Hassetmann and Sohieler 



Similarly, a wavenumber broadening Ak corresponds to a 

 frequency broadening of the Bragg line of order Aw = (dcog/dk ) /^k = 

 (c0g/2k^) • Ak (ignoring capillary effects). The model assumes that 

 this is small compared with the Doppler shift oo^ = - 2K' • u induced 

 by the facet velocity u. For u = O(cl)cC) > where Wg is a typical 

 carrier-wave frequency, this requires 



DK'ut^k'/wi = DK'(k'k*=)'^; » 1 



Substituting Dk*^ « i , this is equivalent to 



K'^k'/k'^)'^^ >» 1 (6) 



Since k'/k*^ » 1 , the frequency condition (6) is less critical 

 than the corresponding condition (5) for the angular resolution. The 

 inequality (5) is normally fairly well satisfied at conventional radar 

 wave lengths for surface -wave heights of order 1 m and higher (except 

 for small depression angles, where the model breaks down, in any case 

 because of shadowing effects). For electromagnetic wave lengths 

 longer than about 1 m the inequality (5) is normally no longer valid, 

 even though the two-scale inequality (4) may still apply. 



The total backs cattered energy is obtained in the wave-facet 

 interaction model by summing over the contributions from all 

 scattering facets. Introducing a facet probability distribution p(X.) 

 with respect to the five basic facet parameters \ = (X.| ,\2»^3»^4»^) » 

 where 



(\,,\2,X.3) = (U|,U2,U2) = facet velocity (= local 



long-wave orbital velocity), 



and (X4,\-) = (8^/9x, ,8t,/9x2) = (n, ,n2) = facet slope, the Doppler 

 spectrum is given by 



^\^- ) t^l/S^^'^d - sojg - Wf)] p(X) d\ 



(7) 



□Tfl, CO- represent, respectively, the Bragg cross section and 

 gravity-wave frequency in the facet reference frame. 



To the modulated Doppler spectrum (7) of the first-order 

 Bragg field should be added the modulated spectrum of the zero'th 

 order field reflected from a plane facet, as described by the specular 

 reflexion model. However, this is important only near vertical 

 incidence and will be ignored in the following. 



Experimentally, the probability distribution p(X.) is found to 

 be approximately Gaussian, in accordance with the theoretical distri- 



372 



