zhakarov hamiltonian

download zhakarov hamiltonian

of 5

Transcript of zhakarov hamiltonian

  • 8/9/2019 zhakarov hamiltonian

    1/5

    STABILITY OF PERIODIC WAVES OF FINITE AMPLITUDE ON THE SURFACE OF A DEEP FLUIDV. E. ZakharovZhurnal Prildadnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 2, pp. 86-94, 1968ABSTRACT: We study the stabil ity of steady n onlinear waves on the surfaceof an infinitely deep fluid [1, 2]. In section 1, t he equations of hydro-dynamics for an ideal fluid with a free surface are transformed tocanonical variables: the shape of the surface ~(r, t) and the hydrody-namic potential ~ ( r , t) at the surface are expressed in terms of thesevariables. By introducing canonical va riables, we can consider theproblem of the stability of surface waves as part of the more generalproblem of nonlinear waves in media with dispersion [g, 4]. The re-sults of the rest of the paper are also easily applicable to the generalcase.In section 2, using a method similar to van der Pohi's method, weobtain simplified equations describing nonlinear waves in the smallamplitude approximation. These equations are particularly simple ifwe assume that the wave packet is narrow. The equations have anexact solution which approximates a periodic wave of finiteamplitude .In section 3 we investigate the instability of periodic waves of finiteamplitud e. Instabilities of two types are found. The first type of in-stability is destructiveinstability, similar to the destructive instabilityof waves in a plasma [g, 6]. In this type of instability, a pair of wavesis simultane ously excited, the sum of the frequencie s of which is amultiple of the frequency of the original wave. The most rapid de-structive instability occurs for capillary waves and the slowest forgravitatio nal waves. The second type of instability is the negative-pressure type, which arises because of the dependenc e of the nonlinearwave velocity on the amplitude; this results in an unbounded increasein the percen tage modu lation of the wave. This type of instabilityoccurs for nonlinear waves through any media in which the sign ofthe second derivative in the dispersion law with respect to the wavenumber ( d2~/dk 2) is different from the sign of the frequency shift dueto the nonlinearity.As announced by A. N. Litvak and V. I. Tala nov [7], t his type ofinstability was independently observed for nonlinear electromagneticwaves.

    1. Canonica l variables. We consider the potential flow of an idealfluid of infinite depth in a homogeneou s gravity field. We choose thecoordinates so that the undisturbed surface of the fluid coincides withthe xy-plane. The z-axis is directed away from the surface. In whatfollows, all vectors are two-dimensional vectors in the xy-plane.

    Let ~(r, t) be the shape of the surface of the fluid and let ~(r, z, t)be the hydrodyn amic potent ial. The fluid flow is described by Laplace'sequation,

    0~q) +Ar =o, (L1)with two conditions at the surface,

    0 ~ 1 . . . . . . . . . . 0 ( I ) [ & I )o t -I 't + V~l~ 0~- t ~ ' =~7- - -WlV@I ~=~, (L2)

    - ~- -l- gq = -- "~" (V(I))"- z=,~ -- ~" k-~7-z} -[- :zV I t -+- Vq'-' 'and a condition at infinity,

    { b - - - > - 0 a S z - + - - e x -.Here g is the accele ration due to gravity and a is the coefficient

    of surface tension.Equations (1. 1)-(1 .8) conserve the total energy of the fluid,

    . [ I * 1- . ( OtD ,o-I+ , : = ~ ' + .. .~ z ,- ) [ ( v , , , ) ~ + - ~ ) , ~ ~ z+ + i -

    v + , , i + , + + , - + - + S ( , + + - + , , + - + ,> + , . +

    The first term in t h i s expression is the kinetic energy and thesecond and third terms are the potential energy in the field of gravityand the potential energy due to surface forces. We introduce thequ an-tity 9 ( r , t) = ~(z, r, t) z=~" Spec ifying the qua ntit ies ~ and g, fuIIydefines the fluid finw since the boundary-value problem for Laplace'sequation has a unique solution. Using the equa tion

    we obtain

    o,e o~v o n o *O t - - O t ~ O t O z z=n'

    i ) T V + ]ot 4-,~,~--~V ],',1 + V+l

    l I { oo ~2 oqS (1.5)2 (V(D)+++ 7 \-0-}-) -- ~- z ( v(D V~)I==~.Equations (1.2) and (1.5). together with Laplace's equation, are

    equivalent to Eqs. (1.1) -(1.3 ). We can prove that Eqs. (1.1) and (1.8)can be put in the form

    07 8E 0iF 5Eo t - - 5 ~ ' o t - - - - T ~ - " ( 1. 6)

    Here E is the energy; the symbols 6E/6~ and 5E/69 denote thevariational derivative.

    Consider first the vari ation of g~. Obviously., the va riat ion of thepotent ial energy is zero. We transform the kinetic energy by meansof Green's formula:

    ., _ = T ~ "I" -N - +~= ---c o s

    t (' ~F 0el)8

    Here ds is a differential surface element . The normal derivative0~/0n is linked with ~ by the Green's function for the boundary-valueproblem of Laplace's equation;

    00 (s) IO n - - a ( s , ,h)~F(,h)dsl , (1,7)Here s and s1 are points on the surface. The function G is sym metric;

    i.e. , G(s, sD ; G(si, s).The variation of the kinetic energy has two terms:

    0@(s) , . 0 80 (s )] ds .8 E * = t f [ S W ( s ) - - T ~ - l- ~ (s ) - g ~ .s

    From (1.7) and the symme try of the Green 's function, we see thatboth terms are the same:

    ~ > o +O7- 1/1 + Vq~"dr 98

    From this we obtain (1.2) instantaneously.Consider now variation in ~ (this simple proof is due to R. M.

    Garipov).Variation of the potential energy at once gives the terms on the

    left- hand side of (2.5). Variation of the kinetic energy gives8E* : - ~- I [ ( V O ) ~+ ( '~ - z ) ~ ] Sq( r ) d r @

    0 , + ,+ 5 , + + , [ ( v + ,- c o

    19 0

  • 8/9/2019 zhakarov hamiltonian

    2/5

    H e r e 6 ~ i s t h e v a r i a t i o n i n t h e p o t e n t i a l d u e t o a c h a n g e i n t h eb o u n d a r y . S i n c e ~ s a t i s f i e s L a p l a c e ' s e q u a t i o n , w e c a n a p p l y t h e G r e e n ' st h e o r e m t o t h e s e c o n d i n t e g r a l :

    l i l ~ 17 8(I3 d s( O(l) \ Off)- ~- V q V ( D ) ~ z = ~ 5 q d r .= i k - - - 3 7 - ~ ( r )

    F i n a l l y w e h a v e

    2 [ a z } 4 7 (V ~1 V q) ) O z ._z z = n 8 r l ( r ) d r .H e n c e w e o b t a i n ( 1 . 5 ) .T h u s , E q s. ( 1 . 2 ) a n d ( 1 . 5 ) a r e H a m i l t o n ' s e q u a t i o n s a n d $ a n d

    a r e c a n o n i c a l v a r i a b l e s , ~ b e i n g a g e n e r a l i z e d c o o r d i n a t e a n d ~] ag e n e r a l i z e d m o m e n t u m . T h e e n e r g y E o f t h e f lu i d is t h e H a m i l t o n i a n .

    T o c l o s e E q s. ( 1 . 1 ) a n d ( 1 .5 ) w e h a v e t o s o l v e th e b o u n d a r y - v a l u ep r o b l e m f o r L a p l a c e ' s e q u a t i o n . W e f in d t h e so l u t i o n o f t h i s p r o b l e mi n t h e f o r m o f a s e r i e s i n p o w e r s o f r / . I f w e a p p l y a F o u r i e r t r a n s f o r m a -t i o n t o t h e v a r i a b l e s x a n d y ,

    t ~ e _ i ( k r ) d r , t I T ( r ) e - i ( k r ) d r ,~1 (k) = ~ d rt (r) ~g (k) = - ~w e o b t a i n t h e s e r ie s i n a m o r e c o n v e n i e n t f o r m .

    O m i t t i n g t h e d e t a i l s, w e i m m e d i a t e l y g i v e t h e r e su l t o f t h e e x -p a n s i o n ( u p to s e c o n d - o r d e r t e r m s ) :

    ( k , z ) = e I k l z { ~ (k ) - [ - ! ~ F (k~ ) "q (kz ) I k~ I ~ (k - - k~ - - k2 ) dk ld ks - -t k- - ~ - I [ ] - - k ~ , + l k - - k 2 ] - - , k l ) X

    I I ~ ( k a n (k._,) ( k ~ ) ( k - - k l - - ~ - - ~ ) ] d k , d k ~ k ~ . ( i . )H e r e 6 d e n o t e s t h e d e l t a f u n c t i o n .I f w e l i n e a r i z e ( 1 . 2 ) a n d ( 1 . 5 ) a n d c o n s i d e r o n I y t h e f i r st t e r m i n

    ( 1 . 8 ) , w e o b t a i n t h e t h e o r y o f s m a l l o s c i l l a t i o n s f o r th e s u r f a c e o f af l u i d , w h i c h d e s c r i b e s t h e p r o p a g a t i o n o f w a v e s w i t h d i s p e r si o n l a w

    (o ( k ) = K g I k I + a l k p .W e n o w c o m p l e t e t h e t r a n s f o r m a t i o n t o th e c o m p l e x v a r i a b l e a ( k)

    v i a t h e e q u a t i o n s1 I k I / '~1 ( r, t ) = 2 ~ S ~ [ a (k ) e (u r ) + a * (k ) e i (k r ) ] d k ,

    i ~ ~oI~ (k )~ F ( r , t ) - -= 2 ~ ] / ' 2 ,) ~ [ a ( k ) e t k r ) - - a * ( k ) e i ( k r ) ] d k . ( 1 . 9 7

    T h e n1 I k ] 5 'n ( k ) - - V 2 m V * (k ) [ a ( k ) + a * ( - - k ) ] ,

    i o % ( k )W ( k ) = - - ] /= g [ k l ,/ - [ a ( k ) _ a * ( _ k ) ] . ( 1 . 1 0 )T r a n s f o r m a t i o n ( 1 . 9 ) c a n b e c o n s i d e r e d a c a n o n i c a l t r a n s f o r m a t i o n

    ( w i t h c o m p l e x c o e f f i c i e n t s ) t o t h e v a r i a b l e s i a *( k ) a n d a ( k ) ; H a m i l t o n ' se q u a t i o n ( 1 . 8 ) b e c o m e s t h e s in g l e e q u a t i o n

    O a ( k ) 5 EO t _ _ - - i S a , (k ) .U s i n g ( 1 . 4 ) , ( 1 . 8 ) , a n d ( 1 . 1 0 ), w e c a n e x p r e s s t h e e n e r g y i n t h e

    f o r m o f a s e r i e s i n p o w e r s o f a ( k ) a n d a * ( k ) :E = I ~ o ( k ) a ( k ) a * ( k ) d k - b f [ V ( - - k , k , , k 2 ) x

    X [a * (k ) a (k~ ) a (k~ ) + a (k ) a * (k~ ) a* (k 2 i ] XX 8 ( k - - k l - - k ~ t d k d kl d k~ +

    - r - ~ - (k , k~ , k2 ) [ a * (k ) a * (k~ ) a * (k~ ) + a (k ) a (k~ ) a (k~) l X

    W e r e

    X 8 (k - - k l - - k . 2 t d k d k l d k ~ ++ t ! ' I V ( k , k ~ , k ~ , k s ) a * ( k ) a * ( k l ) a ( k ~) a (k ~ ) X

    X ~ ( k - - k l - - k ~ - - k s t d k d k ~ d k 2 d k ~ ,

    V ( k , k l , k 2 ) =t { {r (k) o~ (kl) ~ th- - 8 n ' 1 / ' 2 [ ( k k ~ ) + l k l t k ~ [ ] \ ~ 0( k2 ) ) [ ( k k 2 ) - [ I k I ! k " H

    ((o (k) c0 (k2)~V, ( l kx I ~% ,x \ c 0( k~ ) ) I l k l l k ~ l )? 0 ( k ~ ~ ( k ~ ) y / ~ ( I k I y,q+ [ ( k l k ' ) + l k ~ l ] k 2 1 ] \ ~ o(k ) ) \ l k ~ l t k 2 1 ) j ;

    ( 1 . i i )

    ( L .12 )

    W (k , k~ , k2 , k3 )

    -- 32~ " [(o (k) re (kl) o (k~) ~o (k3)] ~ /y (kk l) (k~.ks) +t

    [ L w ( k ~ )( o ( k a) J ( 2 1 k } - [ 2 [ k l i - - l k - - k 2 1 - -- - I k - - k a [ - - ] k , - - k 2 [ - - [ k l - - k s [) +

    + L (o ( k ) o ) ( k ~) j ( 2 1 k 2 ] + 2 [ k 3 [ - -- - ] k - - k 2 [ - - [ k - - k s ] - - l k r - - k 2 l - - I k l - - k : . ~ I ) -

    o (k l ) t o (k~ )] 5 '~ ) ~ ( -k ~3 )j ( 2 l k x l + 2 ] k : [ - -~ l k 2 + k z [ - - ] - - k 2 ] - - l k - - k l I - - ] x + k a [ ) - -

    - k o ) ( k, ) ro ( k 3 )J ( 2 ] k x i + 2 1 k 2 ] - -- - [ k 2 + k s l - - [ k l - - k z l - - I - - k ~ [- - [ k + k s j - -

    W ~ ( k ) ~ o ( k ~ ) l ' / : I k 2 + k s I- - L c o ( k l ) r 9 ( k 2 ) j ( - - ' - -- } - 2 [ k [ - } - 2 1 k ~ 1 - - 1 k - - k l [ - - I k - - k s ] - - [ k - b k 2 I ) - -

    ~ - o (k l ) c o (ka ) t G- - [ ~ ) J ( 2 1 k ' l + 2 l k a l - -- - [ k 2 + k s l - - l k - - k 3 l - - l k - - k s l - - ! k l + k ~ i ) } . ( 1 . 1 3 )

    T h e r e a r e o t h e r fo u r t h - o r d er t e r m s i n a , p r o p o r t i o n a l t o p r o d u c t s o ft h e f o r m a * a a a a n d a a a a a n d t e r m s c o n j u g a t e t o t h e m . T h e s e a r e i g -n o r e d , s i n c e , a s w i l l b e s h o w n i n s e c t i o n 3 , t h e i r c o n u i b u t i o n i s s m a l l .

    W e n o t e t h a t t h e f u n c t i o n s V a n d W o b e y t h e f o l l o v d n g e q u a t i o n s :V ( k , k l , k s ) = V ( k , k ~ , k 0 = V ( k s , k l , k ) ,

    V ( - - k , - - k l , - - k 2) = V ( k , k l , k _o ),W ( k , k l , k ~ , k s ) ~ - - - W ( k l , k , k 2 , k 3 ) = ( z . 1 4 )

    = W ( k , k l , k s , k 2 ) = W ( k2 , k s , k , k O ,W ( - - k , - - k l , - - k 2 , k s ) = W ( k , k l , k 2 , k s ) .

    T h e e q u a t i o n f o r a ( k ) h a s t h e f o r mO a (k )

    - - Ot @ in) ( k ) a ( k ) == - - i ~ { V ( - - k , k l, k : ) a ( k l ) h ( k ~ ) 6 ( k - - k ~ - - k ~ ) +

    1 9 1

  • 8/9/2019 zhakarov hamiltonian

    3/5

    + 2V ( - - k~ , k , k2 ) a* (k~.) a (k~) 6 (k - - k @ k~) ++ V (k, kl , k .~) a* (k~) a* (k~)

    X 6 (k @ k l " k ~ )} d k l d k 2 - -- - i t W ( k , k l , ke , k a) 5 ( k @ k l - - k 2 - - k a ) X

    X a* (k~) a (k~) a (ka) d k ~ d k ~ d k a 9 (3 ..25)W e s e al f r o m ( L 1 . 5 ) t h a t t h e v a r i a b l e s a ( k ) a r e t h e n o r m a l v a r i a b l e s

    i n t h e p r o b l e m o f s m a l l o s c i l l a t i o n s .2 . S i m p l i f i e d e q u a t i o n s . E q u a t i o n ( 1. .1 5 ) i s a n a p p r o x i m a t i o n a n d

    i s v a l i d f or s m a l l n o n l i n e a r i t i e s , r o u g h l y s p e a k in g , f or a / X < < 1 , w h e r ea i s th e c h a r a c t e r i s t i c a m p l i t u d e o f t h e w a v e a n d X i s a c h a r a c t e r i s t i cw a v e l e n g t h . I n t h i s a p p r o x i m a t i o n , w e c a n m a k e a c o n s i d e r a b l e s i m -p l i f i c a t i o n i n E q . ( 1 . .1 2 ) . T o d o t h i s w e w r i t e a ( k ) as

    a ( k ) - : [ A ( k , t ) 4 - t ( k , t ) ] e x p [ - - i o ( k ) t ] . ( 2 . 1 )W e a s s u m e t h a t A ( k , t ) c h a n g e s s l o w l y i n c o m p a r i s o n w i t h f ,

    w h e r e a < < A . W e s u b s t i t u t e a ( k ) , i n t h e f o r m o f ( 2 . 1 ) , i n t o t h e e q u a -t i o n f o r f a n d t h e o n e f o r A . I n t h e e q u a t i o n f o r a w e r e t a i n o n l y t e r m sw h i c h a r e q u a d r a t i c i n k . A s s u m i n g A c o n s t a n t as a v a r i e s , w e i n t e -g r a t e t h i s e q u a t i o n w i t h re s p e c t t o ti m e . T h i s y ie l d s

    ,)~ l '[V -- k , kx , k~) ex p i t [ o ( k ) - - r e ( k l ) - - (0 ( k Q ]/ xa ( k ) - - ~ o ( k z ) - - ~ o (k~)

    6 ( k - - k I - k 2 ) A ( k ~ ) A ( k e ) @+ 2 V ( - - k : t , k , k 2) e x p i t [ ~ 4 - ~ 1 7 6 Xco (k) 4- m (kl ) - - t0 (k2) - -

    X 6 ( k [ - k l - - k ~ ) A * ( k ~ ) A ( k ~ ) ++ V ( k , k ~, k ) _ e x p it [o (k ) 4 - o (k l ) 4 - o (k~ )l co (k) + ~ (M ) 4- ~o (k )

    X 6 ( k 4 - k l 4 - k ~) A * ( k l ) A * ( k Q } d k l d k ~ . (2 02 )I n t h e e q u a t i o n f o r k w e r e t a i n o n l y t h o s e t e r m s p r o p o r t i o n a l t o

    A f w h i c h c o n t a i n t h e m o s t s lo w l y v a r y i n g e x p o n e n ts . O b v i o u s l y , a l lt h e s l o w l y v a r y i n g e x p o n e n t s a r e c o n t a i n e d i n t h o s e t e r m s p r o p o r t i o n a lt o A *A A . G a t h e r i n g a l l t h e s e t e r m s t o g e t h e r , w e o b t a i n

    OA i ) fO t - - - - i 7 ' ( k , k ~ , k e , k a ) (3 ( k 4 - k ~ - - k ~ - - k )

    X e x p it [o (k ) 4 - t o (k l ) - - (o (k2) --- - o (ka)] A * (k~ ) A (k2 ) A (ka) dk i dk~ dk a . ( 2 .3 )

    H e r eT ( k , k l , k 2 , k a ) =

    + k z ) V ( - - k - - k , k , k l ) V ( - - k ~ - - k a , k 2, k ~)_ _ 4 a ( k 2 ~ o ( k ~ 4 - k a ) - - [ o ( k e ) + o ( k a ) l ~"

    o ( k a - - k a ) V ( - - k , k ~ , k - - k ~ ) V ( - - k ~ , k r , k a - - k r )- - 4 ~ ( k a - - k x ) - - [ o ( k s ) - - o ( k , ) l " "

    _ _ 4 o ( k l - - k ~ ) V ( - - k , k a , k - - k a) V ( - -. k ~ , k ~, k ~ - - k l )( 0 ~ ( k ~ - - k l ) - - [ m ( k : t ) - - o ( k ~ ) ] ~ -}-

    + W ( k , k l , k e , k s ) . ( 2 . 4 )O b v i o u s l y , t h e t e r m s o m i t t e d f r o m t h e H a m i l t o n i a n ( 1 .1 1 .) c a n n o t

    c o n t r i b u t e t o ( 2 . 4 ) .I n u s in g ( 2 . 3 ), w e h a v e t o a s s u m e t h a t f m (kz) + E0 (k - - k l ) , ( 2 . 6 )w h e r e k a n d k I a r e i n t h e d i r e c t i o n o f t h e s a m e s t r a i g h t l i n e . I n d e e d ,i f ( 2 . 6 ) h o l d s , b y a d d i n g t o k t c o m p o n e n t s p e r p e n d i c u l a r t o k , w ec a n i n c r e a s e t h e r i g h t - h a n d s i d e o f ( 2 . 6 ) a n d c o n v e r t (2 . 6 ) i n toa n e q u a t i o n . O n t h e o t h e r h a n d , i f t h e i n e q u a l i t y _ c o n v e rs e t o ( 2 . 6 )

    h o l d s , t h i s is a s u f f i c i e n t c o n d i t i o n f o r t h e a b s e n c e o f s o l u t i o n s to ( 2 . 5 ) .F o r g r a v i t a t i o n a l w a v e s , w i t h t h e d i s p e rs i o n l a w

    c o ( k ) = V g l k l ,a n i n e q u a l i t y c o n v e r s e to ( 2 . 6) h o l ds . A c c o r d i n g l y , ( 2 .5 ) c a n n o t h a v ea n y s o l u t i o n s a n d f or s m a l l a/ ,X ( 2 . 3 ) a p p l i e s . F o r c a p i l l a r y w a v e s[ k >> ( g / a ) 1 / 2 ] , w i t h d i s p e r s io n l a w ~o (k ) = a ~ , ( 2 . 6) ho l d s, s o t h a t( 2 . 3) , i n g e n e r a l , c a n n o t b e u s e d ; if i t i s a s s u m e d t h a t t h e w a v e p a c k e ti s s u f f i c i e n t l y n a r r o w , i . e . , i f a ( k ) is n o n z e r o f o r t k - k o i k ~, X ~ i s p o s i t i v e . W e a p p l y t h ei n v e r s e F o u r i e r t r a n s f o r m a t i o n w i t h r e s p e c t t o ? 4:

    b (x, y , t ) = ~ I b ( ' : v , l )

  • 8/9/2019 zhakarov hamiltonian

    4/5

    C a l c u l a t i o n y i e l d s~q (k ) = [ bo l ~ I 4V ( - - 2k , k , k )~ ]9 4 (o ~ ( k ) - o ) " ( 2 k ) - - I V ( k , k , k , k ) =

    4o) ~" k) ~ ~o (2 k) 4 - - t 6o ) (k ) jI n t h e l i m i t , f o r s m a l l k t h i s t a k e s t h e f o r m 9 ~( k) = l / ~ ( k a ) ~ w ( k ) ,

    w h i c h c o i n c i d e s w i t h t h e e x p r e s s io n o b t a i n e d b y S t o k e s i n 1 8 47 . T h u s ,s o l u t i o n ( 2 . 9 ) a p p r o x i m a t e s a p e r i o d i c w a v e o f f i n i te a m p l i t u d e .

    W h e n

    :* = \ 2 g ) 't h e f r e q u e n c y d i s p l a c e m e n t b e c o m e s i n f i n i t e ; f or l a r g e k , i t is n e g a t i v e .

    I n t h e l i m i t , a s k ~ ~o w e h a v et(k ) = - - - i V ( /~ a y - ~ ( / 0 .

    3 . S t a b i l i t y o f s t e a d y w a v e s o f f i n i t e a m p l i t u d e . W e c o n s id e r t h ed e v e I o p m e n t o f s i n a i 1 p e r t u r b a t i o n s a g a i n s t t h e b a c k g r o u n d o f a s t e a d yp e r i o d i c w a v e . W e s e e k a ( k ) i n t h e f o r m

    a ( k ) = b o S ( k - - k o ) e - i ~ + a t (k , t ) e i ~ t , o = o ( k o ) - ~ O ( k 0 ) . ( 3 . 1 )

    W e a s s u m e c ~ (k ) t o b e s m a l l i n t h e s e n s e t h a t! ] a ( k ) l d k ~ l b o [ .

    N o w w e m a k e ( 1 . 1 5 ) l in e a r i n a ( k ) . T o d o th i s , w e c o n s i d e r o n l yt e r m s o n th e r i g h t - h a n d s i d e w h i c h v a r y s l o w l y w i t h t i m e . W e o b t a i n

    0 a ( k ) - - 2 i b o e i Y t V ( - - k o , k 0 , k 0 - - k ) et * ( k o - - k ) ;[Ot - -' r : ~0 (k ) ~ - o (k - - ko ) - - m (k0 ) - - Q (k0 ) . ( 3 .2 )

    E l i m i n a t i n g c d~ (k 0 - - k ) f r o m ( 3 . 2 ) , w e o b t a i n0 t i - t 0c t (k ) ,

    8 = 0 ( k 0 )= ~ e -~ ' r ta~-g ( - - ko , k , ko - - k ) ~ (k ) . ( 3 . 3 )Equa t i on ( 3 . 3 ) has a so lu t i on o f t he f o r m

    ct (k) = c e v t ,

    q = * / ~ i ' r ~ - y l b o [ e U ~ - ( - - k o , k , k o - - k ) - - * / G ~ . ( 3 . 4 )I n s t a b i l i t y w i l l o c c u r i f t h e e x p r e s s i o n u n d e r t h e s q u a r e - r o o t s i g n

    i s p o s i t i v e . I n o r d er t h a t t h e r e s h o u l d b e i n s t a b i l i t y f o r a r b i t r a r i l y s i n a i 1b 0, t h e e q u a t i o n ~ = 0 s h o u l d h a v e a s o l u t i o n . I f w e n e g l e c t t h e s m a l lt e r m a ( k 0 ) i n t h i s e q u a t i o n , w e a r r i v e a t t h e s y s t e m o f e q u a t i o n s ( 2 . 5 ) .A s w a s e s t a b l i s h e d i n s e c t i o n 2 , t h i s s y s t e m c a n b e s o l v e d f o r c a p i l l a r yw a v e s ; t h u s , i n s t a b i l i t y o f t h i s t y p e o c c u r s f o r c a p i l l a r y w a v e s . U n -s t a b l e w a v e v e c t o r s a r e c o n c e n t r a t e d n e a r t h e s u r f a c e c o( k) = w ( k 0 ++ co(k - - k , ) i n a l a y e r o f t h i c k n e s s p r o p o r t i o n a l to th e a m p l i t u d e . T h em a x i m u m i n c r e m e n t i n t h e i n s t a b i l i t y i s o f o r d e r R eq ~ ( k a ) co (k ).

    T h i s t y p e o f i n s t a b i l i t y i s i m p o s s i b l e f o r g r a v i t a t i o n a l w a v e s . H o w -e v e r , f o r th e s e w a v e s s l o w e r i n s t a b i l i t i e s a r e p o s s i b l e . W e u s e E q . ( 2 . 3 )a n d s u b s t i t u t e i n t o i t A ( R) i n t h e f o r m

    A (k) = bo6 (k - - ko) e i a (k~ ~ - c t ( k~ t ) 9I f w e l i n e a r i z e i n c ~( k, t ) , w e o b t a i n

    o ~ ( k )O t - - 2 T (k, k0, k0, k) [ be [- 'a (k) +

    - ~ e ~ i f ~ ( k~ l t T ( k , 2 k o - - k, ko, k o) bo~ * (2ko - - k ) 9T h i s e q u a t i o n c a n b e r e d u c e d t o a n e q u a t i o n o f f o r m ( 3 . 2 ) ; i t h a s

    a s o l u t i o n p r o p o r t i o n a l t o e q t , w h e r e , f o r q , w e h a v eq = ~ / , 2 i6 +_ V ' i b o [ " T ~" k , 2 k o - - k , k o , k o ) '- ' - t / ~':~,

    6 = o (k ) + o (2ko - - k ) - - 20 (ko ) - l - 2 [ bo p~ [T (k, ko, k o, k) -~-+ T (2k o - - k , ko , ko , 2k o - - k ) - - T (ko , ko , ko , ko ) . ( 3 .5 )

    C o n s i d e r f i r s t t h e c a s ed0~ o b[ k - - k o l T ~ - > ~ > ] p . ( 3 .6 )

    T h e n t e r m s p r o p o r t i o n a l t o b -~ c a n b e d r o p p e d f r o m ( 3 . 5 ) . T h e c o n -d i t i o n f o r th e e x i s t e n c e o f i n s t a b i l i t y fo r a r b i t r a r i l y s m a l l a m p l i t u d e si s 5 = 0 , w h i c h i s e q u i v a l e n t t o t h e e x i s t e n c e o f s o l u t i o n s f o r t h e e q u a -t i o n s

    2o (k ) = m (k l ) + m (k2) , 2k = k l @ k2 . ( 3 .7 )O b v i o u s l y , t h e s e e q u a t i o n s h a v e s o l u t i o n s i f

    r ~ > ~ 1 7 6 ( 3 . 8 )( a s u f f i c i e n t c o n d i t i o n ) , w h e r e t h e v e c t o r s k ~ a n d k 2 a r e p a r a l l e l t o t h es a m e s t r a i g h t l i n e . I n e q u a l i t y ( 3 . 8 ) is t h e r e q u i r e m e n t t h a t c o (k ) b ec o n v e x u p w a r d s . F o r g r a v i t a t i o n a l w a v e s ,

    a n d t h e i n e q u a l i t y n e c e s s a r i l y h o l d s.C o n v e r s e l y , f o r c a p i l l a r y w a v e s , t h e i n v e r s e i n e q u a l i t y h o l d s, i n -

    d i c a t i n g t h a t i n s t a b i l i t y of t h i s t y p e i s i m p o s s i b l e .E q u a t i o n ( 3 .7 ) d e f i n e s a s u r f a c e i n k - s p a c e . U n s t a b l e v e c t o r s l i e

    n e a r t h i s s u r f a c e i n a l a y e r o f t h i c k n e s s p r o p o r t i o n a l t o b 2 . T h e m a x i -m u m i n c r e m e n t i n t h e i n s t a b i l i t y f or g r a v i t a t i o n a l w a v e s i s o f o r d e r

    7 ~ ( k ~ V co ( k ) .T h e r e a r e h i g h e r - o r d e r i n s t a b i l i t i e s c o rr e s p o n d i n g t o c o n s e r v a t i o n

    l a w s f o r m :no (k ) = o (k l ) q - (0 (k~ ), nk - - k 1 i - k ~ .

    T h e o r d e r o f t h e i n c r e m e n t f o r s u c h i n s t a b i l i t i e s is y ( k ) ~ ( k a ) n w ( k ) .A l l t h e se i n s t a b i l i t i e s c a n b e c a l l e d d e s t r u c t i v e .W e t u r n n o w t o i n s t a b i l i t i e s fo r w h i c h ! k l - k o l , ~ < ~ . ( 3 .9 )W e s e e f r o m ( 3 . 9 ) t h a t i n s t a b i l i t y i s p o s s i b l e i f w, k < 0 , i n s t a b i l i t i e s

    o n l y b e i n g e x c i t e d f o r s u f f i c i e n t l y s i n a i1 w a v e v e c t o r s~w~~ < ~ -- l b0 [3 .

    W e c o n s i d e r t h e e a s e o f d i f f e r e n t w a v e n u m b e r s f o r t h e s u r f a c ew a v e s .

    1 . I n t h e r e g i o n o f w a v e n u m b e r sI~o < 1 /g ~ 7 ~ - t ( g l ~ ) I ~ ,

    w h e r e w > 0 , X > 0 , a n d ,k it < 0 , t h e d o m a i n o f i n s t a b i l i t y i n t h ep l a n e X y i s b o u n d e d b y t h e i n e q u a l i t i e s 0 < I k l ] ~ r - - X .