Hasselmann and SchieXev 



q| = {(ninj) + ^^-\) [^^^ [K- sw^f - (4]} • 



2 2 ~ ~ .. 



■t. (^H- SCO ) - (cOf) / 8c£jj_ 8tUg\ I ^2j 



T^ ^2 \ 911: 9n;/o ^ 



The subscript refers to values at n = 0. 



To lowest order, the Doppler spectra for vertical and hori- 

 zontal polarisation are identical Gaussian distributions with mean 

 frequency ( w) = scog and variance 



((CO -<(.))') =<4 =2Jk;k^<UjjuJ +(k/(u^)} (14) 



(i^= 1,2) 



The distribution represents an ensemble of Bragg lines of equal 

 energies displaced by their appropriate facet Doppler frequencies 

 co^. 



2 2 



The higher-order corrections q, , qg , ... represent dis- 

 tortions of the Gaussian distribution due to the variations in energy 

 of the Bragg lines associated with variations in the carrier-wave 

 slope. These affect the shape of the Doppler spectrum through the 

 correlation between facet slopes and facet Doppler frequency, 

 Eq. (10). The degree of distortion depends on the depression angle 

 and polarisation. In the cross -polarised case, the zero'th and first- 

 order terms disappear, since (Ty^^)o = (S/Sn; ^^^^ = 0, so that the 

 Doppler spectrum is non-Gaussian already to lowest order. 



Computations of the Doppler spectrum were made for a 

 Pierson-Moskowitz [16] spectrum using a half -plane cosine-to-the- 

 fourth spreading factor. 



^'^ k""^ exp { - P(cOo/co )"} for k, > 



3tt 



Fg(k) = 



for k, <0 



with a = 0.0081, p = 0.74 and Wq = g/U, where U is the wind 

 velocity, assumed parallel to the X| axis. The same spreading 

 factor was taken for both scattering and carrier waves. 



374 



