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FIELD IN ARBITR. UNITS. ARROWS SHOW FREE SPACE FIELD 



Figure 6. Variation of field strength with height, for various distances. 



10 represents a change of one part in 10 5 in the re- 

 fractive index. Now from equation (5), we have 

 by differentiation 



AM • 10 6 = aAa 



(9) 



For a complete reversal of a ray we must have 

 Aa ^ a, and then a is proportional to the square root 

 of AM. In the above case, where AM = 10, we find 

 that a is of the order of 3 ■ 10~ 3 , or about 10 minutes 

 of arc. 



Carrying considerations of this type into a little 

 more detail it is found that the major effects of 

 nonstandard refraction occur only for rays which 

 emerge from the transmitter at an angle of less than 

 3^2 degree. For angles between Yi and 1H degrees 

 the refractive effects produced by the typical non- 

 standard M curves consist merely in minor modifica- 

 tions of the standard coverage pattern, while for 

 angles above \ x /2 degrees the refractive effects are 

 negligible. 



66 SURVEY OF WAVEGUIDE THEORY 



The ray tracing method presented in Section 6.4 

 is only a rather rough approximation to the true 

 solution of the wave equation. It neglects diffraction, 

 which on closer investigation is found to be very 

 important. In order to visualize this, the waveguide 

 analogue was introduced at an early stage of the 

 development. Consider a two-dimensional wave- 



guide consisting, for instance, of two parallel plane 

 sheets of copper of infinite extent. The propagation 

 of an electromagnetic wave in such a guide is some- 

 what analogous to that in a duct. The reversal 

 of the vertical component of the rays by refraction 

 in the duct corresponds to the reflection by the walls 

 in the case of a metallic waveguide. It is well known 

 that wave propagation under these conditions can 

 be described by the methods of geometrical optics 

 only to a very rough approximation. Soon after the 

 discovery of ducts the accurate theoretical treatment 

 of duct propagation was initiated in England. 67 -7o.7i, 

 73,88,94 The g enera i reS ult of these investigations 

 may be summarized as follows. For an atmosphere of 

 arbitrary stratification the field can be formally ex- 

 pressed by the series development, equation (27) of 

 Chapter 5. The constants appearing therein and 

 the height-gain functions involved are, however, 

 different from the standard case and depend on the 

 particular M curve involved. The solution, therefore, 

 consists again of a superposition of "modes" which 

 decay exponentially with distance from the trans- 

 mitter. The height-gain functions do not, in general, 

 increase with altitude all the way up from the ground. 

 In the case of a duct the height-gain functions of 

 the lowest modes have a pronounced maximum in 

 the duct, similar to the curves for the overall field 

 strength shown in Figure 6. This maximum becomes 

 flatter and eventually disappears entirely for the 

 height-gain functions of the higher modes. 



It is useful to supplement the rather complex 



