EQUATION IN TERMS OF HANKEL FUNCTIONS 177 
The characteristic values A then appear as the roots 
of the equation 
U(0) = Aiki [(#) (A — 2ash) 
+ Bihe [ () (A = 2ash) | => 0 . (7) 
The constant factor Bz; appearing throughout is 
determined by the normalization condition (8c), 
which takes a peculiarly simple form in this: case, 
since (3’”) implies the identity 
Paik dU\? 
ete ae | ®) 
Owing to the present lack of adequate tables of 
the H') or h, functions for complex arguments and 
the complicated nature of equation (7), one is forced 
to employ asymptotic expansions to determine the 
A’s so far as possible. Due care must be exercised in 
dealing with the branches of the multiple-valued 
approximations appearing, as well as with the 
so-called “Stokes phenomenon.” Thus in deciding 
between the two rival asymptotic approximations 
* 4 
H® (W) ~ (=r) e7i(W—Sz12) : 
(9) 
H® (W) ~ (<r) Gave + ef W+llm/12)) 
—— arW ’ 
both formally valid in the common domain 0 < 
arg W <7, one employs the first when arg W < 
1/2, and the second when arg W > 7/2. The ambi- 
guity on a “Stokes line” (arg W = 7/2) apparently 
must be resolved by taking the mean of the two 
expressions when their difference is important, as it 
is when strongly trapped modes exist (a; < 0). 
The nature of the results is shown in the important 
case of complete inversion (a; < 0). For simplicity 
p= —ae= (=) (—2a,h)3 . (10) 
Then mee 
I (@+0i—(1-2)_ = 2— 9) 
2 
Ee (9 + n>§] 
ade pe agra — th 
Wace tee Des ets 5) = 0), 
q 
Zs (11) 
(F< arg p<m — i) 
TT 1 + setreent 4 Semel = 0, 
(arg p =z) 
(n a positive integer) 
The corresponding regions of validity are indicated 
in Figure 1, which also shows the dependence of one 
characteristic value on co for a particular ratio a2/ai. 
If one numbers the characteristic values p, in order 
of increasing imaginary part, those for which n is 
greater than some integer are defined by equation 
I. There is an infinite number of these “‘Eckersley”’ 
EREEH 
a 
TRANSITION 
7 60°) IL 
o /—LWSTOKES LINE _ 
=1.0 -0.5 to) 0.5 
Fieuks 1. Diagram showing the locus of p in the com- 
plex plane of o varies. 
or leaky” modes. Equation II joins on smoothly to 
I and defines a finite number of “transitional” or 
“Semileaky”’ modes. 
Equation III defines the strongly trapped or 
Gamow modes for which 0 < In(p) << — Re(p) < 
1. In this case the characteristic values lie almost 
upon a Stokes line (arg p = 7 — 4). The approxi- 
mations valid above the line yield II; the ones valid 
below yield 
IV 1 + ien2iove + ni _Q , or 
(p+ 1)i=(n—*#) ~<1 W=0,1,2- - -). 
Thus in tne Gamow case IV makes J,(p) = 0 while 
II yields almost the same real part, but a very small - 
positive imaginary part for p. The mean approxima- 
tions effectively yield III whose roots are essentially 
the mean of the roots of II and IV. This averaging 
process checks closely with an exact calculation based 
on the (unpublished) WPA tables of Bessel functions 
of order 14 for real and pure imaginary arguments. 
It is to be noted also that IV determines the number 
of strongly trapped modes as the largest integer n 
such that (n — 4) r/o < 1. 
The characteristic value problem may be regarded 
as essentially solved in the cases of leaky modes (1) 
and strongly trapped modes (III). In doubt still is 
the question of the transition between III and II; 
this uncertainty plus the fact that actual determina- 
tion of the roots of II is much more complicated 
than in the other cases, and that in case II the 
arguments of certain of the Hankel functions lie so 
close to the origin as to make doubtful the validity 
of the asymptotic procedure—all these considera- 
tions indicate that resolution of the transitional mode 
question awaits appearance of adequate tables of 
the Hankel functions for complex arzuments. 
