WAVE MOTION 4 (1982) i13-125 NORTH-HOLLAND PUBLISHING COMPANY
113
ON S T E A D Y T H R E E - D I M E N S I O N A L D E E P W A T E R WEAKLY NONLINEAR GRAVITY WAVES Yah-Chow MA Fluid ?vlechanics Department, TR W Defense and Space Systems Group, Redondo Beach, CA 90278, USA Received 6 April 198!
Various kinds of steady weakiy nonlinear gravffy waves are examined. Corrections to the linear phase speed ar,d the direction of modulation are ca~culated.
I. Introduction We examine various types of three-dimensionai steady gravity waves in this paper. For weakly nonlinear waves Zakharov [11]derived an integral equation in wavenumber space for the evolution of the gravity wave fieId up to third order in wave amplitude. We will start with a two-dimensionaI steady wave and study possible bifurcation into three-dimensional steady waves by using the above integral equation. We wilI show that the wavelength and direction of the modulation depend on the amplitude of the two-dimensional steady wave. At special wavelengths the Zakharov equation is not adequate to describe the bifurcation and we use a higher order integral equation [2] instead. We then inerea~se the amplitude of modulation up to the order of the amplitude of the two-dimensional steady wave and find that the phase speed, wavelength, and direction of modulation will be functions of two amplitudes, the amplitude of the two-dimensional steady wave and the amplitude of the modulation. In this case we find that it is possible to have two-dimensional bifurcation even at small amplitude. Recently, Martin [3] used the nonlinear Schr6dinger equation to examine the bifurcation of twodimensional steady waves into three-dimensional waves. The results would be limited to cases where the wavelength of the modulation is close to the wavelength of the two-dimensional wave. Saffman and Yuen [4"...'used a simplified form of the Zakharov equation to examine a similar problem. Their results are valid at various waveie•gths except those special wavelengths mentioned above where the Zakharov equation to third order is not adequate. Their results are valid for weak modulation.
2o The two.dimensional steady wave solution Following Zakharov [1], we introduce a normalized Fourier representation of the free surface by the relationship
rl(x, t) ==2rr ~-o~
[ b ( k , t) e ik'x + b * ( k , t) e -ik'~] dk
0165-2125/82/0000-0000/$02.75 © 1982 North-Holland
(1)
Y.-C. M a / ~,Veakty ~:(;e~.lineaegra~,io: a,a~:es
114
where x = (x, y) are the horizontal coordinates, k = (k~, k~.) the wave vector, and c~ the Iir:.ear frequency of "p~e.scm i ; ~.... :','c o.;,~r~ss,bh,, ~.,~.,,.o :. the s~rface waves (w = ,/ g i'k ' ).'~ o . ...... ~., irrotatioeat fluid of infir~ite depth the Fourier surface displacement b(k~ t) satisfies the fo~owing third order evolution eq,aation:
+2V
~ ;(k., k, ~@b(k~)b*(kj6(k
+ V~*~(k, [¢~, k ~ ) b * ( k , ~ ) b * ( k a ) ~ ( k
-ka +k~) + k~. + k2)] dk.: dka
£ t Cc~
(2} 'There are other csbic terms in (2) but these are unirapo~tant for the time being. T h e c o e ~ c i e n t s V ~ ~, Y ~+~ and W are given iI: C r a w f o r d et ai. [5], T h e t w o - d i m e n s i o n a l steady wave solstion cma be o b t a i n e d from equation (2). To simplify the armlysis that follows we choose the length and time scales s,ach that the gravity g and the w a v e n u m b e r of the two-d.imensionai steady wave are _.normalized to 1. W e cossider a class of solutions for which
42 ~ ,~ which describes the surface p,o_.:k, p r o p a g a t i n g without cha:"ge of shape at speed C in the x directiom Tb.e Fourier surface displacement b e c o m e s
b ( k , 0' = a ~ [ k - - ( 1 ,
b' n~e
~c', =. - a -_, 8 [~k -
> 8 [~k - t - ,2 , 0 ) ] e
2:.c, + ° ' '
,
,a~ v,~
w h e r e a~ is real and [~ositive without loss of generaiity. Substituting ',,4/:,xo '~ :n* {4) ~ we find to o r d e r ~} the following relations: V ~ '[(2, o). (1, m ,
o)]:
2 - v'2 a -2 :
\30)
-2 -,/~ 2
C:I
~4 2.
(5c)
T h e first term on the right-hand side of (5c) represents the linear phase speed of the steady 'wave. The s e c o n d term a ~2/t .d~ 2 c o r r e s p o n d s to the amplitude correction of the phase speed. Higher order correctior;s can be o b t a i n e d but they wili no.* be accurate since e q u a t i o n (2) is valid only to third order.
3. B i f ~ ' c ~ i ~
of ~ twe-dime~.~ion~
w a v e ~nto ~ t h r e e - d ~ m e n s i o ~ a ~ w a v e
T o took for steady waves in three dimensions we put a sma1,i three-dimensio.aai perturbation to e q u a t i o n m this w a v e n u m b e r (4). A D:fu..e~ao:~ point is a s o i n t in the w a v e n u m b e r space such that p e r t u r b a t i o n w.,."~" m o v e s with the same speed as the two-dimensionai steady wave. With perturbation, equation (4) b e c o m e s
K-C. Ma Weaklynonlineargravity waves
115
H e r e b~ represents the t w o - d i m e n s i o n a l steady w a v e [i.e. the right-hand side of e q u a t i o n (4)] and m is an integer. T e r m s involving C~,, etc., are neglected temporarily, Since the steady wave m o v e s with s p e e d C in the x direction we require
c,.(t) = C~(0) e - " ~ + ~ ' ,
d., (t) = &.(0) e -~"~-~c'.
(7)
Substituting (6) and (7) into (2) we obtain an infinite set of h o m o g e n e o u s equations: [(m + p ) C - ~ / ( m + p ) 2 + q 2 - 2 W [ ( m +p, q), (1, 0), (m + p , q), (1,
0)ia~lC~ (0)
= 2 V(-~[(m + p, q), (1, 0), (m - 1 + p, q)]a 1Cm- ~(0) + . , , + W[(m + p, q), ( - m + 2 - p, - q ) , (1, 0), (1, 0)]a~d-*~+2 (0) + "
',
(8a)
and [(-m +2-p)C-4/(-m
+2-p)2+q
~
-2 W[(-m +2-p,
- q ) , (i, 0), ( - m + 2 - p ,
= 2 V ~ - ; [ ( - m + 2 - p , - q ) , (i, 0), ( - m 4- I - p , - ~~; ;a
+ W[(-m +2-p,
2-~ :~ - q ) , (1, 0)]a ~'C_.~+a (0)
~C *,.÷t ( 0 ) + . . °
- q ) , (m + p , q), (1, 0), (1, 0)'_,:a~C~(0) + • • •
(Sb)
Since the off-diagonal t e r m s on the r i g h t - h a n d sides of (8a) and (8b) are smalI, we conclude that to zeroth o r d e r the bifurcation points Iie on curves w h e r e the coefficients of C.~ and C~.~.2 vanish. m + p = ~ ( m +p)2 +q2,
(9a)
m - p = ~¢(rn - p)2 + q ~
(9b)
or
T h e bifurcation curves in the p-q plane are depicted in Fig. 1. Curves satisfying equations (ga) and (gb) are n a m e d class I curves and class II curves, respectiveiy, F o r ciass I curves p i> m - 1 and for cIass II curves p ~ m - 1, and curves in the s a m e class d o n ' t intersect. All the curves are s y m m e t r i c with respect to the p axis. (B)(c) q
(A)
VV/ 3.46
u p
Fig. t. Linear bifurcation curves in wavenumber space. Curve (A): 1 + p = ,~(1 + p )2 + q~ Curve (B): I - p = 4'(t - p )2 + q 2, Curve (C): 2 - p = ~(2-p)2+q ~'.
~:I 6
Y. -C. Ma / Weakly .,~onlineargravi*,y waves
T h e b i f u r c a t i o n curve~ w m b e m o d i f i e d d u e to t h e o f f - d i a g o n a l t e r m s in (So) a n d (8b). W e only s t u d y t h e d e t a i l s of o n e b i f u r c a t i o n curve; t h e class t curve with ra = i a n d q positive. This c u r v e intersects ",vitb class t i curves at (p, q} =
' •
L2'±\ *
2/'
\
+
i
2J
_~
w h e r e n is a n o n - n e g a t i v e integer° F o r p o i n t s (p, q) o n this curve b u t far a w a y f r o m the i n t e r s e c t i o n points, all the c o e N c i e n t s of t h e d i a g o n a l t e r m s of the h o m o g e n e o u s infinite s y s t e m (So) a n d (Sb) a r e finite e x c e p t C~. W e so!ve this infinite s y s t e m t h r o u g h i t e r a t i o n s in t e r m s of t h e a m p l i t u d e ~_ of t h e two-dimer~sional w a v e a n d k e e p t e r m s u p to O ( a ~ ) , E q u a t i o n (9a) is m o d i f i e d to F l+p-w(!+p,q)=[2T[l+,v,q),(1,0),(1,0),'l'
,q
~,/
,~.,
1 \?
2
t-L)
T h e c o e ~ e i e n t T ( k , k , k s , ks) is given in [5]. E q u a t i o n ~:0;'~'" ' can b e s o l v e d e y expa~.4on of q in t e r m s of ~.. F o r fim..,~ ;'~ (p, q~ , ; w e ~nd q = 7 ( i + p)4 _ (! + p,'.: 2(i +p) \-
' lr'J
2Ti
I +p0 , ] ( i 4-.p~ -
\"-
-t , (1, O) "
j~
T h e e r r o r in e q u a t i o n ~ i ,
~
'
p)
~',
--
1+p\ 4,rr.2 ) a , oa
1)
4
is of o r d e r aio
E q . . . . ~o.: ( t t ) is va~id o n i y at p o i n t s ',e, q) a w a y ~rom i n t e r s e c t i o n points° T h e first i n t e r s e c t i o n p o i n t occurs at (p, q;' = (0, 0;t.n ~j = 0 a n d h e a c e for stool1 p a n d a. (th,,~ s~:~4'-.,"~. " mrc'dlar' a r e a m° Fig° 1) the result wilI be d i f f e r e n t f r o m e q u a t i o n (1 D. Small p a n d q c o r r e s p o n d to long w a v e s t e a d y m o d u l a t i o n , which has b e e n s t u d i e d in d e t a i l in [3] a n d r~,~ Coe~cier~ts o~ C; a n d &-.* are smail, and t h e two e q u a t i o n s [(8a) a n d ,Sb~ with m = i ]_ m u s t b e s o l v e d simu~taneousIy with o~h,~r " ~ C~ a n d C~* e x p r e s s e d in t e r m s of C:. a n d '~* ,~ ~ t h r o u g h e x p a n s i o n s in t e r m s of a~. A n a t u r a l scaling of p a n d q is p=a~p,
q=a~4o
(12)
T o o r d e r a 2 t h e s y s t e m (8a) arid ~8.~;' ' ~ has a n o n t r i v i a l so!ution if 1
;
T h e two s o i u t i o n s ( i i ) a n d (i 3) s h o u l d m a t c h in s o m e i n t e r m e d i a t e r e g i o n p = a .2 -,~w h e r e 0 < r~ < 2. F r o m vx _~; we find 2
q 2 O ~ 2-n a .=ZO~ ----~4- . • . ,
(14)
a n d f r o m (13) we find 2
2 ~ q = 2 a { -n
a
~
24r~
a! ...,
(:..s).,
Y.-C. Ma / Weakly nonlinear gravity waves
117
The two solutions match to order a~z-" and the error in q is of order a ~÷3./:, As ~7-* 0 the error in q of the outer solution (4) is of order a~ and the bifurcation curve should be replaced by the inner solution (13). As r/-~ 2, p-~ O(1), the error in the outer solution is of order a~ and hence valid to the order considered. Results are depicted in Fig. 2.
p
2rr2 Fig. 2. Bifurcation curves m order a ~ in wavenumber space. Curve (1): Eq. (90) with m - I, Curve ~2). Eq. t* u,, Curve ~3~. Correcnon te Curve (2) in the inner region p = O(a~).
The second intersection point occurs at p =½, q ~ 1.6771 (n = I) where the class I curve with m = 1 intersects with the class II curve with m = 2. At this wavenumber all the coefficients of the diagonal terms of the system (8a) and (8b) are finite except C~ and ~'*. The Zakharov equation (2) is not adequate here since the coupling between C~ and C* occurs at order a~. Recently, Crawford [2] extended equation (2) to fourth order (see appendix). With this higher order equation the behavior of the bifurcation curve near p = ½, q ~-1.6771 can be readily studied. We first expand the two-dimensionai steady wave solution to order a 3 and there wi!l be waves at wavenumber (3, 0), ( - 3 , 0), and ( - 1 , 0) with amplitudes proportional to al. Next we solve the system (Sa) and (8b) through iterations and keep terms to order a;. Near p = and q = 1.6771 we find [
a~
3o155
1 + .~ + 4--~ (1 +p)-,o(l+p, q ) - ~
]
0.378
3~,
aT. C 1 - - ' ~ W - 2 a l C 2
" [ 2 - D + - -4~ 0.3784~ 2 a~C1+ --7(2-p)-o)(2-p,.
- q j,- ~3.155 al
=0,
~]~, C2
(16a)
=0.
(!6b)
T o obtain a nontrivial solution the determinant of equations (16a) and (16b) must vanish. We define an inner region around p = ½ and q = 1.6771 with a width proportional to a3: tq_ 3 p =~ nip3,
q = 1.6771 ta~q~+alqs. ' 2 3
(17)
Substituting (!7) into (16) we find q2 = -
i..665 2 , 'w
(I8)
and q32
9.799p ~ - = 0.145 4
(19)
lig
Y,-Co M a / W e a k l y ~ordinear gravity waves
t n t h e o u t e r r e g i o n t h e bifurcatior~ c u r v e s a r e g o v e r n e d by e q u a t i o n ( i I) a n d by e q u a t i o n ' ~ w i t h a~ c o r r e c t i o n : q=~/(2-P)4
(2_p)2
,
,
" / ( 2 - p2)24(-2(-2P- )P:);
i 2 T [ i[~/~, z!
4
t
---4,r~ja-,:.
- p~ - ( 2 - p ~
T h e t w o s o l u t i o n s s h o u l d m a t c h in s o m e i n t e r m e d i a t e r e g i o n . F o r •e)> ~ w e 1et .v = ½+ a~~-'~, a n d q = 4: ~.6,"7"3" , z - ( 1 . 6 6 5 / ~ 2 ) a ~: -> ~ a ~-~ Lequa..~on (19)] t h a t ; w h e r e 0 < ~ < 1 • W e find f r o m t h e i n n e r r e g i o n ~ *" ,,z = ± 3 . ! 3 0 a , + O ( a - ~ ' ) .
{21)
From. t h e o u t e r r e g i o n w e find a - 3.~0
~,
from equation (ii),
---3..!304o
f r o m e q u a t m n ,auto
F o r p < : 5 w e let p = ~ -~a : "~ " a n d
(22a)
a=i.6771-(~.,665/~.~}a~-cea:, ,, ~.
~' a n d w e obtain_ t h e s a m e resu.lts
~. ~' a n d t h e e r r o r is o f o r d e r a ,~ +~, F o r r~. -~ 0 t h e ,,,.~-,o r m' It is c I e a r t h a t t h e t w o s o l u t i o n s m a t c h to o r d e r a -~ t h e o u t e r r e g i o n is of o r d e r e 3, a n d t h e b i f u r c a t i o ~ c u r v e s h o u l d b e r e p i a c e d b y t h e i n n e r ~ ~'"' r-~ t._9;~ A s 1 , ~,'~/
2x
4
r~ + i , p ~ g + u ~ a ~j t h e e r r o r in t h e o u t e r r e g i o n is of o r d e r a ~, a n d h e n c e t h e o u t e r s o l u t i o n is v a l i d t o t h e o r d e r c o n s i d e r e d ° R e s u l t s a r e d e g i c t e d in Fi~. 3, N o t e t h a t t h e "e:~mca.,m:, :~ ~ ~' ,, c u r v e with m = 1 of class i
i,677
U
\ ~.p
"-'
2
Fig. 3. Bifurcatio~ curves near V = 2. Curve ~.~;. ~ ~' Bifurcatiion curve (m = !, Class t) te order a 2, Curve (2): Bifurcation curve (m = 2 C~ass II) to order
a 2, C u r v e s
~q~,_,j,,~z.~.'.,. C o r r e c t i o n s
to bifurcation
curves
in the inner
regio;~ p = ~ + O ( a ~
3
).
actua!1y c o n n e c t s to t h e c u r v e m = 2 of class t i at p n e a r ½. F o r p =½ w e find f r o m (19) t h a t q = 1.6771 - d.665/~ , a ~ --=t u . J ~ l ~ )a~.. R e c e n t l y M c L e a n et aL ..26j h a v e s h o w n t h a t f o r ~ n i t e a~ t h e r e is a s t r o n g e r i n s t a b i l i t y L~'--t-~;_.' r c~/,,~". t h a n t h e B e n j a m i n - F e i r i n s t a b i l i t y f o r p = 2~- a n d - - ( 0 3 8 1 / w ; ) < q 3 < * z 2 2 ( 0 . 3 8 i / ~ r 2 ) . T h e y a l s o s u g g e s t t h a t t h e a b o v e b i f u r c a t i o n p o i n t s (p = ~ q = i . 6 7 7 1 - d . , 6 6 5 / w )a,. + 4
2
3",
(0.38z/'z.)a~; experiment
are responsible for the striking three-dimensional
p a t t e r n s o b s e r v e d b y Su [7] in a n
Y.-C. Ma / Weak!y nonlinear gravity waves
1! 9
For n = 2 the third intersection point appears and the analysis around that point (p = i, q = +2•598) involves equation (2) to the fifth order and will not be studied here. Although the above results are strictly only valid for small a~, a recent paper by McLean [8] indicates that even for finite al the behavior for the neutral stability curves remain qualitatively similar.
4, T h e b i f u r c a t e d t h r e e - d l r n e n s i o n a |
waves
AiI the discussions in the previous section are based on small perturbations with respect to some two-dimensional steady wave; i.e., [C,,/alt << 1, and }C,~/a,~[ << I for all m. This leads to a simple formula [e.g., equation (10)] for the determination of the bifurcation point in the wavenumber space. As the steady two-dimensional wave moves off the bifurcation point to the three-dimensional wave, the above !inearization is no longer valid. Assuming IC~]- 0(a~) the three-dimensional wave can be calculated to the order considered. First let us consider a simple case where (p, q) is away from any intersection point (Figure 1). Keeping terms to order a~ and ]C~[z we find the speed of the steady wave and the bifurcation curve in the p - q plane: 2 C = 1 + 4~£2+ 2 T [ ( - , 0), (1 +p, q), t-, 0), (1 (23a) 1 +p -,5(l+p)2+q 2 -- a~(2r[(1
+p, q), (1, 0), (1 +p, q), (1, 0)7
(1 +p)'~
+ Lcd2(T[(I +p, q), (~ +p, q), (I +p, q), (I +p, q)] -2(1 + p)T[(1 +p, q), (1, 0), (1 +p, q), (i, 0)]).
(23b)
In equation (23a) there are two amplitude corrections to the linear speed of the steady wave, i.e. a~ and {C~i whereas in the linearized analysis there is only one amplitude correction (5c). T h e r e are also two amplitude corrections to the bifurcation curve [equation (23b)] whereas ...;- the linearized analysis only an a 2 correction appears [equation (!0)]. A special case where p - 0 . 7 5 , q ~ 2 . 5 1 3 2 is studied here. Let p=0.75+a{/~
and
q = 2.5132 + a a~ ~q.
(24)
Results are depicted in Fig. 4 and 5. For each line in the P-q plane [curves (t)-(4) in Fig. 5] there is a definite amplitude ICI i/al ratio (Fig. 4). Curve (1) corresponds to the linearized analysis. For fixed/3 the value of decreases as 1CII increases. When (p, q) is close to any of the intersection points discussed in the previous section equations (23a) and (23b) are no longer vaIid and a different expansion must be used. We consider fine case where (p, q) is close to (0, 0). Since C,, C~ and a~ are of the same order, we need to include waves at wavenumbers (1, 0), (2, 0), ( - 2 , 0 ) , ( l + p , q ) , ( 1 - p , - q ) , ( 2 + p , q ) , ( 2 - p , - q ) , ( - 2 + p , q), ( - 2 - p , - q ) , (2+2p, 2q), ( 2 - 2 p , - 2 q ) , ( - 2 + 2p, 2q), and ( - 2 - 2p, - 2 q ) in order to carry the expansion to the order considered. Solving equations (8a) and (Sb) through iteration we find , a2 1 +v~I2 C i d ~ . C = 1 - - 2 ~ * 2 - 7 (IC~ 12+ IC,] ~) 4~r 2,r ' 2~
(25)
The last three terms on the right-hand side of equation (25) are corrections to the speed due to finite C~ and C, • Note that C~ C~ must be r,.a, = ' for steady waves.
Y.-C\ ~/.ga/ Weakly ~anii~ear }~R*vitywaves
!20
(4)
/ /
/(3) ./
r
x
F~.g, 4. The amplitude ratio for the bifurcated three--dimens~onM wave.
.....
^
(2) "
"
:7.: ~3, (a)
rag, J. ~ Bifurcatkm lines i~ the ~-4 ~ a n e w~ere " p = t,'. " "1.5 + a } ~ and a=2.5132+a;q.
S~nce p << 1 and. q << 1 w e ~et 2
a.; P =~2P, Equations
_
a~
~ (LOj
q =--q2~r '
g o v e r n i n g C:, a ~ d C;. a r e
---4 ~]x - y = -xa-
/p42
~, 2
;2
2xy,
)
4
1 x--y-~x2=-y2-2xy~ 2
~
2
w h e r e x = IC~,t / a {, y = C,~C~/a ~. T h e q u a n t i t y x is p o s t i v e a n d y is r e a l .
(27a)
(2'7~,)
Y.-C. Ma / Weakly noniinear gravity waves
12I
Several special cases can be studied. Considering the case x << 1 and y << 1, we find that # 82 ~--~--1) x-y-0,
p q~ (-~--~--
1) y - x - 0.
(28)
To obtain a nontrivial solution we require -4
q ,__~_, /52 = 2-2"* "a
;.,2%x
which is the same as equation (13). Results are given in Fig, 6 and there are two branches of solutions. For a fixed value of q there are two values of p with the same magnitude and different signs, For convenience we let 2
2
C = I ' -,- 4---~ a~ + 4----5 a l x Cro
(30)
l~
x
<<
Z
Fig. 6. Bifurcation curves m the fi-~ plane for x <<1.
Cr versus ,6 is given in Fig. 7. For positive p the value of Cr is almost invariant whereas for negative p the value of Cr increases as iPi increases. The nonlinear terms in (27a) and ~27bj ohould modify the above results. With a small value of x equations (27a) and (27b) can be solved through a perturbation expansion. We find that there are three branches of solutions
[,
/31 ~ X["7-+ 2c72- 2 x , "gO#,
.t~4
(31a)
-4-
(31b)
2'
(3ic)
122
Y.-(2 M a / Weakly non!inear gravity waves Cr x<
i!
t Fig. 7, NonH:~ear speed correction of the th.ree-dime.~siona~ steady wave for x << J..
in the ~im.it as x --~ 0 the third branch (31c) disappears. N o t e that for the s e c o n d branch .52 will eventuaIIy b e c o m e positive for a large value of ]q{, no matter how small x is. This suggests that the results in Fig. 6 and 7 are valid only for a positive (and finite negative) value of ft. A special class of soIut{ons with f = '} can be o b t a i n e d f r o m (27)° W e find three possibte solutions: (!) y=x, X:(O~z+g)/]2, (2) y = x, X = - - ~ 2 / 4 , a n d (3) y = - ] , x = ( ~ ± ' , / c e 2 - 4 ) / 2 where c ~ = 3 + q 2 / 4 ~ T h e s e solutions correspm~d to the bifurcated wave perpendicular to the two-dimensional steady wave. T w o - d i m e n s i o n M bifurcation can be o b t a i n e d by putting 4 = 0 in (27). W e find x and y must satisfy the f o ~ l o w b g equation: (x + y)2 = Y(x 2 + Y2 + 4xy ).
./,32)
W e calcuIate two cases here. For x - ~ we find f r o m (27) that the three branches of soIutions coalesce in the p-q plane (Fig. 8):
42 - T = ~.
(33) X~005
"
4
Fig. 8. Bifurcation curve it: the fi-q ,~iane for x = ;~
Y.-C. Ma / Weakly nonlinear gravity waves
i23
Two-dimensional bifurcation occurs at ,5 = 1. Each branch has its own speed (Fig. 9). The second branch [branch (2) in Fig. 91 has a speed close to the continuation of the speed from the linear analysis (x << 1, Fig. 7), whereas the other two branches have much higher speed.
x=0.5
(2)
Fig. 9. Nonlinear speed correction of the three-dimensional steady wave for x = 0.5°
For x = 1 there are three branches of solutions in the ;5-c7 plane (Fig. 10). Two-dimensional bifurcations occur at [5 = - 1 . 3 2 , 0.92, and 6.4. The speed of each branch is given in Fig. 11. Again the first and third branches have much higher speed than the speed predicted from the linear analysis. As x increases we notice that q increases slightly for a fixed value of p and the wave becomes more oblique. A similar analysis can be carried out around p = ½, Due to the algebra involved it will not be pursued here.
S
Fig. 10. Bifurcation curves in the f f ~ plane for x = I.
124
Y. -(2 Me / Weakgy.nenginea~*gravity waves Er
/ /
.//
10 ~ i
/-
/
...............
(~L
i
p
Fig. i i. Nonlinear ~peed correctioa of the three-dh~ensio~a" s~eady wave for x = i.
Cor~ehss~o~ Various possible threeodimensiona~ steady wave so;.ut~.or~s are examined. The e×iste~ce of these waves depend oa their stability with respect to exteraat perturbations°
Appe~dix
We give here eqaatio~n (2) extended to fourth order by Crawford LL3. r,,~
~a(k)b(k)= f .f ~ r V"~(
k.,t) '-. ~ b ( Ot
)/~_ ~ , k~, k 2 )'h' ~ , ' , k z' ~ b ( k j S ( k - k : - k 2") '
+2 V ~ -" (k.., &, k ~ ) b ( k : ) b * ( k 2 ) 8 ( k - k~ + g¢2~~- V ~+; (k, k~, ~2~-" ~ ! ~
3 .; J
~2~.
~ &1 + kz)] dg¢~ d k a
ksjb
+W!(k,k~,k2, " * '
'"~"
'
'
~
~,
+ W ) ( k , k~, k~, k 3 ) b * { k ~ ) b * ( k 2 ) b * ( k 3 ) $ ( k + k~ + ~a + k3) + ~2¢~(k, &~ k2, k 3 ) b ( k ~ ) b { k z ) b ( k 3 ) 8 ( k - k~ - k 2 - k j ] dk~ dk2 dk~
4eo
V ~- ~, V (+~ arid W are gives i~ [5]°
W~(&, kl, k2, &~) = N " ( - k , - k : , - k 2 , k~)+ ~ / ( - & , -k.._, k3~ -~g2)- ~ ( - k ,
k3, - k ~ , -&2)
4- ~ / ( - - k ~, --k2, --&, k:',}'-- W ( - & ~ , &3, -&.. - - k s2}- ~/~,kg,-e"_&~, - k , -&2),
Y,-C. Ma / Weakly nonfinear gravity waves
125
W2(k, k~, k2, k3) = ~V(-k, -k~, - k s , -k~)+ YC(-k~, - k s , - k , -k~), W3(k, k ~ / ¢ 2 , k 3 ) = - [ ~ / ( - k ,
ks, k~, k~)4- rc~/(ki, k2, - k ~ k3),
Y(k, kt, kz, k3, k4) = 2 ? ( - k , - k l , ks, k~, k,) - 2 ? ( - k , ks, - k : ,
k3, k4) -
2 Y(-k, k3, kz, -k~, k~)
- 2 Y(-k, k4, k2, k3, - k t ) - 3 ?(-k~., k2~ - k , k3, k 4 ) - 3 Y ( k z , - k l , - k , k3, k4) +3 ?(k3, k2, -k, -kl, k4)4- 3 ?(k4, k2, -k, k3, -kt), where VVis given in [5] and kkl
4] w(k)w(kl)k2k~k4
•{ 2 k k , - k Z - k ~ - k ~ - k ~ - k 2 4
' + k~l,2 -kik~ + k21t +ik
- k iI,, + x,~i- klkl + k,l- ~!l, + k~i- k~!~, + k~l- k~ !I, + k,t
References [1] V. E. Zakh~ov, Stability of periodic waves ef finite amplitude on the surface of a deep fluid, Y.AppL Mech. Tech. Phys. 2, 190-!94 (1968). [2] D, R, Crawford Unpublished notes (i980). [3~ D. U. Martin, Nonlinear deep water waves. VIII. Two-dimensional bifurcations of Stokes waves, TRW Report 34510-6058-TU00 (1979). ~4] P. G. Saffman and H. C. Yuen, A new type of two-dimensienaI deep-water wave of permanent form, Y. Fluid Mech. 101 (4), 797-809 (1980). [5] D. R. Crawford, P. G. $affman and H. C. Yuen, Evolution of a random inhomogeneous field of nonIinear deep-water gravity waves, WaveMotion 2 (1), 1-I6 (1980)o ~6] J. W. McLean, Y.-C. Ma, D. U. Martin, P. O. Saffman and H. Co Yuen, A new type of three-dimensional instability of finite amplitude gravity waves, Physical Review Letwrs 46(13), 817 (!98!). [7] M. Y. Su, Manuscript (i980). [8] J, W. McLean, Instabilities of finite amplitude water waves, J. Fluid Mech. !14, 315 (1982).