548 



and 



-— (1 + - + i-) 

 kc c c 



3x 



thus \i. = iw , as in fixed coordinates. 

 For the large scale motions, 



and 



".^ 



jw/kc 



]W 



(82) 



With {75b) and (82) , condition (78) for the large 

 scale motion becomes 



U'w + c w' 



w k c = 



at z = 



(83) 



and with (73) and (81) condition (79) for the small 

 scale motion (to 0(kc)] becomes 



U'w + cw' - wkc 



8u ' " dU ,^ .w, 

 -— w - w ;r— [— + 1— J 

 3z dz c c 



k^TC = p„[k{c - u)^ - (g + k'^T))?' 



(88) 



Using (81) to rewrite (88) in terms of w, introduc- 

 ing s = Pa/P„ and using (84) to rewrite the group, 

 cw' - U'w, we obtain the final form of the pressure 

 boundary condition (85) to be satisfied at z ' =0. 



w[-k(l-s)c2 + (g+k^T)] + i|^ skw' 

 - R 

 u 

 w 



isc 

 Rk 



= w[kc^ - (g+k'^T) ] t -g- + iw/c) - w 2cu^ 



3wa 



.•v 



+ s{u,,w'c + iw'c — r— / k + cw U' ( — + ^r-) 



>«' 9z c c 



ic^-,, w w ~ 4w" k^w , _ ,-, 



IT "^'' ^ e {2xkcw + -^ - 2 ^-) = Y2n (w) 



(89) 



We will introduce the variables p,Q,b, and y and 

 rewrite (89) as Pw + Qw + bw" ' = y (w) where 



P = [-k(l-s)c2 + (g+k^T)] 



ic3sk , . / , 

 Q = — ; b = -isc/Rk 



. C"„W 



irw"- 

 k c 



Y (w) at z 

 10 



(84) 



where we have introduced the symbol Y,q(w) for the 

 right hand side of (84) . 



The remaining boundary condition to be satisfied 

 at z = S and then transferred to z ' =0, is the 

 balance of pressure or more precisely of normal 

 stress with surface tension 



32c 



Pw + T 3^ 



(85) 



The viscous normal stress at z = C is given by 



3w 



'^nn R ^nn 



2 n 



R 3n 



where 



3C ^3-3 

 w„ = w - u 7^— and -r — - -r — „ - 

 n 3x 3n 3z dx 3x 



3? 3 



Because viwU'^^(o) = y^U^(o) in the limit V^ "^ °° 

 there is some cancellation in the stress condition 

 and the final result for the large scale is a^^^ = 

 2 w'/R. 



For the small scale, all terms involving the 

 large scale perturbations are negligible for k<<k' 

 so that a^j^ = 2 w'/R. 



The pressure in the air at the surface, z = C, 

 is obtained by expanding the pressure about z' =0. 



p(i;) = p(o) + C If (o) 



(86) 



where p{o) and 3p/3z(o) are available from the 

 momentum equation (67). 



After considerable manipulation we obtain the 

 following formula for p^ - 0^^! 



and Y20 '"' -"-^ ^^^ right hand side of (89) . 



The corresponding boundary condition for the 

 large scale is homogeneous and of the form 



Pw + Qw' + bw" ' 



(90) 



where 



-k(l-s)c2 + (g+k2^)] 



Q = jc3sk/R; b = -jsc/Rk 



To summarize, the long waves satisfy the ordinary 

 Orr-Sommerfeld equation (75) with the appropriate 

 linear boundary conditions for a free water (83) 

 and (90) plus w('°) = w' (<») = <». The resulting 

 homogeneous eigenvalue problem is solved numerically 

 to determine u, w, and c for a given long wave 

 amplitude, C, wave number, k, and R. 



The short waves satisfy a modified Orr-Sommerfeld 

 equation, (75) with the effects of the long-wave 

 perturbations appearing also in the boundary con- 

 ditions (84) and (89). 



To solve this short-wave local-equilibrium 

 problem we resort to techniques that by now have 

 become standard in stability theory for perturbed 

 eigenvalue problems. 



We assume that the short-wave solutions can be 

 expanded about the perturbed solution (no long waves 

 present) in the form 



u = Un + C u. 



w = Wg + C w. 



(91) 



ikipg - a'nn] = p^[- c G' - wU' + — (w"' - 3k'^w') 



,du - -, -, - 3u ,w' 3w 

 dz 3z k 3z 



w ,.,., - 4w" ^k^w . w",, 

 e (2ikcw + - - 2—- xc-)] 



(87) 



We obtain p - k^TC at z' = C' [to 0(Rc)] directly 



w 

 from (41) 



where ? is the large wave amplitude. The eigenvalue, 

 c, is also expanded 



(91b) 



For C = the problem of short-wave dynamics 

 reduces to the ordinary Orr-Sommerfeld equation 

 with free-surface boundary conditions. 



