Chapter 27 



CONVERGENCE EFFECTS IN REFLECTIONS FROM 

 TROPOSPHERIC LAYERS* 



An elevated duct may be treated as a concave 

 spherical mirror whose radius of curvature is 

 a, the effective earth radius. This includes any layer 

 that can act as a reflector to radiation incident at a 

 sufficiently small angle. The problem is here con- 

 sidered as one of geometrical optics only. Ray tracing 

 methods are used, and the phases are assumed to 

 add randomly. This assumption may introduce an 

 error as large as 3 db in the result but is necessary 

 to simplify the solution of the problem. If the reflec- 

 tion coefficient is other than unity, it must be 

 multiplied into the general relation which will be 

 given for C = KLM the net convergence factor. 



27.1 



CONVERGENCE FACTOR 



A bundle of rays leaving a transmitter below the 

 reflecting layer is converged on reflection from a 

 concave surface. The convergence factor K is the 

 ratio of the power density at the receiving antenna 

 after convergence to the power density at the 

 receiver that would be expected after reflection from 

 a plane surface (essentially free space condition). 

 Referring to Figure 1, the convergence factor can 

 be expressed as 



„ _ (x + y) Sdi 



xbdi 



or 



K = 



2/5 0. 



2xy 

 aR sin cj> 



(1) 



(2) 



TRANSMITTER 



Figure 1. Convergence factor K. 



where x = distance from transmitter to point of 

 reflection, 



a By Ensign W. W. Carter, USNR Radio Division, Consult- 

 ant Group. 



y = distance from receiver to point of reflec- 

 tion, 



R = x + y = total range, 



a = effective earth's radius (usually 4,590 

 nautical miles), 



(j> = angle of incidence of radiation at reflec- 

 tion, 

 other angles as shown on Figure 1. 



Equation (2) can be deduced from equation (1) 

 by remembering that 



I = ada = 



and 



5 0! - 5 2 = 2da = 



sin <f> ' 



2.T.5 0i 



(3) 



a sin <t> 



The form shown in equation (2) is the more useful 

 and is similar to the divergence factor for reflection 

 at a convex surface that has been in use for some 

 time. Equation (2) shows that K can grow quite 

 large and even become infinite for certain conditions. 

 Curve 1, Figure 2, shows a plot of the absolute value 



2 



100 y 



s 



80 g 



< 



60 z 



z 



40 < 



200 400 600 



800 1000 1200 1400 Ifioo 1800 

 b IN FEET ► 



Figure 2. Value of K for height of layer (b in ft) versus 

 range (nautical miles). 



of if as a function of b, the height of the layer above 

 the antennas, for a total range of 80 nautical miles. 

 This plot also assumes x = y = 40 miles, which is 

 a necessary condition for a smooth reflector. In this 

 case, K becomes infinite for a layer 1,100 ft above 

 the antennas. Curve 2, Figure 2, shows a plot of 



258 



