REFLECTION THEORY — PROPAGATION BEYOND THE HORIZON 629 



Case 1. Large Layers 



If the layer were a plane, perfectly reflecting surface of unlimited ex- 

 tent, the power at the terminals of antenna ^4^ would be the same as 

 the power received under line-of-sight conditions, 



p _ p AtAr 



If the layer has an amplitude reflection coefficient, q, the received power 

 is. 



^> 



p _ p AtAr 2 



This relation applies when the layer dimensions are large in terms of the 

 wavelength and are large compared with the Fresnel zone dimensions; 

 that is, b > \/2aX/A and c > \/2aX. 



Case 2. Small Layers 



When the dimensions of the layer are small compared with the Fres- 

 nel zone, but large compared with the wavelength, the received power is 

 given by the "radar" formula, 



2 2 



D T> AtAr 2/j • \2 : 



This relation applies when h < -y/'IaX/A and c < \/2aX. 

 terms of Fresnel integrals, is 



Pr = -Pr^VV [C"^") + SHu)][CHv) + SHv)] 

 where u = — ;= and v = 



\a V Xa 



When u and v are very hirge, we have, approximatelj^ 



C(m) = S{xi) = C{v) = S{v) = \ 



and the expression for Pr reduces to that given for Case 1 above, except for the 

 factor q^. 



When both u and v are very small, we have approximately, 



C(ii) = u C{v) = V 



S{u) = o Sin) = 



and the expression for Pr reduces to that given for Case 2. 



When u is large and v is small the expression for Pr given in Case 3 results. 



