46 



FACTORS INFLUENCING TRANSMISSION 



Refraction is of practical importance only when 

 the angle between the rays and the horizontal is 

 small. In the determination of gain as given in later 



TO CENTER OF EARTH 



Figure 3. Refraction over curved earth. 



chapters, the effect of refraction becomes com- 

 pletelj^ negligible \\-hen a is more than a few degrees. 

 For small angles, cos a may be replaced by 

 1 — a-/2. In this case ecjuation (1) is well approx- 

 imated by 



L(. 



a,)-) = 71 — no + 



(2) 



where h is the height above the ground, so that 

 ro = a and r = a -\- h. This is the practical form of 

 Snell's law for the atmosphere above a curved earth. 

 The reference level (see Figure 3) is here taken at 

 the surface of the earth where no is the index of 

 refraction. 



4.1.3 Modified Refractive Index 



In place of the sum {no + h/a) that appears in 

 equation (2), it is customary to define and use a 

 quantity M given by 



M 



- [<» - « ^ 1^] 



10«. 



(3) 



ilf is called the modified refractive index. It gives a 

 unit that is convenient for practical use. The modi- 

 fied index is then said to be expressed in M units, 

 values of which commonly lie in the range of 300 to 

 500. Using this definition, equation (2) becomes 



i(a=-aa=) = (M - Mo)- 



10" 



(4) 



An important special case is that in which the 

 refractive index decreases linearly with height. 



n — no = constant X /(. Then equation (2) may he 

 written in the form 



1.0 .,^ h 



— [oi- — at)-) = — 



2 l:a 



(5) 



where k is the factor mentioned in Section 4.1.1 

 which determines the modified earth's radius ka. 

 Comparing the above expression with equation (2), 

 and differentiating, it follows that 



or 



k = 



(6) 



10" 



1 + a 



dh 



Proof of the fact that refraction is negligible unle.ss 

 the angle is very small may readily be deduced from 

 the pi-eceding formulas. Thus, on differentiating 

 equation (4), 



10"" 

 da = dM . ^^^, 

 a 



and in the standard linear case, by equation (5), 



da = dh/kaa ~ 1.2 • 10''' dh/a 



for k = 4/3. Taking a = 0.05 radians (3°) and 

 dh = 100 meters, one finds da = 0.00024 radians 

 (50 seconds of arc), a very small change in angle. 

 This is the standard deflection which is accounted 

 for by replacing a by ka. The de\'iations from this 

 value experienced with nonstandard refraction are 

 even smaller. The larger the angle a with the hori- 

 zontal at which a raj^ issues from the transmitter, the 

 less the angular deviation. In communication work 

 and for certain radar problems, however, angles of 

 less than one degree are of importance, and da 

 may then become comparable to a. 



4.1.4 Graphical Representation 



Figures 4 to 6 show three different ways of repre- 

 senting rays subject to refraction. Figure 4 gives a 

 true picture apart from the exaggeration of heights. 

 In the case of standard refraction, the curvature 

 of the rays is always conca\'e downwards, the center 

 of ciu'vature being below the surface of the earth. 

 The middle ray shown is the horizon ray and to the 

 lower right is the diffraction region into which rays 

 do not penetrate. Figure 5 shows a diagram with 



