Chapter 16 
INCIPIENT LEAKAGE IN A SURFACE DUCT 
CALCULATIONS FOR THE FIRST 
MODE OF THE BILINEAR MODEL 
RECENT INTERCHANGE of ideas on problems of 
mutual interest with members of the wave 
propagation group of the Radiation Laboratory 
prompted the author to investigate the variation of 
the attenuation constant (or space decrement) «(h) 
of the first mode with the duct height h and negative 
index gradient a of a surface duct (see Figure il). 
h=DUCT 
HEIGHT 
—©Y (X) REFRACTIVE 
INDEX ANOMALY 
—> of (h)/& (0) 
4 FIRST APPROX 
© SECOND APPROX 
1 2 3 4 5 6 7 
—=8 =h(i’b)'/* 
Figure 1. Variation of the attenuation constant with 
duct height. 
The attenuation constant is defined as the constant 
a which occurs in the factor as: (1/+/d) e-% , 
giving the variation of the amplitude with range d. 
The results are shown in Figure 1. In this figure 
the attenuation constant a(h) is expressed in terms 
of the attenuation constant for zero duct height 
a(0). In Figure 1 the curve for 6 < 1 was computed 
from a formula developed by Freehafer and Furry 
of the Radiation Laboratory: 
where 6 = h(k?b) 
Here h = duct height; 
a = —dM/dz inside the duct; 
b = dM/dz above the duct; 
k = 2r/X. 
It was felt that this equation could be used up to 
6 equal to about 1.3 but not beyond this value. 
The curves on the right for 6 > 2, for which a 
condition of nearly complete trapping is approached, 
were obtained as follows. The secular equation for 
the proper value of A(a ~ I,,A), is 
HY(p) _HP(s)H_9(q) + HP)HR@) _ 
Hp) HP)HY@ + HYOHYO 
where 
2k 2k a 
== AG ——— i = =-— 
q=5,\,P 3, (A + ahi s 2h Y= 7 (3) 
is transformed by the substitution 
i 
g=e"ap=@+pip=a(Z) a) 
into fp) — F(a) = 0, (5) 
with 
_ H2%(p) 
f(p) = HY (p) 
U(y2) Viz) + Vive) Ue) @ 
e*/? V(yx) U(x) — U(y2) V(2) 
k@) = 
U(x) = k(x) +e 14), 
V(x) = 1,(2) + e*/81_4(2) . (7) 
Assume now that 
p=ptaA, (8) 
where pp is a constant, which is to be chosen in such 
a manner that A is small in comparison to po in the 
region under consideration. Expanding equation (5) 
in a power series in A, one obtains as a first approxi- 
mation for A: 
es F (ao) — f(po) 
7. J (Po) @) 
and for a second approximation 
dx 
F(a) = 
a Ai f(po) =“ 
bs = Ay) 1 = See 4 
oh = BEE ease] ao 
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