where PjC^,!!) is the Joint probability distribution of 

 surface slopes and displacements. Integrating over the 

 surface, the backs cattered power is thus given by 



^2^r-[^P^(,,n)^^_^d,]dr- (5) 



^^ / --K^ 



For a Gaussian wave field, the surface slopes and 

 heights are statistically independent, since 



<Cn> = <CVC> = V<c^/> = 



2 



on account of the statistical homogeneity. Hence 

 P (^,n) = P (OP (n) , 



, 3 ~ 1 2 ~ 



where 



Pj(?) = (2TT<-?2>)-iexp|-^V2<c2>j 



Substituting in equation (5), the integrations can be pre- 

 formed explicitly yielding 



I = a<c'>5c-M (2TT)-iexp(-riV2) + n(J)(n)} 



(6) 



V. 



where n = ctp/<(;^>^ , 4>(n) =f e " ' " dn 



denotes the error function and a is the same constant as 

 in equation ( 3) • 



We note that the shape of the pulse onset shown in 

 Fig. 9 is the same for all sea states, equation (6) depen- 

 ding on the wave field only through the single scaling 



25-47 



