Chapter 18 



THEORETICAL TREATMENT OF NONSTANDARD 

 PROPAGATION IN THE DIFFRACTION ZONE a 



The assumptions and restrictions underlying this 

 presentation are: 



1. We concern ourselves with problems of the 

 diffraction region only: the field is calculated at 

 considerable distance from the transmitter and not 

 too great height above the ground. 



2. The plane-earth model is used, in which the 

 effect of curvature is simulated by using the modified 

 index M instead of the index of refraction n. 



3. The earth's surface is assumed smooth, and M 

 depends on height only (horizontal stratification) . 



4. Simplified boundary conditions at the earth's 

 surface are used, appropriate to the treatment of the 

 diffraction zone at microwave frequencies. This 

 results in a formula which refers only to a discrete 

 spectrum of modes and makes the calculations 

 independent of polarization. 



5. The directional pattern of the transmitter need 

 not be considered, since only the intensity at the 

 azimuth in question and within 1 degree of the hori- 

 zontal plane is of importance. The problem solved 

 is that of a vertical dipole, electric or magnetic. 



6. The field is described in terms of a single 

 quantity SP, the Hertzian vector being (0,0,^). Then, 

 at a point in the diffraction region, 



where 



jUe*' 



actual field strength 



¥ I 2 • d 2 • E 



(1) 



with d = horizontal distance from source, 

 E = free space field at distance d. 

 An expression for ^r can then be found in the form 



■*(d,z) = & 



\/52j' 



'» U m (h0 V m {z) , (2) 



where hi = transmitter height; 



z = height at which ^ is calculated; 



„ , 2tt 

 co = 2ttJ, k = — . 



y m and U m are characteristic values and functions 

 of the boundary value problem 



^ + [VM*(z) + t 2 ] U = , (3) 



\U(0) = . 



wave moving upward, z 



,(4) 

 (5) 



The modified index of refraction M is supposed to 

 be defined without the factor 10 6 usually included. 



The functions U must be normalized in a suitable 

 way. If we had not agreed to use simplified boundary 

 conditions, the last equation (5) would be more 

 complicated and would depend on the type of polari- 

 zation. Also an integral would appear in addition 

 to the discrete sum in the expression for ty. The 

 actual value for Sl>, for the diffraction zone and 

 microwave frequencies, would not be affected sig- 

 nificantly. 



The quantities y m are complex: 



7m = a m + iPm • (6) 



a. m and (3 m are positive real quantities. It is convenient 

 to think of the terms of the series as arranged in 

 order of increasing a : 



ai < 



< «3 < 



a By W. H. Furry, Radiation Laboratory, MIT. 



These quantities determine the horizontal attenuations 

 of the various modes. For large d only one or at most 

 a few terms of the series are required to give the 

 value of SF. The quantities m are all very nearly 

 equal to k. The slight differences between the /3 m 's 

 determine the phase relations and hence the inter- 

 ferences between the various modes. 



It is convenient to classify the modes into two 

 types: (1) "Gamow" modes which are strongly 

 trapped, so that a is very small; (2) "Eckersley" 

 modes which are incompletely trapped or untrapped. 

 The names "Gamow" and "Eckersley" refer to the 

 men who devised the approximate phase integral 

 methods which apply in the two sorts of cases. For 

 practical purposes, when working within the diffrac- 

 tion region, we need consider only the Gamow modes, 

 or at most the Gamow modes and the first Eckersley 

 mode. 



In order to be able to use the formula to calculate 

 ^ for a given index curve M(z), we must obtain the 

 following information about the modes which are 

 to be used: 



1. The characteristic values. 



2. "Raw" or unnormalized characteristic func- 



226 



