4-5] REFLECTION OF PLANE WAVES FROM THE GROUND 189 



Similarly, the exponential term in Equation 4-27 may be approximated as 

 ^-iiKh sin e ^ I _ j2Kh sin d. (4-34) 



Substituting these approximations into Equation 4-28, and retaining only 

 the first-order terms in sin d, we obtain 



Fh = 2[(e - l)-^/2 -\-jKh] sin d (4-35) 



Fv = 2[e{e - l)-i/2 -\-jKh] sin 6. (4-36) 



Thus, for sufficiently small d, both Fh and Fv are proportional to sin 6. 

 But sin 9, as shown by Equation 4-30, is inversely proportional to range. 

 Hence below the first lobe it follows that F is also inversely proportional 

 to range: 



F oc R-\ (4-37) 



Therefore the received echo power, which is given by Equation 4-1, will be 

 inversely proportional to the eighth power of the range in the region below 

 the first lobe: 



P,i oc R-^ (4-38) 



This is in contrast to the inverse fourth power of the range which holds for 

 free space. The range at which the transition occurs from a fourth-power 

 law to an eighth-power law for a target which spans more than one lobe 

 will be discussed in Paragraph 4-10. 



Equations 4-35 and 4-36 show how the resultant (one-way) field varies 

 with height below the first lobe. Very close to the surface, where the term 

 is small in magnitude compared with the other term in the brackets, Fh 

 and F^ become 



Fy = j^^JyT. si" ^- (4-40) 



Hence the ratio of the fields at the target with vertical and horizontal 

 polarization will be 



Fv/Fh = 6. (4-41) 



If the radar area of the target is the same for these two polarizations, then 

 the ratio of the received echo powers will approach |e|^. This difference is 

 important in the case of sea clutter. 



As the height is increased, the term j2Kh eventually will become large 

 relative to the other term in the brackets in Equations 4-35 and 4-36. Then 

 the field at the target will be approximately proportional to height and will 

 be almost the same for either polarization. 



