16 TECHNICAL SURVEY 
only to a very rough approximation. Soon after the 
discovery of ducts the accurate theoretical treatment 
of duct propagation was initiated in England. 677% 7, 
73,88,94 The general result 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 1. 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 
detay 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 
mathematical development into modes, represented 
by equation (27) of Chapter 1, 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 = Eye est2+) | (10) 
Confining ourselves for the moment to the case of 
the plane earth, it is found from electromagnetic 
theory that 
2 
ke +22 = (=) (11) 
where 7 is the refractive index in the volume element 
considered, and 2 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 
b= cose, 
j= smal (12) 
r 
where @ 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 kK = 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 thé wave 
equation there is some electromagnetic field strength 
everywhere. Consider in particular the region just 
above a duct. There are regions of ‘‘shadaw’’ 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 = 0 and hence J =0. If we 
proceed farther upward in a duct n decreases, and 
it follows from equation (11) that if n decreases 
sufficiently 7 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 beginto increase 
again as we go still farther upwards from the duct 
and there comes a height where / 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 1, 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. 
Tt 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 
