REFRACTION 



47 



modified earth's radius, ka, in which the rays are 

 straight hnes. Figure 6, finally, is a iplane earth 

 diagram; the rays are here curved upwards. 



FiGUHE 4. Ray curvature over earth with radius a. 





Figure 5. Rays in a homogeneous atmosphere 

 (equivalent radius ka). 



These diagrams may be considered as resulting 

 from each other by changing the earth's curvature 

 by an arbitrary factor. From this viewpoint Figure 6 



TRANSMITTGft 



INTCRFERENCE RCCION 



///////, CEOMETBIC PATH 



Figure 6. Rays in a plane earth diagram. (Radius 

 of curvature of rays is — fca.) 



oniciN 



Figure 7. Equivalent parabolic earth diagram. 



represents the limiting case of an infinite earth's 

 radhis. The plane earth diagram is widely used for 

 problems of nonstandard propagation. 



In drawing diagrams for a curved earth of equiva- 

 lent radius fca, it is customary to replace the spher- 

 ical earth outline by an equivalent parabola (see 

 Figure 7). The equation for the surface reduces from 

 the circular form, 



•t/ + ih + fca)- = (fca)-, 



to the parabolic form, 



_1_ . 



2fca ""' ' 



for hs<< .Tj. The height h measured from the 

 surface of the earth, instead of from the x axis,^is 

 given by 



h = /), -| X- , 



2ka 



in which h is laid off perpendicular to the x axis and 

 not to the earth's surface. For clarity in drawing 

 rays or field-strength diagrams, the vertical scale 

 is expanded by an arbitrary factor p, whence 



h 



V 



L+-^-x^ 



\ 2ka ) 



(7) 



This distortion of vertical distances, it can be shown, 

 does not distort angles. The parabolic representa- 

 tion to be reasonably accurate must be restricted to 

 heights in the atmosphere small compared with the 

 extent of the horizontal scale. 



4.1.5 



Curvature Relationships 



The curvature of a ray is defined as the reciprocal 

 of the radius of curvature p. Let -^ be the angle 

 between the ray and a nearly horizontal x axis. 



' d>V 



A B 



Figure 8. Angular relationships of rays. 



By Figure 8, p = — ds/d^, and since i/* is a small 

 angle we may, to a sufficient approximation, put 

 ds = dx, so that 



2 _ _ _# 

 p dx 



