PHYSICAL REALITY OF ZENNECK'S SURFACE WAVE 43 



Since c is to bfj very small it is best to expand goi(c) into an ascending 

 power series in c. 



goi(c) = - 1 H — , - -T^^ 3\ + 



3/2 



_ 2(1 + t2)c^ (8 - 12r^ + 3t^)<:S _ 2(1 + t'')'^^^ _ 



7^(1 - t2) "*" 4t5(1 - t2)5/2 ^6(1 _ ^2^ + 



The recurrence relation then gives us 



_ - 2 (6 - 2t2 - T^)c _ (12 + 6r=')c^ 



g02(C) - _^2(i _ ^2J + ^3(1 _ ^2)3/2 ^4(1 _ ^2) 



(40 - 24t2 - 36r^ + 12t6 + 3t^)c^ 



"^ 2x^1 - t2)5/2 "^ 



- - 6 - 4t- (120 - 727=^ - 108t^ + 36r6 + 9T^)r 

 After dropping all but the leading terms there is left 



" *^ ^2 1 i^2?'2 L tVI - T- J J ' 



£ ^ -^ ^-^^" 1 2(1 + T>r]:^T2^M i 



ri I ife/'2T2(l - r2) J 



The wave tilt near the surface of the ground is then 



E, rvr 



Ez 1 + tVI — rHkz 



This is the wave tilt in the asymptotic field of a quarter wave antenna 

 or flat top antenna. 



If a is not zero but c is small the final field expressions are 



. e-'*^2 r 2 + rV'l - rHk{_w + (2a - w)e^^-'' ''''-'] 



CLo = — too 



^2 I ikr^T-^l 



3 



T— ;; [(w — 2a)g'*2°^/'--' — w] 



^^2 



6 - 6t'' - 3t* + 6tV1 - r^i'y^T^ + 2x^1 - T'-)(ikwy- 



{ikr2yT'{l - t2)3/2 "^ 



