essential directional properties found by both Longuet-Higgins, et aL, (1963) 

 and SWOP I (Cote, et. aL, I960): Waves moving at neatly the speed of the 

 wind (W/c - 1) were concentrated in a rather narrow angular beam about the 

 wind direction (^ =0). The width of this beam increases as the ratio W/c 

 increases^ in the definition of K3, then, the exponent p (cr) wil! be large for 

 W/c ~ ] and diminish as W/c becomes large relative to 1 . The function p (c ) 

 is shown versus W/c in Figure 10 and compares well to the results of Longuet- 

 Higgins, et. aL,(1963) as amended by Cartwright (personal communication). The 

 normalizing factor g (''" ) is chosen so that 



+ Tr/2 



/ K3 (o- , V^ ) d I// - 1 

 -tt/2 



and therefore 



+ tt/2 



g'V) = /cosP('^)^d^ . 



-Tr/2 



in order to estimate the effect of different directional assumptions, the 

 integral equations (7) and (8) were solved for all three of the K's. It should be 

 noted that in carrying through these solutions, the assumption has been made that 

 each K is independent of fetch, 



6.3 Solution of the Integral Equations 



This section proceeds to outline the solution of equations (7) and (8). 

 Although the present discussion concerns only the case of the plane travelling 

 downwind, it should be clear that it applies equally well to the upwind case, 

 provided w , \// , and K ( cr , \^ ) are defined properly. 



Suhsti'-uting for F ( o- , \^ ) and the Jacobians, one has 



E ( «,) = / H {a) [K (cr,^)+ K (o", - >//)] do " {7a) 



Ci o" V sin I tA I 



g 



and 



E (u') = / H (0-) [K {cr,yp)+ K (o- , -v^)] d^p . (8a) 



1 J] + 4<^V cos ^ 



26 



