38 BELL SYSTEM TECHNICAL JOURNAL 



it is not possible to determine Qo with an accuracy better than the 

 discrepancy between the values given for Qq and Hz. 



The above cited facts prove that on the ground the wave-function 

 for a vertical dipole centered on the interface between air and ground is 



n. = (5) + (6) - [(8) - P/2] + [(9) + P/2] = (8) + (9) 

 or 



n. = ((2i + P/2) + ((22 + P/2) - (2i + <22 + P = <2o + (32. 



The function P can only be thought of as follows. The convergent 

 series (5) and (6) comprising the wave function can not be directly 

 expressed as inverse power series; but if the function P/2 is respectively 

 added and subtracted the resulting sum and difference do have the 

 asymptotic inverse power series expansions (8) and (9). 



Part Two — Superseding the Surface Wave 



It has now been shown by theory, by numerical studies and by 

 crucial experiment that Zenneck's surface wave is not a component 

 in the asymptotic expansion of the wave-function for a vertical dipole. 



Since the wave antennas were designed to utilize the horizontal 

 component of the Zenneck wave electric field and do pick up radio 

 signals it is desirable that we explain the success of the wave antennas 

 in some other way at the same time that we throw away the Zenneck 

 wave. 



The object of this part of the paper is to show that the success of 

 the wave antennas can be well accounted for by means of a plane 

 wave theory. It will be shown that if a plane electromagnetic wave 

 polarized with the electric vector in the plane of incidence and in the 

 wave front impinges on a plane solid at a large angle with the normal 

 to the surface then near the surface the ratio of the horizontal to the 

 vertical component of the total electric field is very nearly the same 

 as though the total field were that of a Zenneck surface wave. 



Since the electric and magnetic fields of an antenna ultimately lie 

 in the wave front and since the wave front at any considerable distance 

 is effectively plane for a structure the size of a wave antenna and since 

 the radiation coming down from the ionosphere consists chiefly of 

 that which has been subjected to the minimum number of reflections 

 and the angle at which the radiation arrives at the receiving wave 

 antenna is usually rather low this plane wave theory easily accounts 

 for the success of the wave antennas. 



A plane electromagnetic wave polarized with its electric vector 

 in the plane of incidence falls upon a plane semi-conducting surface. 

 We are interested in the total field. 



