52 BELL SYSTEM TECHNICAL JOURNAL 



The calculation of the field strength as a function of the radiated 

 power requires a knowledge of the effect of imperfect conductivity on 

 the resistance of the antenna. The reader is referred to papers by 

 Barrow ® and Niessen ^ on this subject. In the wave-length range 

 where these curves are of greatest applicability, the practice is to 

 minimize the ground losses by a ground system consisting of a counter- 

 poise or a network of buried wires. When this is done the ground 

 losses are properly part of the antenna losses and the radiated power 

 may rightfully be taken as the rate of flow of energy past a hemisphere 

 large enough to include the antenna and ground system. If this is 

 done, the field strength is given by 



E^'J^n.). (20a) 



where F(x) is the ratio plotted in Fig. 2* 



Part II — Antennas Above the Surface of the Earth 



It is well known *• ^ that calculations based on the physical optics 

 of plane waves give the first approximation to the received field for 

 radio propagation over plane earth. This approximation is accurate 

 enough for all practical purposes if the antennas are sufficiently 

 removed from the surface of the earth.f Under these conditions, the 

 ratio of the received field strength to that which would be received 

 in free space is given by f 



E/Eo = V(l - Kf + 4K sin2 (7/2), (21) 



* In estimating the fraction of the total power input that is radiated the following 

 papers may be helpful: George H. Brown, "The phase and magnitude of earth 

 currents near radio transmitting antennas," Proc. LR.E., 23, 168-182, February, 

 1935 ; and H. E. Gihring and G. H. Brown, " General considerations of tower antennas 

 for broadcast use," Proc. I.R.E., 23, 311-356, April, 1935. 



t This height depends upon the distance, wave-length and ground constants. 

 The range of validity of this approximation is discussed more fully in connection 

 with equation (27). 



J Equation (21) gives the received field strength for either polarization for trans- 

 mission along the ground. In this case the direct and reflected components are 

 oriented in the same direction in space. It may also be used to calculate the effect 

 of the ground for signals arriving at large angles by taking into consideration the 

 space orientation of the components. 



For horizontal antennas the orientation of the electric vector is horizontal for all 

 angles of incidence so that equation (21) applies directly. For vertical antennas the 

 electric vector makes the angle ?2 with the vertical, both in the direct and reflected 

 wave. Hence if the ratio given in equation (21) is taken as the ratio of the vertical 

 component of the received field to the total incident field it must be multiplied by 

 cos ^2. Even if the ground were not present, however, the vertical component 

 would be reduced by this factor so that the effect of the presence of the ground on 

 the field received by a vertical antenna is given by equation (21) as written without 

 the cos ^2 factor. 



