138 N CLniATOLOGY 



By designating Va = rt -\- /1.4, (4.18) becomes 



An 



Ar 





1 - 



rt + hA 



cos Or, 



(4.20) 



By rewriting (4.20) 



cos dr 



(4.21) 



and expanding (1 + hA/rt)~'^ and cos Op, one obtains the expression: 



Lr, 



An _ 1^1 



Ar - ~ ''' Irt + 2hAA 



(4.22) 



by neglecting terms 0p/4! and QiA/rtY, and terms beyond these. For 

 the case of Gp = 0, (4.22) reduces to : 



An _ n 1 

 Ar a a 



157 N units/km. 



(4.23) 



It is seen from (4.22) that the n gradient necessary to trap a radio ray 

 at a given value of dp is practically independent of transmitting antenna 

 height above the earth. For example, a, 6p = ray will be trapped by 

 an n gradient of —157.0 A'' units/km at sea level, where Ut ~ 1.0003, 

 while the necessary n gradient at 3 km above sea level will be —156.9 

 A^ units/km for a,n Ut = 1.0002. This indicates, for all practical appli- 

 cations, that the necessary n gradient for trapping is independent of 

 altitude. Further, by considering the temperature and humidity gra- 

 dients encountered in the troposphere, one is led to the conclusion that 

 ducting gradients would not be expected to occur at altitudes greater 

 than 3 km. In fact, Cowan's [33] investigation indicates that trapping 

 gradients are nearly always confined to the first kilometer above the 

 surface. 



A consideration of (4.22) indicates that the magnitude of the negative 

 gradient necessary for ducting is 1/a for dp = but increases in propor- 

 tion to dp'/2h as dp increases. The gradients necessary for atmospheric 

 ducts as a function of 60 are given for several different profiles in figure 

 4.35 An analysis of radiosonde data indicates that gradients in excess of 

 0.5 N units per meter are seldom exceeded within atmospheric layers. 

 It is interesting to note how rapidly the necessary gradients increase to 



