547 



the study of non-linear coupling in both the air 

 and the water. 



The large-scale motions are taken as 



C jK{x-ct) ~ a jk(x-ct) 

 u = u e ; w = w e 



(74) 



with C = C e 



jk{x-ct) 



To ease the process of working with products of 

 wave perturbations, two distinct complex variables 

 i and j are introduced. 



Under these assumptions, the large-scale mo- 

 tions are governed by the linear homogeneous Orr- 

 Sommerfeld equation 



w"" - 2k^w" + k"** - jkR{(U-c) (w" - k^^) - wU") = 



(75a) 



and u is related to w through the continuity 

 equation 



S = jft'/k (75b) 



where from now on primes will denote derivatives 

 with respect to z. 



In the equation for the small scale, we will 

 keep all terms linear in the small-scale perturba- 

 tions including products of the small-scale and 

 large-scale perturbations. 



When the assumed form for the perturbations (71) 

 is used in (69) together with the continuity equa- 

 tion (73) we obtain the following equation for the 

 small scale 



w"" - 2k2w" + k'*w - ikR[(U-c) (w"-k2w)- wU"] 



= R[W3-£||-] (w"' - k^w') + ikR{(U'^ + u) {w"-k2w) 



- ikR{|-^ + U"'C)w + eRtt^-I (2k2(u-c) + U"]w' 



dZ'^ dx 



- U(w"' - k^w' ) + U"'w - U' (w" - k^w')} 



r2 (w,w,z) 



(76) 



where we have introduced the symbol r2(w,w,z) for 

 the right-hand side of (76) . The various terms in 

 (69) are worked out in Appendix C. 



In deriving (76), terms of 0(k ?) have been ig- 

 nored, however terms such as 3 u/3z in air have been 

 kept since these can be large in a viscous flow. In 

 terms of 0(e3^/3x) a viscous correction has also been 

 neglected since all other terms are proportional to R. 



We are interested in the local equilibrium and 

 more specifically the local growth rate of short 

 waves in the modified wind-water field. Thus in 

 the assumed form of solution for the short waves, 

 for a given k the eigenvalue, c, will be a slowly 

 varying function of space 



terms come from the unsteady lifting and distortion 

 of the small scale flow by the long waves. 



The boundary conditions that are satisfied at 

 the free water surface, z = c. + c,' , will now be 

 derived for both the large and small scale motions. 

 The first boundary condition is that the tangential 

 velocity is continuous at the interface, z' = ?', 



3? 

 "'a 3^ = ^^ 



3x 



Expanding the velocities from (70 and 71) in a 

 Taylor series about z' =0, and keeping terms linear 

 in the large scale and small scales we obtain for 

 the large scale 



U'C + Qa = S. 



at z = 



for the small scale [to (kc ) ] 



3u 



U' <;' +u' -u' =--— C' atz = 

 w 3z 



(78) 



(79) 



The term 3u^/3z = kUy, has been ignored in deriving 

 (79) since it is 0(i/i,) and d2u/dz2(o) has been 

 taken to be zero. 



Conditions (78 and 79) can be expressed in the 

 vertical velocity, w' (and w) , through use of the 

 kinematic boundary condition; that the substantial 

 derivative of the surface displacement function, 

 S(x,z,t), is zero for both the air and water flow 

 at the interface, S = 0. That is, if S(x,z,t) = 

 z-^', then DS/Dt = at S = 0{z ' = ;;') 



where 



D 

 Dt 



irr + U(C) T- + w(C) r- 

 dt 3x 3x 



Expanding the velocity field for both air and 

 water about z' =0, and again keeping terms linear 

 in the large and small scales, we obtain in the 

 long-wave limit for the large scale 



for the air 



3t 







for the water 



3C 



at z 



and for the small scale 



for the air 



ikcc -• [U'C + ulikC - £[U'C' + u'] -^ + w = 



oX 



at z' = 



and for the water 



(80) 



3C 



ikcc - u^^iki; - eu^ -g^ + w^ = at z ' = 

 From (78) to (80) we see that 



at z' 



+ c. 



(X) 



(77) 



and 



where Cq is the eigenvalue of the short wave field 

 in the presence of the wind shear field only; c,(x) 

 will be at most 0((;), the amplitude of the long 

 wave. 



Thus the governing equation for w is the Orr- 

 Sommerfeld equation with additional terms arising 

 from the long-wave perturbations. Some of these 

 terms could be obtained directly by replacing U by 

 U + u in the Orr-Sommerf eld equation; additional 



From (39) and (40) , the velocities in the water at 

 z ' = are related to the displacement i by the 

 expressions 



+ U^)'r' 



-ik(c - u) (1 + ie-rT-)C' 



oX 



k(c - u„) (1 + ieff)?' 



(81) 



