RADIO PROPAGATION 51 



by less than 2| per cent for values of e greater than 4. Accordingly 

 the magnitude of the first factor (which is equal to the first term of C) 

 gives the attenuation factor at great distances with a degree of accuracy 

 sufficient for all practical purposes. If we choose our unit of distance 

 such that 



X = \2tt^(1 - t2)^/X1 



I 



" e/Q 1 + Q' e 1 + 1/(2^ ^ 



all of the attenuation curves will tend to coincide at the greater 

 distances. This is done in Figs. 2 and 3. Figure 2 shows the varia- 

 tion of received field strength with distance for seven values of Q for 

 the case where the impedance of the ground is very different from that 

 of the air.* These curves give the correct attenuation factor for arbi- 

 trary ground constants at the greater distances. At any distance the 

 above assumption introduces a significant error only when Q is large. 

 Accordingly the curves of Fig. 3 have been calculated for various values 

 of the relative dielectric constant when Q is large. f The short vertical 

 line on each curve indicates the abscissa corresponding to a distance 

 of one wave-length. The curves do not depart appreciably from that 

 for an infinite dielectric constant except for distances less than this.f 

 Since the error introduced in applying the curves of Fig. 2 to the 

 general case is greatest for the conditions represented in Fig. 3, the 

 curves of Fig. 2 may be used with confidence. 



It should be emphasized that the curves of Fig. 2 give the ratio of 

 the received field strength to that which would result from the same 

 current in the same antenna on the surface of a plane earth of perfect 

 conductivity. The antenna is assumed at the earth's surface so that 

 the curves are strictly true only for short antennas. The error for 

 half-wave doublets whose mid-points are not more than a half wave- 

 length above the surface of the earth is negligible except in the im- 

 mediate vicinity of the transmitter. The effect of height above the 

 surface of the earth is taken up more fully in the next section. 



* Since the writing of this paper, Part I of a paper by K. A. Norton on "The 

 Propagation of Radio Waves Over the Surface of the Earth and in the Upper 

 Atmosphere" has appeared in Proc. I.R.E., 24, 1367-1387, October, 1936. The 

 curves of Fig. 2 in this paper are similar to those of Norton's Fig. 1, but by presenting 

 the curves as a function of x their validity is extended to include a wider range of 

 ground constants. 



t The writer is indebted to Miss Clara L. Froelich for making these calculations. 



i The ratio E/2Eo is greater than unity at the shorter distances because Eo is the 

 inverse distance or radiation component of the free space field while E is the total 

 field. At distances that are small compared with a wave-length, £/2£o is given by 

 the second and third terms on the right of equation (17) and the effect of the ground 

 is to increase the field by the factor 2/(1 — t*). 



