Chapter 4 

 FACTORS INFLUENCING TRANSMISSION 



4.1 



4.1.1 



REFRACTION 



Survey 



REFRACTION IS caused by the variation of the 

 . dielectric constant (square of refractive index) 

 of the atmosphere. Although the atmosphere is 

 tenuous and the variations of refractive index 

 are small, the effect of refraction upon the field- 

 strength distribution of waves is considerable. As 

 will be shown, refraction under average conditions 

 may be taken into account by using an earth with a 

 modified radius. A representative average value of 

 modified earth radius commonly used is ka with 

 k = 4/3. Under certain conditions, especially in 

 warmer climates, a slightly higher value of k might 

 be preferable. 



The case where a is replaced by 4a./3 is referred to 

 as standard refraction. It corresponds to a linear 

 variation of refractive index with height in the 

 atmosphere. In recent years, more comphcated 

 variations of refractive index in the atmosphere 

 have received considerable attention and have 

 proved to be of great operational interest. This 

 volume, however, is restricted to consideration of 

 standard atmosphere propagation. 



4.1.2 



Snail's Law 



Let ??o and 7?i denote the refractive indices of t-\vo 

 media separated by a plane boundary. The ordinary 

 law of refraction known as Snell's law is then usually 

 stated (see Figure 1), as 



no sin /3o = «i sin /3i, 



where /So and j3i are the angles which the ray makes 

 with the perpendicular to the boundary. It is con- 

 venient to use the complementary angle a, so that 



7io cos ao = ni cos ai. 



For several plane-parallel boundaries, Snell's law 

 generalizes to 



?io cos ao = n.i cos ai = no cos ao = ■ ■ • . 



In the atmosphere, the refractive index is a con- 

 tinuous function of the height. Again, it is usually 



legitimate to consider the atmosphei-e as horizon- 

 tally stratified, so that the refractive index is a 

 function of height only. The case of a continuously 

 variable refractive index is readily obtained by 



Figure 1. Refraction at boundary between two 

 media. 



passing to the limit of an infinity of parallel boimd- 

 aries infinitely close together, Snell's law remaining 

 the same; thus 



n(h) • cos a = ?!o cos ao, 



where now n and a are continuous functions of the 

 height. In place of a discontinuous change in direc- 

 tion, there will now occur a bending of the rays 

 (Figure 2). 





DISTANCE ALONG EARTH 



FiGUKE 2. Refraction in the atmosphere with variable 

 7iih). 



If the boundaries are not plane but spherical, 

 Snell's law must be modified. Analysis shows that 

 over a spherical earth surrounded by an atmosphere 

 in which the I'efractive index 7i is a function of the 

 distance r from the earth's center, the law of re- 

 fraction becomes 



n{r) • r cos a = noVo cos ao, 



(1) 



where a is the angle between a ray and the horizontal 

 (see Figure 3). 



45 



