150 TECHNICAL SURVEY 
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 aistribution 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 simple surface trapping (discussed on page 
145) , the following formula gives the approximate 
maximum value of the wavelength for which guided 
propagation inside the duct can still take place: 
Amax = 2.5d VAM - 10°. 
Here d is the height of the top of the duct above the 
ground in the same units as \z,x, and AM is the 
decrease in M inside the duct. This relationship is 
represented in Figure 25 where, it should be noted, 
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 
h HEIGHT—e— 
‘—o— 
M CURVE 1ST 2ND 3RD 
MODE MODE MODE 
FIELD STRENGTH —= 
Figure 26. Vertical distribution of field strength for 
first three modes in a duct. 
approximately 38 of a cycle of an approximate sine 
wave, followed by an exponential decrease. Higher 
modes have multiples of half cycles added to ‘the 
sinusoidal part. 
How these modes must be combined to give the 
total field strength and its vertical distribution is a 
question which depends on the height of transmitter, 
the distance out to the point where the total field 
strength is to be obtained, the rate of attenuation of 
each mode as a function of the distance, and its 
phase velocity. Since the attenuation and the phase 
velocity are different for the various modes, the 
vertical distribution of the total field changes with 
the distance from the transmitter, and the number 
of modes composing the total field decreases with 
increasing distance. 
Reflection from an Elevated Layer 
This phenomenon has been studied extensively at 
San Diego. The meteorological situation there is 
rather unique in that the warm and extremely dry 
upper air overlies a cooler and very moist lower 
stratum. The transition between the two layers is 
very sharp. This gives rise to an elevated duct of 
the type exhibited by the M curves of Figures 24D 
and 24K. Often the reversal of the M curve takes 
place over an even narrower interval of height than 
shown in these graphs. In such cases there is a 
reflection analogous to the reflection of waves at a 
true discontinuity between two media and which 
cannot be accounted for by the bending of rays. 
At an interface between two media of different 
refractive indices there is partial reflection of radia- 
tion for any angle of incidence, but when the 
phenomenon (partial reflection and partial trans- 
mission) takes place in a layer of finite thickness, 
