Hasselmann and Sohieler 



(2) 

 The scattering function T includes both electromagnetic 



and hydrodynamic interactions at the free surface. For wave lengths 



in the HF range and longer, the hydrodynamic interactions can 



probably be described to fair approximation by classical hydrody- 



ncimical theory, independent of the effects of wave breaking. 



Equation (16) represents the random-field expression of nonlinear 



effects such as nonsinusoldal wave forms' , nonlinear phase velocities, 



etc, , that have been variously suggested as explanation of the observed 



side bands of HF Doppler spectra. 



In the limit of an incident wave short compared with the 

 principcd waves of the sea, the donninant interactions at finite de- 

 pression angles are electromagnetic. The largest contributions to 

 the Integral in (16) arise in this case from interactions in which one 

 of the gravity-wave components, say Jk° , lies near to the peak of the 

 spectrum. Since k° « k', the second component k** is then approxi- 

 mately equal to the Bragg component, cr|jk'' « - 2k''~(cf, Eq, (15) and 

 Fig, 10). The side condition Wj| = (yo<^Q + abCOjj = const (expressed by 

 the 6- function in the integral) defines an integration curve In the k^ 

 plane which Is given approximately by the circle k*" = const. This~ 

 follows by noting that, on account of Eq, (15), the variation Sk" 

 corresponds to an equally large variation ± 6k , But for k° ■<?<: k**, 

 the associated frequency variation 6w|j is generally small compared 

 with the variation Swq, since dwjj/dk « da)o/dk°. Hence the side 

 condition coj = const reduces to coq = const. It is shown below that, 

 at finite depression angles, T'^' Is Independent of k° for k' «k , 

 and the Integration over the directions of k" for fixed k° can then 

 be readily carried out, yielding 



JZ), V (2)+ , . , (2)- , , 

 X (wj) = X (wd) + X (t^cj) 



where (17) 



X^2)S(^^) ^ 2T<21p,^(_g2k')[Eg(ood-scOg) +Eg(swg-Wd)] 



and Eg(co) is the one -dimensional frequency spectrum of the wave 

 field, with {t^) = Jo" Eg(a») doo. (The factor 2 arises through Inter- 

 change of the components a and b In Fig, 10.) 



Thus each Bragg line appears as the carrier of a second- 

 order, two-sided Image of the surface-wave frequency spectrum. 

 Physically, the Doppler continuum arises, as In the case of the wave- 

 facet Interaction model, through the modulation of the first-order 



These Include the often Invoked "higher Interference orders" 

 occurring In the Bragg scattering by a lattice. They are generated 

 only If the periodic scattering field Is not a purely sinusoidal dis- 

 turbance but contains higher harmonics. 



382 



