184 
and 
g = ors (Sm\ wy — A, + y(0) (6) 
Qkh) ’ me ’ 
then 
= f 2 4 4 
Am = —y(hi) = —yX0) + 6 + (Fe) (he) 8 
«|| 4 A Shs) 
ar E (he) 105 cis | 
+500 E (ha)® = 3 (ha) (hs) + gst) a 
hy = —hw + au = a qroce , (8) 
where 
h 
h =ST, 
ah 
=> 
hg = & ,eoe 
Isquations (7) and (8) are of the nature of asymp- 
totic formulas; they should be terminated when the 
individual terms begin to increase, and the error in 
Am or A; is then of the order of magnitude of the 
last term retained. 
The following examples in Table 1 illustrate the 
degree of accuracy obtainable from equation (7). 
As a further check, we treated the case a = +20, 
dX = 0.6356, for which Pearcey and Whitehead!** 
give a value A; = —10.21 + 1.07 X 10~%2. Equa- 
tion (7) yields A; = —10.22, while the imaginary 
part obtainable from Gamow’s formula is 1.24 X 
Omesze 
TECHNICAL SURVEY 
It must be emphasized that the value obtained 
from equation (7) should be verified by carrying 
A ee ae NON 
out the integration of F(hy) = [ Vy(h) + Andh. 
While doing so, one may as well compute 
dF _ 
) [ " dh 
© Jo Vy) = yl)’ 
and then obtain a correction to hi by Newton’s 
method. 
The method of solving equation (3) explained 
above has been found especially useful in the treat- 
ment of substandard refraction, and to a lesser extent 
in the treatment of the trapped modes in case of a 
surface duct. In the latter case one can, of course, 
solve for A,, directly by computing F(h,) by numerical 
integration. The method is not applicable for the 
leaky modes in case of a surface duct. 
So far the discussion has centered on the solution 
of equation (3), which in itsclf is only an approximate 
asymptotic formula valid for large values of k. Let 
the value of A, which satisfies equation (3) be 
denoted by A,,); then an improved value for A, 
can be obtained from 
An = Ain?) 
SOM (OV penal L (DP 2B 
eon 6) Ole Gl a |e 
where 
h 
: dh : dy 
= SS = = GIR, (UA 
0 Vy(h) + Aa’) ~ dhy ie 
and the derivatives of y are to be evaluated ath = In. 
hy 
Tas.e 1. Approximate determination of Ai from equation (7) and the verification that [v y(h) + Ai dh = 2.383.* 
hi 
hy from ————— 
a ON Ax from equation (7) “tes a AG [vat + A, dh vy 
—20 0.6356 12.360 + 9.775% 0.3997 — 0.9518% 2.370 + 0.0042 2.383 
—10 0.6356 4.878 + 6.3027 0.4745 — 1.19827 2.397 + 0.010% 2.383 
—5 0.6356 1.499 + 4.257% 0.5691 — 1.46927 2.393 + 0.0097 2.383 
—2 0.6356 —0.764 — 1.787% 2.363 — 0.0087 2.383 
—0.246 + 2.9027 
y(h) =h+ co, 
