tiii; <: vi, ci i. vi io\ <>!•• \ hiiticm. cox kk\<;k 



10! 



For I ho far point 



in y/m- + 



1 lit n- n- 



These equal ions may lie combined: 



inn \ S/i|- 



1 



</ = ( .; + "o ± - 



Vw- + wm\ 8/i i 



, (77) 



where the plus sign gives the far point and the minus 

 sign gives the near point. The reflection point is 

 obtained by using m = and equation (77) reduces 

 to: 



di = 



■ihr 



(78) 



Thus to obtain the range of the near edge of the 

 first Fresnel zone for the first lobe, substitute n = 1, 

 m = 1 and use the minus sign in equation (77) : 



d„ = 0.688 



hS 



(79) 



The far edge of this zone is obtained by using the 

 plus sign 



fci 2 



d f = 23.3 



Equation (77) is in the form 



/u 2 



= T 



where 



or 



\2?i n 1 n 1 



1 



mm 



Ti = 



diX 



(80) 



(81) 



(82) 



(83) 



If di is taken as the distance of the shoreline, 7\ 

 may be considered as a characteristic site or terrain 

 factor at a particular azimuth and combined with 

 the height and wavelength to obtain the range of 

 any zone of any lobe. 



In order to read the relative intensity and phase 

 lag of the reflected wave from the diffraction graphs, 

 Figure 27 and Figure 28 respectively, it is necessary 

 to have m expressed in terms of v. In Figure 19 the 

 path difference is by definition of m 



Equation (23), with A = <l — b, yields 



A = 



4 ' 



hence 



(85) 



It is also desirable to have an expression for v in 

 terms of n and T. This is obtained by substituting 

 v 2 /2 for m in equation (82) and solving 



ll'ii 1 



n + j, 



(86) 



The width of the zones, that is, along a chord at 

 di parallel to the minor axis of the elliptical rings, 



Figure 60. Width of Fresnel zones. 



may be obtained from Figure 60. Zone m is shown 

 with a chord length b. The distance from the image 

 antenna to the intersection of the chord and ring m 

 is I + mX/2. From this may be written 



mX 

 ~2~ 



= r- + 



(87) 



Neglecting m 2 X 2 /4 since it is small compared to the 

 other terms, 



,2; 



V- + m\l = l- 



A = m 



(84) 



b = \/±m\l , 



? = t = «i = ^r > 



cos ^ n\ 



from equation (78), since ^ is small. 



Where earth curvature effects are appreciable the 



