INCIPIENT LEAKAGE IN A SURFACE DUCT 175 
and 3 that, in contrast to the first mode, these 
branches of the curves for the second and third 
modes do not join on smoothly to the other branches 
which start with standard yalues- at g = 0 and 
approach limiting values for large g 
This duplicity of the solutions, which was doubted 
at first, was substantiated in two ways. The values 
of A, and B, at g = 3 and g = 5 in Figure 1 were 
computed at both branches with increasing accuracy 
(up to 10-5), and it was found that the matching of 
the solutions at the duct height and the degree of 
vanishing of the height-gain function of the ground 
improved correspondingly in both branches. This 
proves that both solutions satisfy the boundary 
conditions. As a second step in testing the reality of 
the limiting points, an’ asymptotic expression was 
derived for the limiting values, and the values com- 
puted therefrom were found to be in fair agreement 
with the. exact values, as is shown in Table 1. 
The physical nature of the duplicity of solutions 
seems to be as follows. The solutions approaching a 
limiting characteristic value for large duct height g 
correspond to the case where the ground sinks to 
great, depths; the other solution corresponds, of 
course, to the limiting case when the height of joint 
rises to infinity. 
The relative importance of the two types of solu- 
tion will depend on the ranges and heights considered. 
At sufficiently great ranges the solutions with the 
smaller value of A,, will predominate, but the greater 
hh 
TNT 
a 
LS) 
‘ 
= Soa a Seas 
1 > Ee ESE Ea 
| Ee Fs Ln ee Ee ee De 
DE soe OE a a a 
SSS Se 
SS s‘= — 2. z = height above 
ground in natural units. 
g = height of joint in natural 
units. 
= normalized wave function. 
oF] 
(°) L 2 3 4 L) 6 
z/9—=— 
Fieure 4. Height-gain functions of the second mode 
for a bilinear M curve. 
——_+ — ——= —— — 
SSS | sees 
a Ss ‘aa ae 
Ba ae a ject || 
a ele Wane 
-—+—+ -— ———=<=s 
SSS as SSS 55 
fas es Se I 
E—— 9* 5 — f= 2) —— 
I aS ss", Oe Tea oe ee ee 
as SSS eA ee, es 
til | a ee, a De Me 
9 Se 
—— i 9S S46 SS SS SS SSS S55 
es 1S eS Eee ES eee eee ee 
[/-—_|- Ff 
{ = SSS SS SS SSS SSS Se 
J VAS aS SS SS SH a Se 
= SS GS Gee ae Ge oe Gee) MD 
A] =n i 7 [ee | en [ae | | ae | ne | ee 
A a a 
py ff 
P—t — ————— 
SSS SSeS 
a a er Es Be es 
fen ies SS en ee ee 
0. ps a a ef LL 
s= —l.z= height above 
ground in natural units. 
g = height of joint in natural 
units. 
a U2 (z) = uormalized wave function. 
O85 1 2 3 4 5 6 
fa 
Ficure 5. Height-gain functions of the second mode for 
a bilinear M curve. 
the height considered the farther must one recede 
from the source before the initial advantage of the 
limiting solution due to a greater -height gain is 
overcome by the stronger horizontal attenuation. 
The greater height gain of the limiting solutions at 
high elevations is illustrated in Figures 4 and 5. In 
these figures, the height-gain functions for the limit- 
ing solutions are drawn in solid lines, those for the 
Gamow solutions in dashed lines; and the unit of 
height is the duct height. It should also be pointed 
out that the normalization condition applied was 
i U2(2)dz = 9; (13) 
so that, if a comparison of height-gain functions of 
solutions of the same class for different values of g 
is desired, the plotted values should be divided 
by V9. 
Taxie 1. Comparison of exact limiting values of D with 
values obtained from the asymptotic formula.* 
s Second mode Third mode 
—1 —0.60 + 2.80i —1.06 + 3.60i Asymptotic 
we —0.59 + 2.837 Exact 
-1/2 —0.78 + 2.744 1.22 + 3.407 Asymptotic 
—0.70 + 2.702 —1.42 + 4.082 Exact 
—2 —1.00 + 2.60: —1.36 + 3.487 Asymptotic 
—0.80 + 2.447 Exact 
*exp (- : >) - ( = =) /8Di =0. 
