48 



ELEMENTARY THEORY OF NONSTANDARD PROPAGATION 



mathematical development into modes, represented 

 by equation (27) of Chapter 5, by a simpler type of 

 analysis which connects it with the ray picture. For 

 the sake of simplicity let the phenomena be two- 

 dimensional, confined to the horizontal x direction 

 and the vertical z direction. If the wavelength is 

 small enough compared to the dimensions of the 

 duct, the electromagnetic field at some distance from 

 the transmitter may, in any sufficiently small volume 

 element, be represented by a plane wave whose wave 

 front is perpendicular to the direction of the rays. 

 Such a plane wave may be written as 



E = E B e iv " e - 1(kx+lz) . 



(10) 



Confining ourselves for the moment to the case of 

 the plane earth, it is found from electromagnetic 

 theory that 



fc 2 



V = 



2 



(11) 



where n is the refractive index in the volume element 

 considered, and X is the free space wavelength. Since 

 k and I are proportional to the directional cosines 

 between the direction of the ray and the x and z axes, 

 we may put 



k 



2irn 



cos a 



I 



2irn 



sin a 



(12) 



where a is the angle between the ray, or the normal 

 to the wave, and the horizontal. 



The further mathematical analysis shows that, 

 for a horizontally stratified medium where n is a 

 function of z only, we have k = constant. In view of 

 equation (12) this gives us n cos a = constant, 

 which is just Snell's law for a plane earth, as enunci- 

 ated before. 



The ray picture, being a rough approximation, 

 gives an electromagnetic field in some regions and 

 none in others. In the rigorous solution of the wave 

 equation there is some electromagnetic field strength 

 everywhere. Consider in particular the region just 

 above a duct. There are regions of "shadow" above 

 the duct caused by the fact that some of the rays 

 are bent downward in the duct. Clearly, at the point 

 of reversal of a ray, a = and hence I =0. If we 

 proceed farther upward in a duct n decreases, and 

 it follows from equation (11) that if n decreases 

 sufficiently I must eventually become imaginary. 

 Instead of a wave component in the z direction we 

 then have an electromagnetic field which decreases 



exponentially as we go upwards. In the top layer 

 of a duct, the decay takes place very gradually 

 because the change in refractive index is extremely 

 slow. Eventually, however, n must begin to increase 

 again as we go still farther upwards from the duct 

 and there comes a height where I is again real and 

 an ordinary wave is again possible. This behavior 

 might be likened to that of a metal foil so thin as 

 to be partly transparent for the waves considered. 

 The duct thus may be likened to a waveguide 

 bounded on one side by a solid reflector, the ground, 

 and on the other by a semi-transparent reflector. 

 The mathematical theory of ducts has therefore 

 often been designated as leaky waveguide theory. 



A closer study of the height-gain functions which 

 appear in the mode formula, equation (27) of Chap- 

 ter 5, shows that in the presence of a duct the leak- 

 age across the upper boundary of the latter is the 

 more pronounced the higher the order of the mode, 

 and that for sufficiently high modes there is almost 

 no confinement of the electromagnetic field within 

 the region of the duct. In consequence of this fact 

 the exponential damping with horizontal distance, 

 which is characteristic of each mode, is more pro- 

 nounced for the higher modes, because for these 

 modes the electromagnetic energy rapidly "leaks 

 away" from the duct. At large distances from the 

 transmitter the field in and near the duct is therefore 

 described by the lowest mode alone. This depends, 

 of course, partially on the relative strength of excita- 

 tion as well as on the attenuation of the various 

 modes. 



Another aspect of the wave theory of ducts which 

 is of great practical importance is the cutoff effect. 

 It is well known that any ordinary metallic wave- 

 guide has a cutoff frequency below which the guide 

 cannot transmit an electromagnetic wave. The mathe- 

 matical treatment of the duct shows that there is 

 a similar lower limit of frequency for transmission 

 through a duct, but, because of the "leakage" 

 phenomenon, it is found that there is no sharply 

 defined cutoff frequency but a gradual decrease of 

 the duct's ability to confine radiation within itself 

 with decreasing frequency. Figure 7 is a graph 

 giving representative values for what may be taken 

 as the cutoff frequency of a duct as a function of its 

 height in feet and AM, the decrease of M in the in- 

 version layer. These values are the result of a some- 

 what crude approximation and should not be taken 

 to indicate more than the order of magnitude of the 

 frequency at which this effect occurs. 



