Chapter 14 
THEORETICAL TREATMENT OF NONSTANDARD 
PROPAGATION IN THE DIFFRACTION ZONE* 
hae 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 VW, the Hertzian vector being (0,0,¥). Then, 
at a point in the diffraction region, 
actual field strength = |W |?-d?-H), (1) 
with d = horizontal distance from source, 
E, = free space field at distance d. 
An expression for Y can then be found in the form 
(dz) = J em Una) Ute), @) 
™@ 
where h,; = transmitter height; 
z = height at which W is calculated; 
2r 
=F, bsa=. 
w af, d 
ym and U,, are characteristic values and functions 
of the boundary value problem 
oo +eM2) +r1U=0, — @) 
Vet wave moving upward, z—> wo, (4) 
U(0) = 0. (5) 
The modified index of refraction M is supposed to 
be defined without the factor 10° usually included 
The functions U must be normalized in a suitable 
where 
@By W. H. Furry, Radiation Laboratory, MIT. 
166 
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 V. The 
actual value for W, for the diffraction zone and 
microwave frequencies, would not be affected sig- 
nificantly. 
The quantities y, are complex: 
Ym = Om + 1Bm - (6) 
Om and Bm are positive real quantities. It is convenient 
to think of the terms of the series as arranged in 
order of increasing a: 
a1 < ae < a3 < ag: 
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 YW. The quantities 6, are all very nearly . 
equal to k. The slight differences between the £,,’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 
W 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- 
tions, which satisfy the differential equation and the 
boundary conditions but still require multiplication 
by suitable normalization factors. 
3. The normalization factors. 
There are three methods of attack on the problem: 
1. Numerical integration of the differential equa- 
tion, accomplished in practice by the use of a 
differential analyser. 
2. Phase integral methods. 
3. Use of known functions and tables, for suitably 
chosen M curves. 
