204 



TROPOSPHERIC PROPAGATION AND RADIO METEOROLOGY 



wave equation must be used. Ray 3 is reflected from 

 the ground and in crossing some of the other rays 

 produces the phenomenon of interference. In connec- 

 tion with Figure 21 it must be emphasized that the 

 height scale is tremendously exaggerated and that 

 all the rays shown come from a small group which 

 are propagated in a nearly horizontal direction. 



Sometimes it is convenient to express the path of 

 the ray in terms of ray curvature. The true curvature 

 of a ray as it appears on an un distorted (curved 

 earth) diagram is different from the curvature exhi- 

 bited by a ray on a plane earth diagram. The true 

 curvature of a ray is given by 1/p, where p is the 

 radius of curvature, and it can be shown that, for 

 nearly horizontal rays, this is related to the gradient 

 of n by 



1 = - J . (20) 



p dh 



However, the relative curvature of the earth with 

 respect to that of a ray is (1/a) — (1/p). Now let 

 us set this equal to the curvature 1/ka of an equiv- 

 alent earth. Then 



1 _ 1 J_ 



a p ka 



(21) 



and, introducing equation (20), 

 1 1 



k = 



1 



1 _L dn 



(22) 



This amounts to a definition of k which is more 

 general than the one introduced in Section 17.1.6 

 but reduces to the latter when the index curve varies 

 linearly with height. 



For a plane-earth diagram, M is used in place of n. 

 Since 



M = In + 



1 10 6 



dM __ 1 / dn 

 dh a\dh 



1 10 6 



Substituting the last equation into equation (22) 

 gives 



a dM 



(23) 



and shows that k, in its most general form, is propor- 

 tional to the slope of the M curve. Reference to 

 Figure 20 shows that fc assumes negative values for 

 a range of altitudes whenever a duct is formed in 

 the atmosphere. 



These relations may also be expressed in terms of 

 m, where 



m = 



(24) 



is the ratio of the radius of curvature of a ray to 

 the radius of the earth. From equation (22) it follows 

 that 



JL 



in 



1 



(25) 



Both k and m vary with height except in the special 

 circumstance that the M curve is linear. Table 2 

 gives a number of corresponding values of k and m 

 and indicates their significance. 



Table 2. Relation of k and to. 



The Duct — Superrefr action 



When the M curve has a negative slope, fc is 

 negative ; the curvature of the rays is concave down- 

 ward on a plane earth diagram, and the true curva- 

 ture of the rays is greater than the curvature of the 

 earth. Hence rays which enter the duct under suffi- 

 ciently small angles are bent until they become 

 horizontal and then are turned downwards. This 

 particular form of refraction is called superrefrac- 

 tion. Such rays will be trapped in the duct, oscillating 

 either between the ground and an upper level, or 

 between two levels in the atmosphere. These condi- 

 tions are illustrated by Figure 22 for the case of a 

 ground-based duct and by Figure 23 for an elevated 

 duct. 



The detailed construction of a ray diagram in the 

 case of an elevated duct is shown in Figure 23. It 

 is assumed, for illustration, that the transmitter is 

 placed at the point which produces the maximum 

 amount of trapping, and this point turns out to be 

 located at the maximum of the bend in the M curve. 

 The vertical line for Mi corresponding to hi is drawn 

 as shown, and again the line 1 is drawn to the left 

 of Mi at the distance ai 2 /2, to represent ray 1 which 



