56 TROPOSPHERIC REFRACTION 



3.6. Linear or Effective Earth's Radius Model 



The classical method of accounting for the effects of atmospheric 

 refraction of radio waves is to assume an effective earth's radius, a^ = ka, 

 where a is the true radius of the earth and k is the effective earth's radius 

 factor. This method, advanced by Schelleng, Burrows, and Ferrell [8], 

 assumes an earth suitably larger than the actual earth so that the curva- 

 ture of the radio ray may be absorbed in the curvature of the effective 

 earth so that the relative curvature of the two remains the same, thus 

 allowing that radio rays be drawn as straight lines over this earth rather 

 than curved rays over the true earth. This method of accounting for 

 atmospheric refraction permits a tremendous simplification in the many 

 practical problems of radio propagation engineering although the height 

 distribution of refractive index implied by this method is not a very 

 realistic representation of the average refractive index structure of the 

 atmosphere. This section will consider the refractive index structure 

 assumed by the effective earth's radius model and how this differs from 

 the observed refractive index structure of the atmosphere. Further, the 

 limits of applicability of the effective earth's radius approach will be ex- 

 plored and a physically more realistic model, the exponential, will be 

 described for those conditions where the effective earth's radius model is 

 most in error. 



It is instructive to give a derivation of the expression relating the curva- 

 ture of radio rays to the gradient of refractive index. In figure 3.2 a 

 wave front moves from AB to A'B' along the ray path. If the phase 

 velocity along AA'isv and v + dv along BB', then, from considering the 

 angular velocity, 



'- = '-^ (3.14) 



p p -j- dp 



or 



^ = ^ (3.15) 



V p 



where p is the radius of curvature of the arc A A'. Now, since the phase 

 velocity, v, is 



V = - (3.16) 



n 



where c is the velocity of light in vacuo, one obtains 



dv^_dn (3^^) 



V n 



