552 



experiment for waves longer than about 10 cm. This 

 is close to the first possible long wave that can 

 strongly interact with a short wave whose group 

 velocity is equal to the long wave phase velocity. 

 Thus the results we have obtained to date indi- 

 cate that the long waves can receive energy due to 

 their interaction with wind driven short waves. 

 The interaction mechanism we have investigated 

 requires the presence of the wind and the variation 

 of the short wave growth rate along the surface of 

 the long wave due to changes in the local wind 

 field caused by the passage of the long wave. Of 



course further work remains to be done to explore 

 the full implications of these results, to complete 

 the calculations and to make fuller comparison with 

 experiment . 



5 . ACKNOWLEDGMENTS 



This work was supported by the National Science 

 Foundation under Grant ENG7617265. We acknowledge 

 many stimulating discussions with our colleague, 

 Professor E. Mollo-Christensen. 



APPENDIX A. 



DERIVATION OF STOKES ' DRIFT MODULATION FROM 

 KINEMATIC WAVE THEORY 



which, with S' = (c-u)s'^/k', is found to agree 

 with (53). 



Kinematic wave theory, modified to allow for small 

 dissipation or growth due to energy interchange 

 with the wind, gives the following conservation 

 equation for the wave action density, A' , of the 

 train of short waves: 



— ' + — (c'A-) = 2n:A' 

 3t 3x g 1 



(A.l) 



where Q'. is the temporal growth rate. The wave 

 action density for waves on a current is given by 

 [Bretherton and Garrett (1968)] 



A' = E'/fi' 



(A. 2) 



where E' is the energy density and H' = k' (c' - u) 

 the frequency relative to the fluid at rest. By 

 introduction of 



^' (C 



u)2k'P 



(C 



u)S' 



(A. 3) 



(A.l) may be cast as a conservation equation for 

 the Stokes' drift 



3t 3x^ g ' ^ k- ^3t g 3x ' i 



(_ lii + 2n')S' 

 3x 1 



(A. 4) 



With n: = k'g' (l-akC)/2 and expressed in the vari- 

 ables T and 5 this takes the form 



APPENDIX B. 



THE EXTENDED SOLVABILITY CONDITION 



We first determine the adjoint to the homogeneous 

 problem for a shear flow over a water surface. 

 This problem is written 



L(w) =0 



Wj = cw + (U' - kc) w' 



W2 = Pw + Qw' + bw"' = 

 w(<») = w' (") = 



at z 



(B.l) 



where L is the Orr-Sommerfeld operator. 



The adjoint to the Orr-Sommerfeld equation is 

 [Sturart (I960)] 



£{v) = v"" + av" - 2ikRU'v' + [k'* + ik3R(U-c)]v = 



where 



a = -2k2-ikR(U-c) 



(B.2) 



From the Lagrange identity [Ince (1926) pp. 210, 

 214] 



J 



{vL(w) - wL(v)}dz = P(w,v) 



[5 T^ + (c- - c) ^ ]5.n S- = e(l-akC) 

 dT g at. 



(c'-u) g 



2u -,-T-^, [c'-c)V]., + T]} 

 (c' -c) '^ 



9 (A. 5) 



By neglecting the variation of the left-hand side 

 with T one finds from this 



S" (Cg-e) (c'-e) (c"-u) g 

 k' 



2u 



(c'-c) K-'-'-k. +^]> 



(A. 5) 



where P{w,v) is the bilinear concomitant. The 

 boundary conditions on v that will complete the 

 statement of the adjoint problem are found by the 

 requirement that P{w,v) be zero at both end points. 

 Since w and its derivatives are zero at z = oo, this 

 leaves the conditions on v to be found for z = 0. 

 P(w,v) is written in bilinear form as 



P(w,v) = {v v' v" V'" } 



= V. [U] .w 



ikRU' a 1 



-a 0-10 



+10 



-1 



< > 

 w'" 



{B.3) 



The free surface boundary conditions for w may be 

 written 



