RADIO PROPAGATION 49 



For an antenna on the surface of a perfectly conducting plane this 

 function may be written * 



/7"/g-2irifli/X 



n = 2no = 2 — -i-^ amperes, (12) 



47r/<i 



where Ri = ^d? + z^ is the distance. For an antenna on an imper- 

 fectly conducting plane 



n = 2Wn^, (13) 



where W is the ratio of the Hertzian potential due to an antenna on 

 an imperfectly conducting plane to that on a perfectly conducting 

 plane. W may be expressed as the sum of two infinite convergent 

 series, A and D, which are defined in Appendix I (page 70). 



W = A + D. (14) 



The series D becomes unwieldy for distances greater than the order 

 of a wave-length. In order to facilitate computation, D may be 

 transformed into an asymptotic expansion to which it is equivalent 

 at large distances, so that 



W ^ A - BI2-{-F. (15) 



At still greater distances A also becomes unwieldy and it may in turn 

 be replaced by its asymptotic expansion, which contains the term 

 5/2, so that 



W = C + F. (16) 



When the impedance of the ground is very different from that of the 

 air,t F is small compared with A — 5/2 « C. If the conductivity 

 of the ground is not zero, F is exponentially attenuated so that it 

 may be neglected in comparison with C in equation (16). Even if 

 the conductivity is zero and the relative dielectric constant is as 

 small as 4, the only effect of F in equation (16) is to produce oscilla- 

 tions in W of approximately 3 per cent from the magnitude of C. 

 Even under these extreme conditions the received field strength may 

 be calculated from equation (15) neglecting F without introducing 

 an error greater than 3 per cent. As the transmitting antenna is ap- 

 proached, the approximations involved in equation (15) become poorer 

 and poorer but at the same time the field strength becomes independent 

 of W so that at no distance is there an appreciable error introduced 



* The factor 2 occurs in equation (12) since IIo and £o refer to transmission in free 

 space. 



t This is true when the so-called "complex dielectric constant," e — 2/<t//, differs 

 sufficiently from unity. 



