194 



TROPOSPHERIC PROPAGATION AND RADIO METEOROLOGY 



17.1.4 



Refraction — Snell's Law 



17.1.5 



Refraction over a Curved Earth 



The bending of rays in the atmosphere depends 

 upon the refractive index n which is a function of the 

 temperature, pressure, and moisture content of the 

 air. The manner in which these quantities control the 

 index of refraction is explained in Section 17.2.1. To a 

 first approximation, assuming horizontal stratifica- 

 tion of the atmosphere, the index may be considered 

 to be a function only of height above the ground. The 

 corresponding case, familiar in optics, is that of two 

 media, such as water and air, with different refrac- 

 tive indices ni and n 2 (Figure 8 A) . If ai and ai are the 

 angles between the rays and the plane of the bound- 

 ary, Snell's law of refraction states that 



iii cos ai = n 2 cos «2 . 



In the atmosphere the refractive index changes 

 continuously with height. The simplest case, often 



BOUNDARY 



Figure 8. A. Refraction at a sharp boundary. B. Re- 

 fraction through a layer with variable n. 



encountered in practice, is that of a refractive index 

 which decreases linearly with height. This is known 

 as standard refraction. Snell's law applies here also, 

 since the atmosphere may be divided up into an in- 

 finity of parallel boundaries, the change of refractive 

 index from one boundary to the next being infinites- 

 imally small. Instead of a sudden change of direction 

 there is then a gradual change or bending of the rays 

 (Figure 8B). Snell's law may then be stated gen- 

 erally as 



n cos a = n cos ao , 



where now n and a are continuous functions of height 

 and the zero subscript on the right-hand side refers to 

 any fixed reference level. The curvature of the re- 

 fracted rays is downwards or upwards according to 

 whether the refractive index decreases or increases 

 with height. 



In reality the surfaces of constant refractive index 

 are not planes but are concentric spheres about the 

 earth's center. In this case Snell's law assumes a 

 slightly different form. Instead of using angles re- 

 ferred to the plane surfaces it is now necessary to 

 refer the angles to horizontal planes tangent to 

 spheres about the center of the earth (see Figure 9) . 

 The new form, as given in Section 17.4, is 



nr cos a = ano cos ao 



(4) 



where r and a are values of the radius vector from the 

 center of the earth to a point in the atmosphere and 



Figure 9. Refraction through a curved layer. 



to the earth's surface, respectively, a now stands for 

 the angle formed by the ray with a plane normal to 

 the radius vector, ao and no are the values of a and n 

 at the ground surface. 



If h is the height above the ground surface, so that 

 r = h + a, the above equation may also be written 

 in the form 



1 + 



cos a = no cos ao 



(5) 



h/a is a very small quantity, and n differs from unity 

 by only a few parts in 10,000. Under these conditions 

 n(l + h/a) may be replaced by n + h/a with neg- 

 ligible error. The quantity n + h/a is called the 

 modified refractive index, or the modified index for 

 short. Equation (5) then assumes the form 

 I 



(» + ;) 



cos a = no COS ao 



(6) 



As a result of general agreement it is customary to 

 use, instead of n + h/a, the symbol M defined as 

 follows : 



M = 



(-H 



10 6 



(7) 



At the surface of the ground M reduces to 



Jlfo=(n„-l) 10 6 . (8) 



