NONSTANDARD PROPAGATION IN DIFFRACTION ZONE 167 
The method of numerical integration is being used 
intensively in England by Booker, Hartree, and 
others. It encounters considerable difficulties in con- 
nection with the fitting of the boundary condition 
at z— o and also in the determination of normali- 
zation factors. These difficulties have been overcome 
by special and fairly elaborate procedures. In this 
country the feeling has been that we should direct 
our efforts toward the use of the other methods. 
If either method (2) or method (38) is to be readily 
applied to a variety of cases without a prohibitive 
amount of labor, the M curves must have a suitable 
form. The form indicated turns out to be the same 
in both eases. It consists of portions, each of which 
is a straight line. If enough such portions are used, 
any actual M curve can be accurately represented, 
but it is impractical to use more than a very few. 
Present efforts are directed toward dealing with 
cases where there are just two straight-line portions 
and there is no prospect of soing bevond the cases 
with three (Figure 1). 
fe 
M ——e 
Ficure 1. Schematic straight-line M curves. 
At first sight these curves look overly artificial, 
but there are considerations which indicate that 
they are really an altogether reasonable choice. First, 
some actually occurring curves have very much thir 
sort of appearance. Second, the sharp breaks in the 
curves have no really strong effect on the results. 
Third, practical considerations severely limit the 
number of parameters which can be used in specify- 
ing the curve, so that a meticulous reproduction of 
every actual curve is out of the question. Fourth, 
the assumption of horizontal stratification is usually 
not well enough justified to make highly precise 
results reallv significant. 
The phase integral methods were pushed first, 
because the calculations-:are quite easy and do not 
require special tables of functions. Unfortunately 
the gaps between the regions of validity of the 
different phase integral approximations turn out to 
be extremely wide and to cover just the more inter- 
esting ranges of slope and duct height. This makes 
it necessary to resort to the exact solutions to deter- 
mine characteristic values and normalization factors. 
The phase integral methods provide limiting cases 
which can help in guiding the exact computations. 
Also the phase integral formulas are usually quite 
adequate for the computation of the “raw” charac- 
teristic functions, once the characteristic values are 
known. 
In order to make the exact calculation, we need 
tables for complex arguments of the solutions of the 
equation 
These solutions can be expressed in terms of the 
Airy integrals, but for greater convenience the solu- 
tions have been standardized in the form 
y 
ho) = G) A #952) Ge iD), 
The tabulation of these functions for |z| < 6, on 
a square mesh 0.1 unit on a side is being done on the 
automatic sequence-controlled calculating machine 
at Harvard University. Work was begun in the latter 
part of August 1944, under authorization from the 
Bureau of Ships. Photostats of about one-fourth of 
the tables were obtained by November 1944. 
The present objective is to produce charts from 
which a; and #6: and the normalization factor for the 
first mode can be obtained for any M curve made up 
of two straight portions, the upper one being of 
standard slope. After this, similar charts for the 
second mode, and perhaps the third and fourth, will 
be undertaken. When this has been done, the 
approximate determination of field strengths and 
coverage will be possible by a definite routine 
procedure. 
