atmospheric stratification and refraction 



207 



are extended slightly, other times slightly decreased; 

 (c) for a transmitter above the duet no excessive 

 changes in field strength occur below the duct — this 

 can be deduced from (b) by using the reciprocity 

 principle; (d) there is no appreciable superrefraction 

 when the transmitter lies appreciably above the duct. 

 For some time after the discovery of superrefraction 

 it was thought that the concentration of radiative 

 energy in the duct might result in a decrease of the 

 amount of radiation above the duct and hence in a 

 reduction of coverage there. The cases illustrated in 

 Figure 24, at least, are not in accord with this 

 presumption. In spite of the great increase in ranges 

 in the duct the amount of energy trapped is small 

 compared to the total energy of the radiation field. 



17-2-6 \^/a Ve Picture of Guided Propagation 



It must be realized that while ray treatments give 

 accurate results under certain conditions, there are 

 features of the propagation problem which can be 

 satisfactorily discussed only on the basis of the 

 electromagnetic wave equations. As an aid to under- 

 standing the wave treatment the close analogy 

 between the functioning of a duct and a hollow metal 

 waveguide (or dielectric wire) may be used. In both 

 cases the field which is being propagated may be 

 represented as the sum of an infinite number of 

 terms (modes). Each waveguide mode is propagated 

 with a separate phase velocity and an exponential 

 attenuation factor and has a field distribution over 

 the wavefront that is independent of distance in the 

 direction of propagation. 



In a metallic waveguide a finite number of modes 

 are propagated with very small attenuation, while 

 the remaining modes, infinite in number, have 

 attenuations so high that they are, practically speak- 

 ing, not propagated at all. The same division of 

 modes into those that are freely propagated and 

 those that are highly attenuated is found for duct 

 propagation. In the duct, however, the difference 

 between the two types of modes is less pronounced 

 than in a hollow metal tube. 



As the frequency is decreased, the number of 

 transmission modes decreases both for the hollow 

 metal tube and the duct until the cutoff frequency 

 is reached, below which neither serves as a wave- 

 guide. For the case of simple surface trapping (Section 

 17.2.3) the following formula gives the approximate 

 maximum value of the wavelength for which guided 

 propagation inside the duct can still take place: 



X„ mx = 2.5d \/AM • lO" 6 . 



Here d is the height of the top of the duct above the 

 ground in the same units as X max , and AM is the 

 decrease in M inside the duct. This relationship is 

 represented in Figure 25 where, it should be noted, 



50 100 500 1000 



d IN FEET 



Figure 25. Maximum wavelength trapped in simple 

 surface trapping. Duct width d in feet. AM is total 

 decrease of M in duct. 



the duct width is given in feet and the wavelength 

 in centimeters. When the wavelength exceeds the 

 critical value obtained from this graph, guided 

 propagation is no longer to be expected. M curves 

 of different shapes will require slightly different 

 numerical factors in the formula. 



The main difference between the modes is found 

 in the vertical distribution of field strength. The first 

 three modes for a simple ground-based duct are 

 illustrated in Figure 26. The lowest mode has 



M CURVE 



FIELD STRENGTH — — 



Figure 26. Vertical distribution of field strength for 

 first three modes in a duct. 



