546 



(6 - A')? = (2c - B')^, 



From (63) we find 



C = ^qSKP 



J 



B-A' 

 — r — ;— 'i'^i 

 2c-B' 1 



(63) 



(64) 



By use of this, C^- and c,^^ may be expressed in 

 terms of C and an eigenvalue relation obtained by 

 substitution into (62) . This then gives 



c = /g/R + RT +. 



(65) 



with correction terms proportional to the mean 

 square slope of the short waves and to the air-to- 

 water density ratio, both of which are likely to 

 be small corrections of little importance. The 

 major result of the analysis is that given by (63), 

 (64) namely that 



d 

 dT 



(in?) = 



B-A' 

 2c-B' 



26^ 



aB'c' 



2c(c'-c) 



g 



C.2 



(66) 



i.e., the growth of the long-wave amplitude is 

 given by the sum of the growth due to direct action 

 of pressures in the manner of Miles (1957, 1962) 

 and the indirect growth due to the Stokes ' transport 

 by the growing short waves. The second term may be 

 large compared to the first term, if c' is close 

 to c. However, the analysis presented does not 

 hold in the immediate neighborhood of Cg = c but a 

 separate (and nonlinear) analysis is then required. 

 For waves running against the wind, c', c', and a 

 will be negative, so that the presence pf the short 

 waves will always increase the decay rate of the 

 long wave. 



THE WIND- INDUCED GROWTH OF SHORT WAVES IN THE 

 PRESENCE OF LONG WAVES 



where velocities are scaled to free stream velocity 

 outside the boundary layer over the water, and 

 lengths scaled to boundary- layer thickness 6 ; R 

 is the Reynolds number based on 5 . (In Section 2, 

 lengths were scaled to k' , the short- scale wave 

 number) . 



To derive the Orr-Sommerfeld equation, these 

 equations are cross-differentiated and subtracted 

 to eliminate the pressure. Some use is made of 

 the continuity equation and the result is 



u 



3^w 

 3z3t 8x3t 



uV^w + wV^u 



i V2-^ 

 R 3z 



+ 1 v2-^ 

 R 3x 





 (69) 



The flow in the air is taken to be a horizontal 

 shear flow plus two wave perturbations of disparate 

 scales: the fast scale, x and t; and the slow 

 scale, X = ex, and t = et where e = k/k' . The 

 variation with z is set by the shear profile and 

 viscous effects and will be taken to be the same 

 order for both wave fields. The long wave field 

 is a function of x,z, and t only; the short wave 

 field is a function of x,z, and t and in addition 

 will be influenced by the long scale waves so that 



u = U(z) + u(x,z,t) + u' (x,z,t,-x,t) 



w = w(x,z,t) + w' (x,z,t;x,t) (70) 



The surface deflection is taken as 



5 = C (x,t) + C' (x,t;x,t) 



The major effect of long waves in a parallel 

 shear flow on the behavior of the short waves will 

 come from changes in the local growth rate and 

 convection velocities as well as an unsteady lifting 

 of the small scale as the large waves pass. There- 

 fore the small scales will be assumed to be of the 

 form 



- ~ . '■ . ",-- r>i ik{x-ct) ~, ,, 16 

 w(x,z,t;x,t) =w[z-5(x,t)]e =w(z)e 



The perturbation equation governing the modification 

 of short waves on the wind-water interface by the 

 long-wave field is derived from the momentum equation 

 by the procedure used to derive the Orr-Sommerfeld 

 equation. Additional effects arise because the 

 short waves see not only the mean wind field, U(z), 

 but long-wave fluctuations, u and w. The large- 

 scale field is governed by a linear equation, 

 the small-scale field by an equation linear in u' , 

 w' which also contains terms linear in u and w. 



As in 2, we take the water to be inviscid and 

 the flow potential but we consider the air to be 

 viscous: with no surface current, and continuous 

 tangential velocity between air and water, this 



corresponds to the limit jj^ 



with y^ 



and v^ finite, justified by the large density ratio 

 between water and air. Both fluids are taken to 

 be incompressible. 



We begin with the Navier-Stokes equations for 

 two-dimensional flow in the air 



3u . 3u 3u 



31 + "33r + '^ 



3p 



3 -^\''- 

 3x p R 



3w 3w 3w 

 "^ * "3^ * "3l 



3w _ 3£^ 



3z p R 



(67) 

 (68) 



-~, -, ~,^-,,ik (x-ct) - , , , 19 ,.,, , 

 u' (x,z,t;x,t) = u[z-;(x,t)]e = u(z')e (71) 



where z' = z-C and c = c(x,t). Changes in the wave 

 number, k, are 0(e3u/3x); such terms will be ignored 

 in this local analysis. For this assumed form of 

 w' and u' , the continuity equation becomes 



3w' 



3z 



+ iku' 



3u' 3;; 



3z 3x 



which to lowest order becomes 



(72) 



(73) 



since £C~ = -w/c. The presence of c, in the assumed 

 form for w' and u' introduces several terms into 

 the equation for the small scale. In addition, 

 the velocity perturbations, u and w, also appear. 



The equation for the large scale is obtained by 

 a phase average of (69) written with the assumed 

 form (70) and (71) . The non-linear coupling of the 

 small-scale motions will not be included although 

 the corresponding effects in the water are the 

 main subject of this paper and are worked out in 

 Section 2. We anticipate further work to complete 



