HIGH INTIAL ELEVATION ANGLES 



53 



3.3. An Approximation for High Initial 

 Elevation Angles 



A method may be derived for determining ray bending from a knowl- 

 edge only of n at the end points of the ray path, if it is assumed that the 

 initial elevation angle is large. Equation (3.2) in terms of refractivity, 

 N, is equal to 



Ti,2 = - / cot d dN ■ 10" 



(3.5) 



assuming w = 1 in the denominator. Integration by parts yields: 



Tl,2 = 



Ni.e., 



cot 8 dN ■ 10" 



A^,,e, 



N coid ■ 10" 



JA^i.fli 



i.A^j 



Bx.Ni 



N 

 sin^0 



dd ■ 10' 



(3.6) 



Note that the ratio, N/s,m^d, becomes smaller with increasing 6 for values 

 of 6 close to 90°. If point 1 is taken at the surface, then di = ^o and 

 A^i = A^,. Then for ^o = 10°, No = and do = 7r/2, the last term of 

 (3.6) amounts to only 3.5 percent of the entire equation, and for the same 

 values of ^"2 and 62, but with ^0 = 87 mrad (-^5°), the second term of (3.6) 

 is still relatively small (--^10 percent). Thus it would seem reasonable 

 to assume that for 



do > 87 mrad (-^5°), 



the bending, n ,2, between the surface and any point, r, is given sufficiently 

 well by 



or 



Tl,2 = — 



A cot X 10" 



^r.»r 



-I AT ,,.9(1 

 Tl,2 = Ns cot do X 10-"^ - Nr cot Or X IQ-^ 



(3.8) 



The term —Nr cot dr X 10~^ is practically constant and small with re- 

 spect to the first term, for a given value of do and r, in the range ^0 > 87 

 mrad. Thus ti,2 is seen to be essentially a linear function of N s in the 

 range do > 87 mrad. For bending through the entire atmosphere (to a 

 point where Nr = 0), and for do < 87 mrad, (3.8) reduces to 



T = N, cot ^0 X 10- 



(3.9) 



