328 REFRACTION AND REFRACTIVE INDEX MODELS 



Then, when 6 is small, one may expand (8.14) [as in chapter 3, (3.50) to 

 (3.58)] and obtain the convenient expression: 



1^2 ^ 2(h, -h,) _ 2(^^ _ ^^) ^ jo-ejy^ 



(8.15) 



where all values of 6 are in milliradians. 



After obtaining r by use of (8.12) and (8.15), one may determine the 

 distance, d, along the earth's surface that the ray has traveled from: 



rfi,2 = a [ri,o + (62 - di)]. (8.16) 



Thus by successive application of the above formulas, one may trace 

 the progress of the radio wave as it traverses its curved path through 

 the atmosphere. Normally the use of these equations is quite straight- 

 forward. When considering horizontal changes in n, however, one must 

 satisfy these equations by iterative methods. In the present application, 

 since n had to be determined by graphical methods, it was felt to be 

 sufficient to assume a constant distance increment of 250 to 500 m, solve 

 for appropriate height increment from 



Ah = Ad tan di 



1 + ' 



a. 



(8.17) 



graphically determine N for the point di + Ad, hi + Ah, and then deter- 

 mine do and Ti,2. 



This latter type of ray tracing was done for various rays of initial 

 elevation angles between 261.8 mrad (15°) and 10 mrad (^0.5°). The 

 calculations were not carried to smaller elevation angles, since this type 

 of ray tracing is not valid within surface ducts for initial elevation angles 

 below the angle of penetration [16]. 



8.2.5. Comparisons 



Although both of the calculated ray paths consisted of an oversea 

 itinerary with coastal transmission sites, they are quite different in other 

 aspects. Canterbury Plain is located southeast of the 10,000-ft chain 

 of the Southern New Zealand Alps at a latitude of 44°S (the equatorward 

 edge of the westerly belt of winds in November). Cape Kennedy is 

 located on a sea-level peninsula at 28°N (the poleward edge of the north- 

 east trade circulation in May). While the Canterbury profile showed 

 superrefractive tendencies, the Cape Kennedy profile illustrated sub- 

 refraction at the surface counterbalanced by an elevated trade wind 

 inversion layer, indicating that the total bending values of Canterbury 



