Chapter 20 
SOME THEORETICAL RESULTS ON NONSTANDARD PROPAGATION" 
PROPAGATION IN TILK OCEANIC 
SURFACE DUCT 
IK ANALYSIS SECTION of Columbia University 
Wave Propagation Group undertook a theoretical 
study of propagation in case of surface ducts, which 
have recently been reported to be of common oc- 
currence in oceanic areas. The M curve chosen was 
M(h) = 346.4 + 0.036h + 43e-%” (1) 
where the height h is expressed in feet. This curve 
has an M deficit of 43 units and a duct height of 48 
ft and is considered to be representative of condi- 
tions prevailing around Saipan when the wind is of 
the order of 10 to 20 mph. 
The analysis was based on the phase integral 
method. The standard W.K.B. (Wentzel-Kramers- 
Brillouin) version of the asymptotic solutions of the 
wave equation’ had to be extended in two ways. 
One was in the adoption of Langer’s form of the 
asymptotic solutions, 449 which enables one to bridge 
the “gaps” around the turning points. The other, 
and more important, development was in the exten- 
sion of Langer’s method to handle a case with two 
_ turning points. This was accomplished by joining 
the solutions from cach turning point at the duct 
height. The resulting solution agrees with Gamow’s 
for completely trapped modes but deviates from it 
when leakage begins. lor leaky modes the standard 
Langer solution is adequate. 
Coverage diagrams were computed for the S and 
X bands and for transmitter heights of 16 and 46 ft. 
In case of the § band, it was found that the first 
mode was nearly trapped, while the second mode 
was considerably leaky with a decrement of about 
3 db per nautical mile. The two modes were com- 
bined, and coverage diagrams were computed over 
ranges and heights such that the second mode 
contributed no more than 25 per cent to the total 
field. 
In the case of the X band, it was found that the 
first two modes were completely trapped, the third 
mode nearly trapped, while the fourth mode was 
leaky with a decrement of over 3 db per nautical 
mile. In computing the coverage diagrams for the X 
band, the four modes were combined over such 
ranges and. heights that the fourth mode did not 
contribute more than 25 per cent to the total field. 
“By C. L. Pekeris, Columbia University Wave Propagation 
Group, Analysis Section. 
183 
CHARACTERISTIC VALUES FOR A 
CONTINUOUSLY VARYING MODIFIED 
INDEX 
In the theoretical treatment of nonstandard prop- 
agation by the method of normal modes, one is 
confronted with the task of solving the differential 
equation for the height-gain function U(h) given by 
equation (2), which, it will be noted, is identical with 
equation (8) in Chapter 21. 
Um(h) ae k? [y(h) ar An] Un(h) = 0, 
Qn 
k = a0 , 
by asymptotic methods, the characteristic value 
A, is determined, to a first approximation, by the 
condition that 
1 
i) ®) 
(4) 
In order to solve equation (3) one has to find a 
value hi, which is generally complex, such that when 
y(h) is substituted in the radicand and the integral 
( (2) 
.y(h) = 2X 10°°M(h) , 
hi 
hk] Vy(h) = y(hi) dh =n =o (m - 
0 
Am = —y(hi) . 
hy 
Val) = y(hi) dh = F(hi) (5) 
evaluated, the result should be purely real, and equal 
tO Um/k. In ease of a surface duct, /(hy) is real and is 
a continuously increasing function of its argument 
for real values of h; ranging from zero up to the 
duct height hy. In order for #'(hi) to increase beyond 
the value F(h,) and still to remain real it is found 
that h; must be complex; i.e., the path in the complex 
hy-plane along which F(h) is real consists of the 
portion of real axis 0 Sh, Sh, followed by a curve 
in the fourth quadrant. 
In case of substandard refraction, F(hi) is real 
only for complex values of fi, and the method of 
solving equation (3) to be explained presently is 
particularly helpful in this case. 
Let 
y(h) = y(O) + bik + beh? + +++ +bh™ + ---, 
—2b2 
be? 
ii AY (—6bsbi1 + 12627) 
ie 15 ‘ 
_ (120b:bsbs — 24b1%4 — 120bs") 
h Bit 5 
etc. 
