Chapter ]5 
CHARACTERISTIC VALUES FOR THE FIRST MODE 
FOR THE BILINEAR M CURVE* 
i [aan MODEL of an M curve composed of straight- 
line segments suggested itself to workers at the 
Radiation Laboratory early in 1944 as one in which 
phase integral calculations could be carried out very 
rapidly. At about the same time Lt. Comdr. Menzel 
suggested the use of this model together with tables 
of Hankel functions to obtain exact solutions. In the 
fall of 1944 it became evident that phase integral 
methods were not of much use with this model. 
Tables of the required Hankel functions, essentially 
standard height-gain functions, for complex argu- 
ment were prepared at the Harvard Computation 
Laboratory, and considerable effort was directed to 
the obtaining of exact solutions. 
The units, notation, and model are given by the 
following formulas and illustrated in Figure 1. 
oo 
yah) 2mxi0 
Figure 1. Models and units. 
H = (egy = (=) - (BY 
with H (feet) = 7.24 [A(em)]} , 
with L(thousand yds) = 6.69 [A(cm)}} , 
h d @U 
poe on GED te ea 2) = Ws 
The natural units of height and distance represent 
two different compromises between wavelength \ 
and earth radius a, so that \ + H + L + a form, 
very roughly, a geometric progression. It is seen that 
for microwaves, heights and distances occurring in 
practice are fairly small numbers of natural units. 
The M curves are plotted in terms of the height z 
in natural units and of a quantity Y which is simply 
M multiplied by a suitable wavelength dependent 
factor. The standard part of the curve then has 
slope unity. In the bilinear model the anomaly 
consists of a segment with slope s* times standard, 
or, in these diagrams, simply slope s?. For negative s 
Zz2= 
*By W. H. Furry, Radiation Laboratory, MIT. 
168 
there is a duct; s positive but less than 1 gives transi- 
tional cases; and s greater than 1 gives substandard 
cases. 
The essential quantity Y used in calculating the 
field is given by: 
W = (cit-2ni dih—imit ) PVs xt xX 
L 
Veta te Unie) Um(e2) - 
The power density is equal to the free space power 
density multiplied by W?d?. The characteristic values 
are complex: D = B + 7A. For the standard case: 
D, = —1.17 + 2.027. (For } = 10 em this corre- 
sponds to an attenuation of 1.22 db per thousand 
yards.) VW consists of three factors: one, that for a 
plane wave, which can ordinarily be omitted; the 
second, a constant factor which depends on wave- 
length through Z, the natural unit of distan¢e [this 
factor can be replaced by just 2+/x if x?(= d?/L?) 
instead of d? is written in the first line]; and finally 
the critical factor written in terms of natural units 
only and involving characteristic values and charac- 
teristic functions. The imaginary parts of the charac- 
teristic values are the coefficients of horizontal 
attenuation, and the characteristic functions are the 
height-gain functions. 
It is seen that for a typical microwave frequency 
the horizontal attenuation of the first standard mode 
(g = 0) is rather sizable. The plot of the height-gain 
curve shows that if both transmitter and receiver 
are at about 200 ft there is a gain of 50 to 60 db. 
zf 
FIRST STANDARD 
MODE 
FIRST MODE 
TRAPPED 
(QUALITATIVE) 
SE-1.32 
921.9 
25 20 15 10 5 
> 20 Log,lul 
Oo -5 
Figure 2. First standard and first trapped mode. 
In discussing the behavior of U in relation to the 
Y curve, it is best to plot U or | U| rather than 
decibels. It is also helpful to draw a vertical line at 
the abscissa —B, and this is then usually used as 
the axis in plotting U or | U|. The diagram for a 
trapped mode shows that | U| shows exponential 
decay in the “barrier” region where the Y curve lies 
