m-io6 



However, p' = $ , hence p' = I = $, and therefore 



A (X,u) = |$(X,u)|^ . (III-128) 



Consequently A(\, u) is the absolute square of the amplitude of the scattered 

 wave traveling in the {\, u, v) direction, and therefore, A(>^, |a) is proportional 

 to the power scattered in the (\, u, v) direction. 



It is now necessary to obtain an expression for $ (X , n) . We introduce 

 the nondimensional coordinates 5 , r\ given by 



kx =5 , ky = ri (III-129) 



Let the surface be described by S(x, y) . (The explicit form of this func- 

 tion will not be required.) In terms of these new variables, the surface can be 

 described by the normalized function, C,{^, r\), such that: 



kS(x,y) = a C (? , ri) (III- 130) 



where J = kh and h^ = <S^ (x, y)> denotes the mean square height of the surface. 

 Then 



00 00 



p^Jx,y,S(x,y)] = j ( e^^^^+^^^^^^(^'^)^$(X,u)dXdM (III-131) 



-00 -co 



As in the other developments, we have the boundary condition that 



^inc -^' ^' ^^^' ^^^ ^ ^sc ^^' ^' ^^^' ^^^ " ° 



CO 00 



^nhm 2l.1LtttlcJnf. 



S-7001-0307 



