Chapter 17 
THE SOLUTION OF THE PROPAGATION EQUATION 
IN TERMS OF HANKEL FUNCTIONS* 
ce CALCULATION Of the field strength in the atmos- 
phere depends upon finding a solution of a wave 
equation incorporating the propagation properties of 
the atmosphere and satisfying the boundary condi- 
tions at the surface of the earth and for large heights. 
This chapter shows how the wave equation, for cer- 
tain specified conditions, may be solved in terms of 
Hankel functions. 
Let z = height of receiver above earth’s surface, 
h, = height of transmitter, 
d = great circle distance between source and 
recelver, 
\ = wavelength, k = 27/n, 
f = frequency, w = 2rf, 
M = modified index of refraction, 
a = radius of the earth 
Under the simplifying assumptions of horizontal 
stratification, slight variation of refractive index in 
a wavelength, smooth earth’s surface, the plane earth 
representation, and the use of the simplified boun- 
dary condition Y = 0 for z = 0, which eliminates 
the polarization of the source, the field of. a (dipole) 
source is described by a scalar wave equation: 
AW + k? My = 0, (1) 
plus appropriate boundary conditions. Separation of 
equation (1) in cylindrical coordinates leads to the 
formal expansion for the field of a dipole source: 
V = eit — in H® (kd cos an) Un (2) U,(hi) , (2) 
where Re (cosa,) > 0 
Here the characteristic values sin’2, and the 
(normalized) characteristic functions U,(z) satisfy 
the equation 
= Kaine 6 WA W =O, (3) 
plus the boundary and normalization conditions: 
U(0) =U (3a) 
eiwtU(z) represents an outgoing wave for large 
positive z . : (3b) 
lim i, * U'de = 1. 
7 (k)—> 0 J, 
Usually sin’a is small, kd is large, and one has 
| H® (kd cos an) 
(3c) 
mw 
D Hy 
a (ea cos =) 
9) 4 
dk cos &, 
eni(kd cos an) 
2 
en tka. — (1/2) sin’ an) 
IIe 
(4) 
176 
The exponential decay factor of the horizontal waves 
thus has the form exp [( —kd/2) In (sin? an)] , 
and the sin2a, values evidently lie in the upper half 
of the complex plane. 
The problem is then to find the characteristic 
values and characteristic functions of the system (38) 
for a given dependence of modified index of refraction 
upon height. For a ground-based duct of height h 
with an M curve made up of two line segments, the 
upper having standard slope, equation (3) becomes 
TO +RUAtYAIU=0,  @) 
y(z) = 2ai(fz —h) ,0O<2z2<h, 
y(z) = 2ax(z — h) ,2zZ2h, 
A = sin? a@ + 2ach , 
where 
a= >. 
4 
ga 
The linear change of variable 
hm NE 
mn = (5 a) [A + 2a:(z — h)] 
inside the duct, and 
4 
Le = (5 a) [A + 2a2(z — h)] 
above the duct reduces equation (3’) to 
2U 
Ge + zU = 0 ) 
whose general solution is 
ii = cee + Byho(2x1) 
Achi(x2) + Boho(x2) . 
The “‘h,’”’ functions are expressible in terms of Hankel 
functions of order 4: 
ne) < (2) ot HY 2 ae 
y\%) = 3 v + 3 wT, 9 as 1,2) 9 (5) 
Condition (8b) is satisfied by setting A, = 0. A; and 
By are determined by the requirement of continuity 
so = nde ( (a) a el (29 
@) ng | (zis) i] be Lan) sf (6) 
Pode) (Cy PG) 
[(ze) | [(e) 4] 
8By Lt. W. F. Eberlein, USNR, Office of the Chief of Naval 
Operations. 
(3”) 
fs 
