545 



on the basis of quasilinear theory, whereafter the 

 effect of wind field modulations may be extracted 

 from the results. In Section 3, we derive the 

 governing equations for the local growth rate for 

 short waves in the modulated flow of the long waves. 

 Ni^merical results for a are presented in Section 4. 

 Consistency of the two-scale expansion requires 

 that the wind-induced growth rate is small, which 

 is indeed the case, since it is proportional to 

 the air-to-water density ratio. Accordingly, we 

 shall set, formally. 



a' + 13' 



e (a+ ig) 



(43) 



Substituting (42) and (43) into (41) and remembering 

 that all the quantities involved are real, we find 

 the following pair of relations: 



g + k'-T - k' (c'-u)2 + ek'a(c'-u) (1 -akC) = (44) 

 [c(c'-u)^ + (c'-c) (c'-ul + (c'-u)u^ 



+ k'T 



;(c'-u) (1 - akC)]C' 



+ 2c(c'-u)(;' + [ (c'-u) (c'-2c + u) 



+ 2k'T]C' = 



(45) 



From (44) it thus follows that 



Since only the terms which are linear in the large- 

 scale perturbation are to be retained, one may 

 ignore the variation of k' with E, when carrying 

 out this integration. 



,1 ^ rB ak i ^^^h 



A = 



(c^-c) 2(c^-c) 



k'u 



2v(c'-c 



g 



^ [(C -c)Vj^, + T]} 



(52) 



where C is a constant to be determined from the 

 initial value of C • By inserting this expression 

 into (37) one finds 



S' = - S 



~i2 



^i^ +— i— (c- + 

 (c'-c) k' (c'-c) ^ g 



L g g ^ 



-2u 



j^rz^ [(C -c)v^, +T]} 



where 



S' 



2 =i|k'^^'|2 



(53) 



(54) 



is the mean-square slope of the short waves. (In 

 Appendix A an alternative derivation, based on 

 kinematic wave theory, is given.) In the second 

 bracketed term u may be expressed in terms of i, 

 by the use of the linearized expression 



+ /g/k' + k'T + 0(E) 



u + v(k') 



0(e) 



u = kci; 



(55) 



(46) 

 Inspection of (35) and (36) reveals that the long 

 waves receive their growth both directly from wind 

 pressure and indirectly from interaction with the 

 short waves, the latter effect being proportional 

 to the mean-square slope of the short waves. Thus, 

 since the variation of long-wave parameters with 

 time is small, little error is incurred by taking 

 u in (46) to be a function of 5 alone. Furthermore, 

 the frequency of the short waves must then be 

 constant in a coordinate system travelling with the 

 long waves so that 



k' (c' - c) = uA 



(47) 



which, together with (46) determines how the wave 

 number for the short waves varies along the wave 

 train. Differentiation of (47) gives 



3k' _ "^'"S 



3C 



c'-c 



g 



(48) 



where c' is the group velocity. 



Thus, S' may be written 



S^ = A'C + B'<:^ 



where (ignoring terms which are nonlinear in C) 



A' = -S 



~2 aB'v 



(c'-6) 



g 



(56) 



(57) 



S'^c 



, , ,, {-c' - c' + 2u 



(c'-c) g 



k' 



(c'-c) 



g 



[(c'-c)v^' + t]} 



(58) 



The boundary conditions for the long waves may now 

 be written. Substitution of the solution for the 

 short waves, and (24), into (35) and (36), gives 



c(C - iC) + A'C + iB'i; 



(59) 



c ' = k ' Vv I + V + u 



(49) 



and where v(k') is defined by (46). With the aid 

 of (46)- (49), (45) may thus be written 



2{c'-il)[5^ + (c^ - i)V^] = {(c'-ii)B(l - akC) 



- (c'-u)u^ + [(c'-c)V]^' + T] 



k'u^/(c^ - c)}c 



(50) 



This equation may be readily solved by integration 

 along the characteristic line 



T = T^ ,- d?/dT^ = (c^ - c)/c 



(51) 



— = - (g + k2rp)J - kc($ - i$) + 0(e2) 



(60) 



For the wave-induced pressure an expression similar 

 to (42) is used, namely 



kc(a + ig)(; 



(61) 



Substitution of this and (59) into (60) and separa- 

 tion into real and imaginary parts yields 



= [g + k^T - kc^ - kc (B + 5) ] c - kc^g + kcC 



T 



(62) 



