549 



2k^w" + k^w - ikR[(U-c)(w" -k-^w^ )-woU" ] = 



"^o^o + 'i^- '^^o' "O = ° 



PWn + Q "6 + ^ K' 



at z' 



and 



Wa {") = w' (<») 











(92) 



The eigenvalue, Cq, which determines the growth 

 rate of the short waves in the parallel shear flow, 

 is determined by a numerical solution of these 

 equations for a given k and R. 



The equations governing the modification to the 

 flow due to the long-wave perturbations are derived 

 by using (91) in (76) and equating terms of 0(C). 



In this operation the as-yet-unknown correction 

 to the eigenvalue, c, , will appear multiplying the 

 lowest order solution, Wg . The resulting problem 

 for Wj^ is written 



w'j" - 2k2w'^' + k'*w^ - ikR[(U-CQ) (w" -k2w ) -w U"] 



= c,r, (w,z) + r^(w,w,z) (93) 



where r, (w,z) and r2(w,w,z) are known functions of 

 the long-wave perturbation, w(z) , and the lowest- 

 order short-wave perturbation, w^^ (z) . From (84) 

 and (91) 







/ 







vrdz = 



(97) 



where v is the solution to the adjoint problem 

 Z(v) = 



with v(0) = v' (0) = v(<») = v' (<><=) = (98) 



Condition (97) is then used to determine the modifi- 

 cations of the flow due to non-linearities. 



The present problem differs from that in (96) - 

 (98) in that the boundary conditions of (95) involve 

 linear combinations of the derivatives of w, at z 

 = and are non-homogeneous. 



In Appendix B, we show that the adjoint boundary 

 conditions that replace (98) in the determination 

 of V are 



v' (0) = v' (") = V (») = 



(99) 



and [B-ikRcU' + o(U'-kc)]v + [U'-kc]v" + cv" ' = 



at z = 



where 



a = - [2k2 + ikR(u-c)] ; 



§P-f(U--kc) 



and the extended solvability condition for non- 

 homogeneous boundary conditions is 



r, (w,z) = - ikR(w[J - k^Wg) jg^j. 



and r2(w,w,z) is defined by (76) with the long 

 wave perturbations normalized by c; . 



TOie boundary conditions for wj have homogeneous 

 operations that are identical to those for Wj, but 

 the equations are non-homogeneous with terms that 

 depend on w , the long-wave perturbations, w, and 

 the unknown correction to eigenvalue, c, . 



,dU 



^0«I + ("dz 



kCg) Wj 



Y„ c^ + y^^ (w,w ) 



Pw + Q wj + b w'" 



and w, (■») = w' ("=) 



^21 ■=! + ^20 <*'''0) 



(95) 







where Yjq (w) and y^^g (w) are defined in equations 

 (86) and (90) with the long wave perturbations 

 normalized by C 



and Y^^ = - w- + kw^ 



rvdz = J [Cj r^ (z^) + r^ (z)]vdz 



(Y c +Y ) [— (0) + (— -r2_)v(0)" 



11 1 10 CQ CQ bCQ 



+ (Y c + y ) [r- (0)] 

 21 1 20 t> 



(100) 



The solvability condition (100) is then used to 

 determine Cj^ and thus the correction to the local 

 growth rate due to the presence of the long wave 

 perturbations . 



A yv CO 



Y [r^(0) + (— - -2-)v(0)]+Y J(0)- [r (z)vdz 

 10 =0 CQ bCQ 20i> -'n 2 



Ci = - 



Y [— (0) + ( r:^)v(0)]+Y -(0)- rr,(z)vdz 



11 CO CO bCQ 2lb Jq 1 



(101) 



Y21 = 2Cgk{l-s)Wg 



ik3s/Rw^ + is/kRw" ' 



This problem is similar to that considered by 

 Stuart (1960) and many others in later studies. 

 In Stuart (1960) we have the problem 



L(wp = r(z) 



After c has been determined from (101) , the 

 normal stress on the small-scale waves in the water 

 due to the air flow may be determined. The sim- 

 plest approach is to use (88) and infer p^ - a^^ 

 directly from p,^ using the momentum equation in the 

 water (or Bernoulli ' s equation) . 



Retaining the terms linear in the large-scale 

 quantities, we have 



with w (0) = w' (0) = w ("») = w' («>) = 



(96) 



where L is the Orr-Sommerfeld operator. 



The solvability condition [Ince (1926) pg. 214] 

 for this problem is 



Pa " °nn = Pw^* ^^ '■'' ""w^ ^ " (g + Tk^)^] (102) 



where c is given by (91b) with c from (101) . The 

 correction to the growth rate, c,, is doubly complex 

 in that it has both real and imaginary parts (c 

 and c^) that are in phase and out of phase with 5. 



