NONSTANDARD PROPAGATION IN THE DIFFRACTION ZONE 



227 



t.ions, which satisfy the differential equation and the 

 boundary conditions but still require multiplication 

 by suitable normalization factors. 



3. The normalization factors. 



There are three methods of attack on the problem : 



1. Numerical integration of the differential equa- 

 tion, accomplished in practice by the use of a 

 differential analyser. 



2. Phase integral methods. 



3. Use of known functions and tables, for suitably 

 chosen M curves. 



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 2 — > co 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 (3) 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 cases. 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 going beyond the cases 

 with three (Figure 1). 



Figure 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 this 

 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 really significant. 



The use of the jointed-linc model for phase integral 

 work was decided on last winter in the Radiation 

 Laboratory. 



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 



'2V 



v3y 



} h (z) 



Z h H i tt2 ; 



3 \3 



U = 1,2) . 



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 ai and /3i 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. 



b The use of the solutions for this case in terms of Hankel 

 functions was suggested by Lt. Comdr. Menzel. 



