however, the plane was going much faster than the fastest wave being observed. 

 Furthermore, the plane was "seeing", what was for all practical purposes, a 

 pure wind sea. Thus, it is most reasonable to assume that nearly all of the wave 

 energy was confined within ±90° of the mean wind direction. These facts indi- 

 cate that the plane will always be overtaking the waves as it moves in the down- 

 wind direction. From (5), oj will be negative and it is convenient to redefine 

 wand (J as 



ui = - a + y a cos \p 



1 + 4Vaj 



cos 4' 



a= 1 + 



2V cos^ 



g 



I.Kf 



I ^l<l 



(6a) 



This corresponds to the case of the plane travelling downwind. A similar re-defini- 

 tion of w and tr may be made for the case where the plane is travelling upwind. 

 These definitions are made merely to maintain consistency in the analysis of any 

 particular run. 



6.2 Specifications of the Integral Transform 



The relation between the apparent spectrum E (ui) and the real two-dimen- 

 sional spectrum, relative to the plane heading F {a,\f), may be expressed in either 

 of two forms: 



E(u>) = /F(cr,^) 

 c. 



8»/' 



9w 



dcr 



(7) 



E (u^) = / F (o-,<//) 

 ^1 



a T 



9 u) 



6^1 



(8) 



where ci indicates that the integration is to be carried out over all u and «// which 

 can yield the specified value of w (Cartwright, 1963). In the case of the plane 

 travelling downwind, o- and ^ together must satisfy equation {5a) and (6a) with 

 w fixed. The corresponding Jacobian's would be 



24 



