Chapter 23 



APPROXIMATE ANALYSIS OF GUIDED PROPAGATION 

 IN A NONHOMOGENEOUS ATMOSPHERE* 



THE MILITARY IMPORTANCE of guided Or 

 "anomalous" propagation in a stratified atmos- 

 phere is now well known. Unfortunately, or perhaps 

 fortunately, the problem cannot be treated with the 

 aid of known and tabulated functions except in some 

 special cases because the exact field distribution with 

 height is a function of a function, namely a function 

 of the distribution of the modified index of refraction. 

 For each distribution of this index with height we 

 should have a curve for the field distribution. These 

 curves will look similar in a general way and yet 

 they will differ in detail; but in this particular 

 problem we are not much concerned with details. 

 Even if we had exact solutions we should still want 

 some generalized way of expressing pertinent infor- 

 mation. 



An approximate analysis of field distribution in 

 terms of master curves, depending on one, or at 

 most, two parameters, will be discussed. For example, 

 if we have atmospheric conditions favoring forma- 

 tion of a guiding layer immediately above the ground 

 or sea level, then we can try to represent the field 

 distribution with height with the aid of the master 

 curve shown in Figure 1 . This curve depends on only 

 one parameter, H, so chosen that in the layer between 

 Q/S) H and H, the field intensity does not deviate by 

 more than 6 db from the maximum. 



0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 

 h/H 



Figure 1. Master curve for field distribution with 

 height inside a duct. 



This particular curve is chosen for the first trans- 

 mission mode, and it has been suggested by the exact 



"By S. A. Schelkunoff, Bell Telephone Laboratories. 



2 



l 

 3 



analysis of guided waves in a homogeneous layer. 

 In this case of sharp discontinuity in the index of 

 refraction the field distribution curves are sinusoidal 

 in the layer and exponential outside. The position 

 of the maximum of the sinusoidal portion of the 

 curve and the relative rate of decay of the exponen- 

 tial part depend on the ratio of the wavelength to 

 the thickness of the layer and on the amount of 

 discontinuity in the index of refraction. In Figure 2, 

 curve 1 is identical with the curve in Figure 1 ; curve 

 2 shows what happens if the wavelength is doubled; 



1.0 



0.8 



0.6 



E 

 0.4 



0.2 





 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 



h/H 



Figure 2. Master curves for wavelength X (1), 2X (2), ' 

 and Y 2 X (3). 



and curve 3 corresponds to the case in which the 

 wavelength is halved. If the wavelength is (3tt\/2) /4= 

 3.3 times as large as the wavelength corresponding 

 to curve 1 or larger, no guided waves are possible 

 with the field intensity vanishing at the ground or 

 sea level. 



The situation is different if the index of refraction 

 is allowed to vary continuously and to diminish 

 indefinitely. Suppose, for instance, that the lapse 

 rate of the index of refraction is constant. We don't 

 expect any critical wavelength in this case; as the 

 wavelength increases we expect the field to spread 

 out more and more. In fact, we expect the shape of 

 the field distribution curve to remain the same, 

 namely to be determined by that solution of 



d-E 

 dh 2 



= W 2 



o)-/j.e 



(h)) E 



(1) 



which vanishes at h = 0. In this equation 



244 



