544 



where 9 is the phase, 



k' = 



(27) 



where 



S' = R.P.{|'c'*(i*' + £$•)} 



(37) 



is the wave number, 



(28) 



is the frequency (measured in a fixed coordinate 

 system) , and 



]c' (z 



U. 



(29) 



and the star denotes complex conjugate. The velocity 

 potential must satisfy Laplace's equation. Substi- 

 tution of (26) -(29) into (2) gives 



k'2{$"^,^, - $') + i£k'[ k^itT' + 2k' $' 



+ 2(z'k^ - l^]i'^)<f'^,] 



+ 0(62$"') = (38) 



The assumption of a slowly varying wave train allows 

 one to regard k' and id' as functions of the "slow" 

 variables , T and 5 • 



An approximate asymptotic solution for the short 

 waves if sound by expansion in the small quantity 



£ = k/k" 



(30) 



(That e thus defined is a slowly varying quantity 

 causes no special difficulty) . Substitution of 

 (25) - (29) into (18) and (19) and omission of all 

 terms of order |c'|^, l?^?'|f and higher, as well 

 as of terms of order e and higher, gives the 

 following boundary conditions for the small-scale 

 motion to be satisfied at z' = 0: 



*^ = - i;'(c'-u) + euC^ + ec(i;^ - Cg) + iekCg*"' 



+ (31) 



^ = - (g + k'^T)!;"' + ik'(c'-u)$' 

 p _-_"- 



+ iek' [iu*' + ic('i>| - 4>^) + 2k'TC' + k ' TC ' ] 



k'^i. [w - k5(£ - Cf)] +. 



(32) 



This equation may be solved approximately by series 

 expansion in e. One finds in a straight-forward 

 manner that 



..2 k' 



k'z'^ )]A - iezA^ + 0(e2A)} 

 (39) 



1 z' ^ t ~ ^ 



e {[1 - ie{ —- v'T'r lla - Hr^a, -i- nic^r 



2 k' 



where 



where A = A(5,f) is to be determined by aid of the 

 kinematic boundary condition (31) . By substitution 

 of (39) into (31) and expanding in powers of £ one 

 finds 



A = -i{c'-u)c"' + E{(c'C')g + c(C| - ip} 



+ O(e^O) (40) 



Combination of (40) and (32) yields 



^w 



— = -[g + k'^T - k' {c'-u)2];f' + iek'{ [c{c' - u) ^ 



+ (c'-u)ug + (c'-c){c' - u)^ + k^Tlc' 



+ 2c(c'-u)(f' 



T 



+ [ (c'-u) (c'-2c + u) + 2k'T]c'^ } (41) 



u'/k' 



(33) 



u and w are the perturbation velocities, 't^ and 

 $2 respectively, of the large-scale-flow evaluated 

 at 2. = C, and p^, is defined by 



The induced surface pressure due to the wind may be 

 assumed to be related to the small-scale surface 

 deflection in a quasilinear manner that takes into 

 account the modulation of the wind field by the 

 long waves. The following expression is chosen: 



^w 1 ~ ie 

 — = — p ' e 



p p '^w 



(34) 



The terms neglected as being of higher order in £ 

 include the term C^t^ ' ^" (19)/ which expressed in 

 the slow coordinates becomes 



E-^k' 



3t 



^)2 I 



and is hence negligible compared to the term k' Tc;'. 

 For the long waves one finds similarly 



w = k[c(C., 



iC) + S^] +. 



(35) 



— = -(g + k^ T) i - i.c{i^ - ii) 



1 r 2 " 2 / 3 



I"^ ^ '37- 3S 



2-2,.^ 1_)2 1^, I 2 



4k'2[ |i$' + £$■ 



'■] +... 



(36) 



P' 



— = k" (c'-u) (a' + iE 



P 



) (1 - ak^)C' 



(42) 



where a' and B' are aerodynamic coefficients (having 

 the dimension of velocity) giving the in-phase and 

 out-of -phase components, respectively, of the induced 

 pressure. The modulation of the wind field due to 

 the presence of the long waves is accounted for by 

 the factor (1 - akt;) . For long waves running with 

 the wind and having a phase velocity less than the 

 wind-speed, the air flow at the crest will slow 

 down in the region below the matched layer where 

 U = c, and the small-scale growth rate will thus 

 be reduced in this region. Conversely, the air 

 speed will increase over the troughs leading to an 

 increased growth rate there. Hence, the coefficient,- 

 a, will be positive for such waves. For waves run- 

 ning against the wind, however, or for waves with 

 c greater than the wind speed, a will be negative. 

 To determine the numerical value of a, one must 

 carry out calculations based on the Orr-Sommerfeld 

 equation. First the wind field modulation due 

 to a long wave of small amplitude is calculated. 

 Then, the pressure on the short waves is computed 



