III-75 



As stated in (III- 42), the y^ must be chosen so that 



Then 



Y s if 1 - a= s (III-63) 



n n 



i Y > if 1 - a^ < 

 n n 



A (z) = A e^^'^n^ (III- 64) 



n n 



The scattered wave may therefore be written as 



P (x,y) = > A e n n (III- 65) 



The components corresponding to values of n for which 1 -ttj^ s: are plane 

 waves propagating downward with direction cosines (a^^, Yn)- For n such that 

 1 - ttj^ < 0, the corresponding component of scattering is a plane wave propagat- 

 ing in the x direction but with an amplitude decreasing exponentially as z de- 

 creases. 



In order to solve the scattering problem in this setting, it remains 

 to obtain the coefficients A . There are several ways of doing this, depending 

 upon the approximations used and the assumptions concerning scattering in a 

 neighborhood of the surface. One usually studies the sinusoidal surface 



z = S(x) = h cos px (III- 66) 



irradiated by an incident wave 



= ^ik(ax+Yz) ^jjj.g^j 



inc 



artbur m.littUJnt. 



S-7001-0307 



