543 



+ ... (10) 



P' 

 w 



P 



-(g + 



C )C' 

 tt 



i)' + TC' 

 t XX 



z z 



1,~ 



-(P„ + p;) = -g(c + c;) - (*t + *;) 



?• {< 



tz 



t [<*x ■^ *i) + <*z + *z) ] 



i'2 



X 



f'2) + i 

 z ' 2 



-(?■ 



3t^ 



■ ?'2) 

 (19) 



+ T(K + C' ) + 



XX XX 



(11) 



In (11) , P^, is the surface pressure in the absence of 

 the short waves and p^ the additional surface pres- 

 sure added due to the presence of the short waves . 

 In deriving (10) , partial use has been made of (2) , 

 which holds for $ and $' separately. To arrive 

 at equations for the long wave, (10) and (11) are 

 averaged over the large scales. This is most 

 conveniently done by taking the ensemble average 

 of a large number of realizations differing only 

 as to the phase of the short waves , which is assumed 

 to be randomly distributed among the members of 

 the ensemble. This procedure yields 



*z = ^t + *x^x + S ■ + 



(12) 



= gc 



2 X X XX 



both to be applied at z = C- The last bracketed 

 term of (18) and the last two of (19) will give 

 rise to higher harmonics. Their contribution to 

 the large-scale motion will be of higher order, 

 and they can hence be neglected. In deriving (19) , 

 use was made of the linearized boundary conditions 

 for the long waves. Thus, for example, the term 

 ^^-^C ' in (19) arises from replacing $^2 i" dD ^Y 

 ^^^, which will give a negligible error to within 

 the approximation employed. 



Since the major aim of the analysis is to deter- 

 mine the lowest-order effect of the short waves on 

 the growth rate of the long waves, it is sufficient 

 to retain only linear terms. However, all terms 

 linear in the large-scale motion which modulate 

 the small-scale wave train must be retained. The 

 long wave will be taken as a uniform, infinite 

 wave train of wave number k = 2tt/A . Its phase 

 velocity differs from the linearized value. 



3t^ 



(C'2) - \('^'^ - 1>'2) + 

 2 x z 



(13) 



at z = C, where the tilde denotes the average over 

 the large scales and 



S' 



(14) 



is the Stokes' drift due to the small-scale motion. 

 In deriving (13) , use has been made of the linearized 

 boundary conditions for the small scales, for 

 example. 



^'*;z = ^'^"tt" ••• 



3t 



3 (ft^) - c;2 + 



2 -1 2 



(15) 



The long waves are to be determined as a solution 

 of Laplace's equation 



V2$ = (16) 



subject to the boundary conditions (12) and (13) 

 and the condition that disturbances vanish at large 

 depths , i.e.. 







for z 



(17) 



The corresponding boundary conditions for the short 

 waves are obtained by subtracting those for the 

 long waves from the full equations. 



*' 

 z 



C' + it'C + $ C' +'C 



t XX XX 



3x 



(t'^ 



,.,.) 



(18) 



/g/k + kT 



(20) 



by terms proportional to the square of the small- 

 scale wave slope, and by terms due to the wind, 

 which are proportional to the density ratio between 

 air and water both of which may be expected to be 

 small. The short waves driven by the wind may also 

 give rise to slow growth, or decay, of the large- 

 scale waves. For the subsequent analysis, it is 

 convenient to introduce the following nondimensional 

 "slow" variables: 



C = k(x - ct) 



T = k c t 



The solution for the long wave is sought in the 

 form (real part always implied) 



C = C(T)e 



>(T)e 



iS 



ig+kz 



(21) 

 (22) 



(23) 

 (24) 



The variation of the surface deflection and potential 

 with the "slow" time, x, allows for the effects of 

 wind, and the presence of the short waves, to have 

 a weak influence on the growth rate, and the phase 

 (and consequently also the phase velocity) of the 

 long waves. __ Without the wind and the short waves 

 both c, and # would be constants. 



For the short-waves, on the other hand, both 

 the phase velocity and wave number will vary slowly 

 along the long wave because of the modulation by 

 the latter. We therefore set 



K' = C (C,T)e 



ie(x,t) 



(25) 



,^ , , ie(x,t) 

 (S,z',T)e (26) 



