RADIO PROPAGATION OVER SPHERICAL EARTH 



479 



Equation (4) states that the field strength at a point on the surface 

 of a conducting sphere is less than that on the surface of a conducting 

 plane by a factor which is a function of the quotient of the distance 

 along the surface by the cube root of the wave-length. 



This factor is plotted in Fig. 1. For small values of x = (f/Vx it 

 approaches unity so that the Watson solution for radio propagation 

 over the surface of a perfectly conducting sphere merges into the 

 Abraham solution for propagation over a perfectly conducting plane 

 at short distances. In order to depict this graphically the curves that 

 result from neglecting all terms except the first, first two, first three, 





 -2 

 -4 

 -6 



tn -8 



_i 



u 



5 -10 



o 



Ld 



Q -12 



Z 



~ -14 



uT -16 



-18 



-20 



-22 



-24 



50 100 



X = d/\/X (d 8. X IN KILOMETERS) 



500 



Fig. 1 — Successive approximations to Watson's formula for the ratio of the field 

 received over perfectly conducting spherical earth to that over perfectly conducting 

 plane earth. 



etc., have been plotted. It will be noted that as more terms of the 

 complete Watson solution are added the resulting curves more nearly 

 approach the Abraham solution for the shorter distances. When x 

 equals 160 the curvature of the earth reduces the field strength one 

 decibel. At this point the error in neglecting all of the terms except 

 the first results in an error of a decibel. For larger values of x the 

 first term approximates the complete series with increasing accuracy, 

 as shown in the curves of Fig. 1. In other words, no error greater than 

 one decibel is incurred if the Abraham solution is used when d/\^'^ < 160 

 and only the first term in the Watson solution is employed when 

 d/\"'i > 160. 



