44 



ELEMENTARY THEORY OF NONSTANDARD PROPAGATION 



whole length of the transmission path. This is a 

 severe restriction, but it has proved indispensable 

 up to date in order to make the problem susceptible 

 to mathematical treatment, and it is reasonably often 

 fulfilled in practice. 



6.4 



RAY TRACING 



In order to understand the mechanism of trans- 

 mission of radiant energy in a duct the course of 

 rays issuing from the transmitter is traced according 

 to equation (5). Note that for the small angles with 

 the horizontal at which these phenomena occur, 



dh 

 a = dx' 



(6) 



where x designates the horizontal distance. Hence 

 from equation (5) 



/?"/' 



dh[a - + 2 (M - Mo) ■ 10- 



(7) 



Since M is a given function of height, equation (7) 

 gives in integral form the relation between distance 

 and height, where a is the angle with the horizontal 

 of the ray emitted by transmitter and M is the 

 value of M at the transmitter height. 



Practicable graphical methods of ray tracing have 

 been developed and used extensively to compute 

 actual coverage diagrams. 66,68 ' 69 ' 71 ' 76,82,98,99 Three 

 schematic pictures of ray tracing, showing the main 

 phenomena of interest, are presented in Figures 2, 





upward curvature of the rays. On the left-hand side 

 of each diagram the M curve is plotted. By equation 

 (5) we have 



a = , 

 if 



M = Mo - | a 2 • 10 6 . . (8) 



The vertical lines drawn on the M curve diagram 

 are the values of M — \ a 2 • 10 6 for the rays 

 selected. Wherever this line intersects the M curve 

 the corresponding rays become horizontal and there- 

 after reverse the sign of dh/dx. In the case of Figure 

 3 these reversals combine with reflections from the 

 ground to make a family of rays oscillate between 

 an upper limit, different for each ray, and the ground. 

 The limiting angle of emergence beyond which re- 

 versal no longer occurs is designated by (2 or 2') 

 in Figures 3 and 4. The duct is the vertical interval 

 cut out by the intersection of the vertical line desig- 

 nated by 2 with the M curve or with the ground. 

 The terms trapping, superrefradion, or guided pro- 

 pagation are often employed to describe these phe- 

 nomena. 



A word might be said here about the substandard 

 case which, although much less frequent than the 

 duct, is of operational significance. It is readily seen 

 that in this case the rays undergo a strong upward 

 curvature in the layer in which there is a substandard 

 slope of the M curve. As a result of this the apparent 

 horizon distance is reduced, and the ranges of radar 

 and radio equipment for targets or receivers near 

 the ground are greatly diminished. M curves of the 



OR SEA LEVEL M, M 



Figure 2. Rays in the standard atmosphere. 



3, and 4. These figures are plane earth diagrams in 

 which the ordinary downward curvature of the earth 

 has been eliminated and replaced by an additional 



substandard type occur often when fog is present but 

 are not uniquely correlated with fog. 



In order to compute coverage diagrams on this 



