BACKGROUND 137 



sufficiently great to direct a ray either back to earth or in a circular path 

 at a constant height above the earth; i.e.: 



noVo cos do ^ , 



nr ~ ' 



This condition then allows one to obtain the value of do which divides the 

 rays into two groups: those that penetrate the duct and those that are 

 trapped within the duct. This particular value of ^o, called the angle of 

 penetration and designated dp, is obtained by noting those instances where 



Mo > 1 (414) 



nr 



and solving for the value of ^o such that (4.13) is equal to unity. 



It is instructive to consider the order of magnitude of refractive index 

 gradient needed for trapping for several commonly observed refractive 

 index profiles. If we rewrite Snell's law, 



ritrt cos dt = HaTa cos dd, (4.15) 



where the subscripts t and A refer to the values of the variables at the 

 transmitter height and the top of the trapping layer, then trapping 

 occurs when 



rit Tt 



riA Va 



> 1, (4.16) 



and the angle of penetration at the transmitter, dp, is given by setting 



cos dp = 1. (4.17) 



nt Vt 



riA Ta 



The maximum permissible n gradient for a given value of dp is then given 

 by: 



^ ^ ru^^UA ^ 



t^r ta — Tt 



where ua must satisfy (4.17); i.e.: 



_ Villi cos d^. (4.19) 



Ta 



