II NDAMENTALS OF PROPAGATION 



195 



Hence M is the excess of the modified refractive in- 

 dex above unity, measured in units of one millionth. 

 This unit is called an .1/ unit [MU]. Values of M for 

 the atmosphere lie in the range of 200 to 500. Cus- 

 tomarily M is referred to simply as the modified 

 index of refraction. 



Using the numerical value for the radius of the 

 earth, 6.37 X 10 6 m(21 X 10 6 ft), the rate of increase 

 of M with height, owing to the term h/a, is (I /a) 10 6 , 

 which is equal to 0.157 MU per meter (0.048 MU per 

 foot) . As the result of a large number of experiments, 

 carried out chiefly in the northern temperate lati- 

 tudes, the rate of decrease with height of the re- 

 fractive index has been found, on the average, to be 



*» 1 6 = _ 1 1 io« 



dh 4 a 



= -0.039 MU per meter. (9) 



This is the rate of decrease assumed for the standard 

 atmosphere. 



It will be noticed that the average rate of decrease 

 of n with height is one quarter of the rate of increase 

 of the term h/a which results from the curvature of 

 the earth. The fact that these quantities are of com- 

 parable magnitude is of great importance, as will be 

 seen later. 



Consequently the vertical gradient of M for the 

 standard atmosphere is 



dM _ fdn 1 

 dh \ dh a 



10 6 



3 1 



4a 



10 6 



1 



GO 



io 6 



(10) 



which has the value 0.118 MU per meter (0.036 MU 

 per ft). The value of M at any height, relative to the 

 surface value M Q , for the standard atmosphere, is 

 equal to 



M — M = 0.118 h; h in meters, 



M - M = 0.036 h; A in feet. (11) 



17.1.6 



Equivalent Earth Radius — 

 Flat Earth Diagram 



An important conclusion may be drawn from 

 equation (11). As will be shown in Section 17.2.4, 

 dn/dh is the negative of the curvature of a ray in the 

 atmosphere, and l/o is the curvature of the earth. 

 The algebraic sum of these two quantities (their 

 numerical difference) is the curvature of the ray 

 relative to that of the earth. The net result is this : if 



the earth is replaced by an equivalent earth with an 

 enlarged radius equal to 4a/3 the rays may be drawn 

 as straight lines. To state the result in another way: 

 using the equivalent earth with radius equal to 4a/3 

 corresponds to replacing the actual atmosphere, in 

 which the index n decreases with height, by a homo- 

 geneous atmosphere with an equivalent index n' 

 which is independent of height (see Figures 10, 11, 13, 

 14, and 15) . This transformation of coordinates great- 

 ly facilitates the calculation and interpretation of cov- 

 erage diagrams for the standard atmosphere. 



More generally, if the rate of change of n with 

 height differs from the value- (1/4) (1/a) 10 6 MU 

 per meter given above, which may be true in certain 

 parts of the world, the equivalent earth radius de- 

 parts from the value 4a/3. In general the equivalent 

 earth radius is designated by ka. For a steeper drop 

 of refractive index with height, k increases and be- 

 comes infinite when the curvature of the ray is just 

 equal to the curvature of the earth. 



In the general case, when k is not equal to %, 

 equation (11) must be modified to the form: 



M - Mo = f- 10 6 , 



= 0.157 



= 0.04S 



h in meters , 



h in feet 



(12) 



to account for a linear moisture gradient correspond- 

 ing to a different value of k. 



Since the change of the earth's radius takes care of 

 the variation of refractive index and substitutes a 

 homogeneous atmosphere for the actual atmosphere, 

 it follows that in a diagram in which the earth is 

 given a radius ka, the radiation propagates along 



TRANSMITTER 



Figure 10. Ray curvature over earth of radius a in an 

 actual atmosphere. 



straight lines. The difference is illustrated in Figures 

 10 and 11. In Figure 10, which shows the true geo- 



