299, 59-75 (1967). G.B. Whitham, "Nonlinear dispersion of water waves", J. Fluid Mech. 27, 399-412 (1967). R, Barakat, "Instability of periodic capillary-gravity waves on deep water", Wave Motion 6, 155-165 (1984). W. Pierson and P. Fife, "Some nonlinear properties of long crested periodic waves with lengths near 2.44 centimeters", J. Geophys. Res. 66, 163-179 (1961). [6] C,V. Easwaran and S.R. Majumdar, "Instability of capillary-gravity waves using an averaged Lagrangian", University of Calgary, Preprint. [7] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York (1974). [8] T, Kawahara, "Non-linear self modulation of capillary-gravity waves on liquid layer", J. Phys. Soc. Japan 38, 265-270 (1975).
483
INSTABILITY OF CAPILLARY-GRAVITY WAVES IN WATER OF ARBITRARY UNIFORM DEPTH* C.V. E A S W A R A N and S.R. M A J U M D A R Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada T2N IN4 Received 17 July 1986, Revised 9 February 1987
The Benjamin-Feir instability of periodic capillary-gravity waves on a liquid layer of arbitrary uniform depth is investigated. When surface tension is present, there is always instability for some wavenumber and liquid depth and bounds on the sideband frequencies for unbounded amplification are derived. The results are compared with the slow modulation theory using an averaged Lagrangian.
1. Introduction This p a p e r investigates the effect o f surface tension on the B e n j a m i n - F e i r instability o f small amplitude surface waves on a liquid layer o f arbitrary u n i f o r m depth h. The p r o b l e m o f u n b o u n d e d amplification o f sidebands (the " n o i s e " ) propagating along with a primary wave has first been considered by Benjamin and Feir [1] for a liquid o f infinite depth. Subsequently Benjamin [2] extended the analysis for a liquid layer o f uniform depth h. His result s h o w e d that u n b o u n d e d amplification o f sidebands occurs if the w a v e n u m b e r k o f the main. wave satisfies kh > 1.363 . . . . This particular limit has been f o u n d in a different context by W h i t h a m [3]. The slow m o d u l a t i o n equations for water waves changes their character from hyperbolic to elliptic for kh = 1.363 . . . . The effect o f surface tension on B e n j a m i n - F e i r instability for an infinitely deep liquid has been investigated by Barakat [4]. He finds that capillary-gravity waves are unstable to sideband m o d u l a t i o n s in a deep liquid and gives b o u n d s on the sideband frequencies for the growth o f instability. O u r aim here is to extend the analysis to a liquid layer o f uniform depth h. The analysis closely follows [2] and we retain m a n y o f its notations for easy reference. The main difficulty is the excessive algebra involved. Even the process o f deriving Stokes-wave solutions with surface tension up to s e c o n d - o r d e r terms in ka (where a is the amplitude, small but finite) is formidable. We have been able to use the capabilities o f the algebraic m a n i p u l a t i o n system M A C S Y M A to handle the algebra t h r o u g h o u t this paper. We first briefly recall the B e n j a m i n - F e i r analysis. The primary wavetrain is assumed to have amplitude a, w a v e n u m b e r k and frequency to. Two perturbing wavetrains with wavenumbers k~ and k2, frequencies to~ and to2 respectively are introduced and allowed to interact nonlinearly with the primary wavetrain, It is assumed that k, = k( l + k'),
tot = to(1 + to'),
k2 = k ( 1 - k'),
to2 = to(1 - to'),
(1)
* This paper was presented at the Tenth U.S. National Congress of Applied Mechanics held in Austin, Texas, June 16-20, 1986. 0165-2125/87/$3.50 O 1987 Elsevier Science Publishers B.V. (North-Holland)
484
C. V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves
where k', to' are small compared to 1. The amplitudes of these sidebands, e~ and e2, are taken to be small compared to a, and slowly varying functions of time. The arguments of the fundamental harmonics of the sidebands are ( i = 1,2),
~=k,x-to,t-3,,
(2)
where Yi (i = 1, 2 ) a r e unknown slowly varying functions of time. As the nonlinear interaction between the main waves and the sidebands evolves, the product of sideband terms and the second harmonics of the main wavetrain give rise to terms with arguments 2~--ffl,2-----~2,1+("/1+"/2). NOW if the condition 0 = y l + 3,2~ constant ( S 0 , rr) is satisfied, these two modes become resonant with each other and the amplitudes of the sidebands can grow exponentially in time with the resultant distortion of the main wavetrain. The main task of the Fourier-mode analysis is to show that the condition yl + 3'2 -~ constant is possible.
2. Capillary-gravity Stokes waves
In this section we derive the Stokes-wave solution for periodic capillary-gravity waves in a liquid of arbitrary uniform depth h. We take the x-axis horizontally on the liquid surface, the y-axis vertically up, with the orgin at the undisturbed liquid surface. We denote the surface by y = ~7(x, t).
(3)
Considering an irrotational inviscid liquid, the velocity potential q~(x, y, t) satisfies (suffix denoting partial derivative) q0xx+ q0yy= 0,
t > 0,
throughout the liquid,
(4)
subject to the no-slip condition at the bottom qby = 0,
(5)
y=-h
On the free surface we have the two boundary conditions rh + r/xqbx- q0y = 0
(6)
o n y = ~7(x, t),
( ~ t ..~- ~(q~x+qb~)-rr/xx(l+~72)-3/2+g~7 1 2
=0
o n y = r/(x, t).
(7)
In (7), ~" denotes surface tension per unit density. We look for small amplitude periodic wavetrain solutions (Stokes waves) for (3)-(7). If a denotes the amplitude and k the wavenumber, we require that ka is small. To facilitate the analysis we introduce the nondimensional groups p = zk2/g,
(8)
H = kh.
Then by successive approximation the following Stokes-wave solution is found up to O(k2a 2) terms: rl = ka2A + a cos ~'+ k a 2 P cos 2~',
q0 _ toa cosh(ky + H ) sin ~+ toa2Q c o s ( 2 k y + 2 H ) k sinh H
(9) sin 2if,
(10)
C. V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves
485
where ~"= k x - tot,
(11)
A = -½(1 +p) cosech (2H),
(12)
p =
(13)
1 - 2 sinh 2 H ) sinh(2H) + sinh(4H) tO2o' 4 sinh 2 H{2togcosh(2H) - gk(1 + 4 0 ) sinh(2H)}
( 1 - 2 sinh 2 H)tog+ g(1 + 4 p ) sinh(2H) Q = 4 sinh 2 H{2tog cosh(2H) - gk(1 + 40) sinh(2H)}'
(14)
to 2= gk(1 + p ) tanh H.
(15)
The dispersion relation up to O(k2a 2) is found by suppressing secular terms at the third order: to 2 = tO2o(1+
k2a2D2),
(16)
where D 2=
2Q cosh 2 H + P cosech(2H) + 2A cosech(2H) + 1.
(17)
The perturbation solution breaks down when the denominators in (13) and (14) vanish; this occurs when (1 + 4 p ) sinh(2H) = 2(1 + p ) t a n h ( H ) cosh(2H).
(18)
This breakdown is associated with second harmonic resonance and solutions valid at and near the critical wavenumbers can be constructed using modified scales similar to the PLK method [5]. In this paper we confine ourselves to nonsingular wavenumbers where the solutions (9)-(10) are valid.
3. Perturbation analysis As is usual in similar situations, the perturbation scheme proceeds by introducing a small disturbance and investigating its asymptotic behaviour in time. Thus we set
n =n,+,~,
(19)
(~ ~ (~1 + ~,
(20)
where ~1, ~1 are the main waves given by (9) and (10) and ~, ~ are small perturbations whose nature is to be determined. To derive the governing equations for ~ and ~ we substitute (19) and (20) into (6) and (7) and reduce the resulting equations evaluated at y = Th to equations evaluated at y = 0 using a Taylor series about y = 0. After substituting the expressions (9) and (10) for ~ and ~, into these equations, we obtain the following two linear equations governing the perturbations ~ and ~: ~ , - ¢~yly=0+ a[-k{C~x + ~ coth(H)} sin ~ + {-t~yy + ~x coth(H)} cos ~']y=o
+ a2[{kto~x(Q sinh(3H) cosech(H) - Q+½) -¼~yy - Pk~yy} cos(2~') + { k 2 w ~ ( 2 Q - 2 Q sinh(3H) c o s e c h ( n ) - 1 ) - ½ k ~ y - 2 P k Z ~ x } sin(2~) + ½kzo~x- ~''~ryy- kAq~yy]y=O= 0,
(21)
486
C. V. Easwaran, S.R. Majumdar / Instability of capiUary-gravity waves
~t[y=O-'l'~xx + g~ + a[to~y sin ¢ + {__to2~ .at_to(~x coth(H) + ~ty} cos
~']y=O
+ a2[{kto~y(½Q sinh(4H) cosech2(H) - 2Q coth(H) +½ coth(H)) +½tot~yy} sin(2¢) + {QkzoZ~(7coth(n) - sinh(4H) cosech2(H)) +/aoqbx(½Q cosh(4H) cosech2(H) +½Q cosech2(H)(1-2 cosh(H)) +~)1..t_~tot~xyl ~ coth(H) +a~i)tyy + °1
kP~ty } cos(2~')
+½kzo2~ coth(H)+½kzo~ "[-~£O~xy ' " coth(H) "~-4(~lyy ' " "~ kAtib,y]y=o " = 0.
(22)
The next step in the analysis is to introduce the right form of the perturbing wavetrains ~. Following Benjamin [2] we take this as ~ = ~1 + ~2 with ~, = e, cos ~, + ka{A, cos(~"+ ~',) + B, cos(~" - ~',)}+ O(k2a2e,)
(i = 1, 2),
(23)
where
~, = k,x - tod - % (i = 1, 2).
(24)
As (23) shows, the perturbations consist of a pair of sideband modes together with the product of their nonlinear interaction with the primary wavetrain. As shown in [1], the neglected terms with arguments 2ff + ~'~ are nonresonant and need not be considered while those with arguments 2 ~ - ~ may be merged with those having argument ~. It is further assumed that ei, y, are slowly varying functions of time, with (25)
"Yi= O(tok2a2),
di = O(tokEa2ei),
to', k'= O(ka).
(26)
I[ will also be necessary to assume that the unknown coefficients Ai, B~ are O(1). The perturbing velocity potentials q~, are such that V2q~, = 0, (q~,)y = 0,
(27)
y=-h.
(28)
These are then assumed to be of the form ~ _ cosh ki(y + h) {e,(to~L, + "~,M~) sin ~ + i,N~ cos ~,} ki sinh(k/h) coshl(k + ki)(y + h)l
+ toae, Ci
sinhl(k + k,)h I
c o s h l ( k - k,)(y + h)l sin(~+~',)+D,
sinhl(k_k~)h I
I sin(~-~',).j
(i=1,2). (29)
Here, too, the coefficients L~, M~, Ni, C~, D,, whose natures are unknown, will be taken to be O(1). We now proceed to determine the equations governing the coefficients in ~ and t~. Towards this end, equations (23) and (29) for ~ , ~i are substituted in (21) and (22) and all terms are reduced to simple harmonic components. The process is quite laborious and the excellent abilities of MACSYMA were amply used. Once the simple harmonic components are separated, we require that each of the components satisfy the equations independently. Thus separating the components with arguments ~ - ~ and ~"+ ~ we obtain the following 8 linear equations for A~, B~, C~ and Di (i -- 1, 2):
k( to + toi)A~ - to( k + k,)C~ = ¼(k + k,) cosech(H) cosech(k~h){(to + to~)(sinh((k + k~)h) + sinh((k - k~)h))}
(i = 1, 2),
(30)
487
C. V. Easwaran, S.R. Majumdar / Instability of capillary.gravity waves
k[g + r ( k + k~)2]A, - to(to + to,) coth((k + k J h ) C, = ~ cosech(H) cosech(k,h){(to2 + to2) cosh((k + ks)h) - (to + to,)2 cosh((k - ki)h)}
( i = 1,2), (31)
k(to - to,)B,- to( k - k,) D, = ¼cosech(H) cosech(k,h){(to + to,) sinh((k + k,)h) - (to - to,) sinh((k - k,)h)}
(i = 1, 2),
(32) k[g + z k ( k - kJZ]B, - to(to - to,) coth((k - k J h )D, = ¼cosech(H) cosech(kih){(to -to,)2 cosh((k + k,)h) - (to2+ to2) cosh((k - k,) h)}
( i = 1,2). (33)
Before proceeding it is necessary to consider the ratio of wavenumber and frequency perturbations, to' and k' respectively. For this purpose we note that to a first approximation the sidebands may be expected to satisfy the linear dispersion relation at wavenumber k. Noting that k', to' are assumed to be much smaller than k and to, we have, to a first approximation, toto' d kk' = group velocity at wavenumber k =
[ ( g k + rk 3) tanh(kh)],
or say, (34)
t o ' / k ' = ½{(1+ 3p)/(1 + p ) + 2 H cosech(2H)} = A. The expression (34) has been obtained by omitting O(kZa 2) terms. The approximate solutions of (30)-(33) are now readily obtained, in the limit to, ~ to, k, -~ k:
(35)
A, = 2P, c o t h ( n ) +½H cosech2(n) ~ l* H coth(n)-A 2 J' L
(36)
B, = - f
C,=2Qsinh(2H)
D1 •1 = +H{ ½)t
(37)
( i = 1,2),
cosech2(H) + coth2(H) ~ j.
(38)
Next we separate components at wavenumbers k, from (21) and (22) with (23) and (29) substituted in. Taking approximations to O(tok2a%J and O(to2ka2eJ terms, we obtain the pair of equations (39) and (42): e,[to,(1 - L,) + 3~,(1- M,)] sin ¢, + g,(1 - N~) cos t', = ~tok2a2[e,R sin ~', + (6,1 e2 + 6,2el)S sin(C, + 0)]
(i = 1, 2),
(39)
where R = 3 + (2A + A + B) c o t h ( H ) + 2 C coth(2H) - I D I / H ,
(40)
S = 3+ ( P + B) c o t h ( H ) + 2Q c o t h ( 2 H ) - I D I / H ,
(41)
and A. B. C, D are given by (35)-(38) and 0 = Yl + Y2. e,[to.l(gk, + rk~) tanh(k,h) - to,L,- 3)i(1 + M,)] cos ~', + d,(l + N J sin(¢J = ltok2a2[e,U cos ¢/+ (6/le2 + 6,2et) V cos({,+ 0)],
(42)
C. V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves
488
where U = --~+ (2/t + A + B) t a n h ( H ) - 2C cosech(2H) + IDI/14, V = -¼+ ( P + B) t a n h ( H ) - 2 Q +
(43)
IDI/H.
(44)
One observes that the forms of these equations are similar to those of Benjamin [2] (except for a change of sign in front of IDI which is probably a misprint in [2]) into which they reduce if surface tension is made zero. Once again we require that the separate simple harmonic components satisfy (39) and (42) independently. Hence upon equating coefficients of cos ~ and sin ~i in these equations we obtain ~,(1 - N~) = ½wk2a2{(8,1e2 + 6,2e~)S sin 0},
(45)
e~(1 + Ni) = -½¢ok2a2{(6~le2+ 6~2e,) V sin 0}.
(46)
Adding these two equations, we have
ei = (½cok2 a2X sin 0)(t~i,ezq- t~i2el) ,
(47)
x = ½ ( s - v ) = 1 + ( p + B) cosech(2H) + 2 Q coshE(H) - I D I / H .
(48)
where
To obtain equations governing 0 we separate sin ~ components from (39) and cos ~ components from (42) and get, respectively, ei[~oi(1 - L,) + 3~i(1- M~)] = ½~ok2a2[eiR + (8ile2+ 6~2el)S cos 0]
( i = 1,2),
(49)
e i [ o ~ ( g k i + zk 3) tanh(k~h) - w~L~- ~i(1 + M~)] =½ojk2aZ[e~U+(6~2e2+8~zeOVcos 0]
( i = 1,2).
(50)
Subtracting (50) from (49) results in
~/~=~¢071[(gk~+,k3)tanh(kih) - w~]+~wk2 ~ 2a 2[~(R1 _ U)4 6ile2+ Bi2el X cos 0. 6iael + 6i2e2
(51)
Adding the two equations contained in (47) gives 0 = ½[w~-'{(gk, + zk 3) tanh(k~h) - w~z} + wz~{(gk2 + rk 3) tanh(kzh) - ¢o2}] +½~ok2a2[ (R-U)+e~+e2Xe,e2 cos 0].
(52)
If we denote
f ( k,) = [ (gk, + Tk 3) tanh( k,h ) ] ~/2,
(53)
then it is easy to see that, [2], to a second approximation, fZ(k,) - ro~z = f2(k) - toz - w2~o,2 Y, where
Y = 1-½[f2(k)]"/[f'(k)] 2, where the primes in (54) denote k-derivatives.
(54)
C. V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves
489
Using this expression and noting that ½(R - U) - / ) 2 = X,
(55)
we get from (52) O = tok2 a2 X ( l + e 2 + e2 } 2ele: cos 0 -toto'2Y.
(56)
This expression is crucial to the whole stability issue. If only dispersive effects were present, we will have, if cg is the group velocity at wavenumber k,
~0+ cg~ = (k'k)2ff(k) = to'2ff(k)ff(k)/[f'(k)] 2, Ot
8x
(57)
where, here, f(k) is the disperson relation to = f ( k ) . Thus the property 0 ~ constant cannot be realized unless there are effects counteracting the dispersion. This is precisely the first term on the right of (56) which represents the nonlinear effects balancing the dispersive effects given by the second term.
4. Instability criteria Integrating (47) we find
e,(t)=e,(O)cosh{½tok2a2X f~sinOdt}+[Si~e2(O)+ei2el(O)]sinh{½tok2a2 Io'sinOdt},
(58)
so that if O = constant ( # 0, rr), as t-~ ~ , ei(t)-~ oo, so that the sideband instability can be achieved. To obtain precise criteria for instability we need to uncouple the differential equations for el and e 2. This is achieved as follows: Defining T = tok2a2t, we have from (56), ~,2 y )
d(cos 0) - e l e 2 - d- T -
X - - - k - ~ a2 ele2sin 0 + ~1( e l2+ e2)Xsin 2 0 cos 0,
(59)
and from (47),
d~ d~ dT
(60)
- - -- X e l e 2 sin 0. dT
Using (60) in (59), we get cos
--0,
(61)
or, say, e 18 2 COS O + a e 2 = constant = tr,
(62)
oe= 1 - to,2y/(k2aEX)"
(63)
where
C. V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves
490
From (60), we have el2 - e 2 = constant = 2ira (1 - g),
/~ being a constant.
(64)
From (60), upon using (62) and (64), we get ~--~/ = X2{(1 - ot2) e 4 + 2cra/xe 2 - or2}.
(65)
The stability characteristics of the sidebands are determined from the quadratic in e 2 on the right of (65). We refer to Benjamin and Feir [1] for a detailed discussion of this aspect. For the present purpose it is sufficient to distinguish the following three cases: (i) - 1 < a < l: instability, (ii) a = - 1 : marginal instability, (iii) a < - 1 : stability. Using the definition of a given by (63), case (i) gives (66)
2k2a2X > 00'2 Y
as the criterion for instability. Therefore it is seen that when X ~ Y < 0 there is no instability and when X ~ Y > 0 the sidebands grow unbounded so long as the perturbation frequency satisfies co ,2 <~2k2 a2 X / y..
(67)
5. Relation to slow modulation approach
As pointed out in the introduction the wavenumber limit kh = 1.363... for gravity waves also follows from Whitham's slow modulation approach using an averaged Lagrangian. We have shown that an average Lagrangian for capillary-gravity Stokes waves is given by, [6],
ttg~rK
k2E 2
/
-3cr 2
Tk2+rgv
)Kio
-V
1} 3-rg
2to / 4r o j.
(68)
Here the notation is that of Whitham [7, Section 16.6], with E = l(g + ~.k2)a 2 (the energy density of capillary-gravity waves) To = tanh(kho),
U=
(g + zk2)(3 - T°2) 4To( T2(g + rk 2) - 3 ~'k2)"
In the above expressions, ho is the undisturbed water depth, h is the surface h(x, t) and 3',/3 are related to the mean velocity and height of the surface. Using the variational principle L:~ ---0 (see [7, Chapter 16] one can deduce the dispersion relation
(O = O)O-t-
k2E CO
2Co0ooD2 2Co-½Co ~--~ ~-~-5] g + rk 2
kho(gho - C 2)
1 kho
+O(E2) '
(69)
C.V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves
491
where co, Co are the phase and group velocities respectively:
Co=[(g+zk2)k-lTo] '/2,
C -'c rg+3rk2 2kho ] o - ~ oL g + r k 2 sinh(2kho)J"
Then the characteristic velocities are ( k ~ @ ) Co± tog(k)/22
1/2
,
(70)
where/22 is the coefficient of k2E/co in (69). Thus the waves are unstable to slow modulations if tog/22 < 0 and stable for tog/22>O. In the next section we indicate how this approach and the Benjamin-Feir instability compare on a stability diagram.
6. Discussion of results
Fig. 1 is a stability diagram based on the Fourier-mode perturbation analysis and is valid when ka can be considered small. The significant effect of surface tension is quite evident from the diagram. One observes that the neutral stability curve starting at kh = 1.363... for p = 0 tends to lower values of kh as p increases. The branch (a) corresponds to Y = 0 and branch (b) corresponds to equation (18). In both situations the perturbation analysis is invalid and modified scales need to be introduced. 5.0
4.0
3.0
i~ i!i!i!i!i!¸iiili
ii i i i i i i i i i i i i i iiiiiiiiiiiiiiiiiiii i i i i i i i i i i i i iiiiiiiiiiiiiiiiiiii!iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii i i i i i i i i i iiiiiiiiiiiiiiiiiiiiii !!i iiiiiiiiiiii
kh 2.0
1.0
0.O
o.o
~.o
2.0
-rk2/g
3.0
4.0
~.o
Fig. 1. Stability diagram based on Fourier-mode analysis. Shaded areas indicate instability.
Fig. 2 is a stability diagram based on the slow modulation approach of Section 5. This is very similar to Fig. 1 except for the appearance of two additional branches (c) and (d). (c) corresponds to gho = C~; this is where the group velocity Co coincides with the phase velocity of long waves, gv~o. In this situation 1"22becomes singular. The branch (d) corresponds to 122 = 0. Once again near (c) the perturbation analysis is invalid. The branches (c) and (d) are missing in Fig. 1 because as shown by equations (56) and (54) the dispersive effect balancing nonlinearity is taken only up to the first order in the Fourier mode analysis. As seen from (69) the branches (c) and (d) in Fig. 2 arise from the second order effects of dispersion.
492
C, V. Easwaran, S.R. Majumdar / Instability of capillary-gravity waves 5.0
4.0
iiiiiiii!ii!!iiiiiiiiii,ii i iiii iiii!iiiiiiiiiiiiiiiiiiii ( d ) iiiiiiiiii!ili!iiiiiii!iiii
kh 2.0 :::::::::
::::::::::
:::
1.0
O.O
o.o
,~.o
2.o
3.o
4.o
5.o
-r k2/g Fig. 2. Stability diagram based o n the averaged Lagrangian. Shaded areas indicate instability.
This instability diagram is identical to that of Kawahara [8] who uses a multiple scale formalism to derive a nonlinear Sehr6dinger equation governing the self-modulation of capillary-gravity waves. It is interesting to note that if surface tension is not zero there is always instability at some value of kh. Apart from surface tension, viscosity is expected to have a significant effect on the Benjamin-Feir instability, particularly at small wavelengths. Inclusion of viscosity significantly modifies the Stokes-wave expansion and the resulting analysis of instability poses extreme algebraic difficulties. We hope to use MACSYMA to overcome this difficulty in future work.
Acknowledgment This work was done with the aid of MACSYMA, a large symbolic manipulation program developed at the MIT Laboratory for Computer Science and supported from 1975 to 1983 by the National Aeronautics and Space Administration under grant NSG 1323, by the Office of Naval Research under grant N00014-77C-0641, by the U.S. Department of Energy under grant ET-78-C-02-4687, by the U.S. Airforce under grant F49620-79-C-020, and since 1982 by Symbolics Inc. of Cambridge, MA. This research was supported by grant A5294 from the Natural Sciences and Engineering Research Council of Canada.
References [ 1] [2] [3] [4] [5]
T.B. Benjamin and J. Feir, "The disintegration of wavetrains on deep water, Part 1: Theory," J. Fluid Mech. 27, 417-430 (1967). T.B. Benjamin, "Instability of periodic wavetrains in nonlinear dispersive systems", Proc. Roy. Soc. London A 299, 59-75 (1967). G.B. Whitham, "Nonlinear dispersion of water waves", J. Fluid Mech. 27, 399-412 (1967). R, Barakat, "Instability of periodic capillary-gravity waves on deep water", Wave Motion 6, 155-165 (1984). W. Pierson and P. Fife, "Some nonlinear properties of long crested periodic waves with lengths near 2.44 centimeters", J. Geophys. Res. 66, 163-179 (1961). [6] C,V. Easwaran and S.R. Majumdar, "Instability of capillary-gravity waves using an averaged Lagrangian", University of Calgary, Preprint. [7] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York (1974). [8] T, Kawahara, "Non-linear self modulation of capillary-gravity waves on liquid layer", J. Phys. Soc. Japan 38, 265-270 (1975).