Chapter 22 
FIRST ORDER ESTIMATION OF RADAR RANGES 
OVER THE OPEN OCEAN" 
1 ese MOST STRIKING nonstandard propagation 
conditions are for the most part associated with 
meteorological conditions which can exist only over 
those portions of the sea which are contiguous to 
extensive land masses. At large distances from the 
coasts, however, low ducts exist which, though they 
never produce strongly locked modes at the usual 
radar frequencies, nevertheless modify radar ranges. 
The problem of the low duct has the great advantage 
that conditions are sufficiently near standard that 
numerical solutions can be found in convenient form 
by an extension of the perturbation methods of wave 
mechanics. 
At appreciable distances from land the temperature 
of the air is essentially that of the sea, and the air 
is in neutral equilibrium. Montgomery has pointed 
out that under these conditions there is much 
evidence to support a logarithmic distribution of 
specific humidity. 
The logarithmic distribution of water vapor leads 
to an M curve given by 
2 on (22 =m 2 
where d is the duct thickness, z is the height coordi- 
nate, and a is the radius of the earth. If we plot the 
function in the brackets, we obtain the dashed 
curve of Figure 1. 
This type of M distribution is inconvenient because 
(a) the logarithmic term which represents the modi- 
fication does not approach zero as the height increases 
as a modification term should; and (b) In(z/d) 
becomes infinite when z = 0. Accordingly it is pro- 
posed to replace the function in the brackets by the 
first two terms of its series expansion about the 
minimum. This amounts to substituting for the 
logarithmic curve a parabolic curve which has the 
same minimum point and the same radius of curvature 
at the minimum point as the original distribution. 
At twice the duct height the parabola has a standard 
slope, and it is continued from that point upward as 
a straight line of this slope (AB in Figure 1). 
The modification term is now represented entirely 
by the departure of the parabola from the line 
M—M, 
AB, ie., 
—_ mu, = 21092 Dea z 
M—M,= 109811 +3(3 1)'], 0<5<2 
yp eee z 
as lg = 7 UO ip SF: 
191 
When the duct is low, the modes leak and are not 
far different from the standard ones. Thus it seems 
Zapazs 
a Png 
Figure 1. Schematic M curve for ground-based duct. 
reasonable to employ the well-known methods of 
perturbation theory for calculating the characteristic 
values and functions of the parabolic atmosphere in 
terms of departures from standard. 
If we brush aside mathematical questions of a 
delicate nature, it is possible to obtain an approxi- 
mation for the characteristic values which leads to 
the following expression for the fractional change in 
the attenuation constant (i.e., the real part of ym) 
Re (Fm) — Re (Ym) 
Re (Ym) 
5 
-f (¢ — 6) Im [h2? (¢ + en)] at 
5 [ha’ (€m)]? Im (em) 315 
Here, Re and Im designate the real and imaginary 
parts, and 
2d 
jas, 
L 
L is an abbreviation for (ad?/6z?)} and is equal to 
33 ft for \ = 10 cm, 
Ym is the characteristic value for the standard case, 
‘Ym is the characteristic value for the parabolic case, 
5 ON? 2 
ha(z) is () AH,0(5#), 
H,® is the Hankel function of second kind, order 
2 
14, of the argument Ge), 
€m’s are roots of he(f) = 0. 
The expression above has been evaluated for the 
