182 TECHNICAL SURVEY 
occupies a “mean” position among other curves of 
this type. 
There are two methods for selecting H. In one 
method H is defined as that value of h for which the 
coefficient 8? — wue(h) in equation (1) vanishes. 
This value of h separates the region in which the 
solution of equation (1) is “more” or less sinusoi- 
dal” from the region in which the solution is “more 
or less exponential.”” This definition leads to one 
equation connecting H and B. Next, the stratified 
region 0 < h < H is replaced by a homogeneous 
region in which the dielectric constant is equal 
to the average value of ¢«(h) in the interval (0,h). 
If we impose the requirement that curve 1 repre- 
sents the exact field distribution under the new 
conditions, we obtain the second equation for H and 
B. Eliminating B and expressing the result in sym- 
bols approved by the wave propagation committee, 
we have 
H 
nfm dhe He M(H) = = <— ». (2) 
0 
If the lapse rate'of M is constant, this mia gives 
H = 65x (- een (3) 
If M(h) is proportional to h?, then 
mei 
H = 18 "| : (4) 
If the lapse rate of M is constant, the exact solution 
may be expressed in terms of Bessel functions. 
Figure 3 shows the exact and approximate solutions. 
The second method is based on the fact that the 
solutions of equation (1) minimize and reduce to 
zero the following function: 
see Sal 
+[(F (Fi) aD: (5) 
In deriving this equation we should remember that 
~~ E? dh 
(e) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 
Ey = (const) Ut [J4(U) + J_3 (U)] 
843 
U=*/, { i= 3a, } 
Ficure 3. (1) Exact solution normalized to have min- 
imum value of unity. (2) Approximation. 
Ey =sin p p< = 
Ey — Alesse? p> & 
we are concerned with solutions which vanish at 
h = Oandh = wo. Hence, if we wish to approximate 
this solution by a function of one parameter H, we 
eliminate H from the following two equations 
I=0,=5 =0. (6) 
If, for instance, we wish to approximate the field 
distribution by the master curve in Figure 1, we solve 
, oP _ 9 X 105 
aH L28 if 
v, (7) 
where 
H 
., omh 
es 2 oe 
i, M sin Z Ho 
—3ah 
— 55.5 M —— dh. 
iL el (8) 
By this variational method the numerical coefficient 
in equation (8) is found to be 64 rather than 65. 
The great advantage of the variational method 
lies in the fact that, if we wish, we can increase the 
number of parameters in the approximating func- 
tion. For example, we can assume 
E(h) in(F),0<h<H 
sin aexp| -v "a", >H, (9) 
without specifying that 9 = y = 37/4 as we did 
in obtaining the curve in Figure 1. We should then 
calculate H, 6, and y from 
ol ol ; 
I=0, Fy as = 0, —— ay =). (10) 
However, aside from the labor of solving these 
equations and having to deal with more complicated 
results, we shall lose the advantage inherent in a 
description of the field in terms of only one easily 
understood parameter. The most we could hope for 
from an analysis of these equations is a somewhat 
better choice of the master curve for the type of 
atmospheric conditions which are the most likely 
to occur. 
The obvious general conclusion from equations 
(7) and (8) is this: if M(h) is multiplied by a con- 
stant factor, the effect on H is the same as that 
obtained if we divide \ by the square root of this 
factor. If M is proportional to h”, then H is propor- 
tional to \”“*”), Since the gain of the guided wave 
over a free space wave is proportional to \p/H?, 
where p is the distance from the transmitter, the 
gain is independent of the wavelength when M(h) 
is proportional to h?. For a uniform lapse rate the 
gain varies inversely as one-third power of the 
wavelength. 
