140 BELL SYSTEM TECHNICAL JOURNAL 



8500 km. instead of 6370 km. Since in the new system of coordinates, 

 the ray is straight, the new equivalent dielectric is to be assumed 

 constant and equal substantially to unity. 



Refraction can therefore be taken into account as follows: In 

 making calculations, we start with the topographical features of the 

 path and construct an equivalent profile ^^ of some sort plotted from 

 known elevations of points along the path. If refraction were to be 

 neglected, the actual radius of the earth would be used. To take 

 refraction into account, the process is exactly the same except that 

 the fictitious radius re(= 1.33ro) is now used. Reflection and diffrac- 

 tion calculations are then based on this equivalent profile, in which 

 account has already been taken of refraction by means of the fictitious 

 radius. 



It follows from the discussion given in Appendix III, that this 

 transformation is not limited to optical paths. The discussion applies 

 to the amplitude of the disturbance set up at one point due to a 

 radiating source at any other point, whether that source be an actual 

 antenna or one of the elementary reradiating oscillators of Huyghens. 

 Under all circumstances where Huyghens' principle applies, the signal 

 is passed on from one intermediate plane to another by the repeated 

 application of the principle. Since this transformation is justified for 

 determining the effect that any elementary oscillator at one point 

 produces at a second nearby point it is justified for the process as a 

 whole provided only that the line connecting the two points is inclined 

 to the horizontal by only a small angle. 



Optical Path Transmission 



Let us now consider the application of these concepts to the case of 

 transmission along an optical path. It has been pointed out that in 

 many cases we would expect to find a well-defined reflected wave 

 superposed on the direct wave. The two will, therefore, interfere 

 constructively or destructively depending on phase relations. In 

 other words, a set of Lloyd's fringes will be set up. 



The airplane measurements over New York Bay gave direct evidence 



of the existence of these fringes. In order to check this quantitatively, 



the data are presented in Fig. \\-A. Vertical polarization was used. 



1^ The elevations above sea-level involved are so small compared with the distances 

 along the surface of the earth that they cannot be plotted on the same scale. This 

 difficulty can be overcome within limits by increasing the scale used in plotting 

 elevations, and at the same time decreasing the scale used in plotting the radius 

 of the earth by the same ratio. When this is done, a line which in the actual case 

 is straight remains approximately so even with these distorted scales. This gives a 

 general picture of the profile but due to the slight curvature introduced, all distances 

 involved in the calculations of this paper have been determined analytically. The 

 scales of the profiles shown have thus been altered by a factor of about 50. 



