RADIO PROPAGATION FUNDAMENTALS 597 



and the confusion about the surface wave can often be neglected. For 

 near grazing paths, R is approximately equal to —1 and the factor A 

 can be neglected as long as both antennas are elevated more than a 

 wavelength above the ground (or more than 5-10 wavelengths above 

 sea water) . Under these conditions the effect of the earth is independent 

 of polarization and ground constants and (3) reduces to 





^Pf =2sin^ = 2sin^^ (4) 



2 \d 



where Po is the received power expected in free space. 



The above expression is the sum of the direct and ground reflected 

 rays and shows the lobe structure of the signal as it oscillates around the 

 free space value. In most radio applications (except air to ground) the 

 principal interest is in the lower part of the first lobe; that is, where 

 A/2 < 7r/4. In this case, sin A/2 == A/2 and the transmission loss over 

 plane earth is given by: 



(5) 



It Avill be noted that this relation is independent of frequency and it is 

 shown in decibels in Fig. 2 for isotropic antennas. Fig. 2 is not vaUd 

 when the indicated transmission loss is less than the free space loss shown 

 in Fig. 1, because this means that A is too large for this approximation. 



Although the transmission loss shown in (5) and in Fig. 2 has been 

 derived from optical concepts that are not strictly valid for antenna 

 heights less than a few wavelengths, approximate results can be obtained 

 for lower heights by using hi (or h-i) as the larger of either the actual 

 antenna height or the minimum effective antenna height shown in Fig. 

 3. The concept of minimum effective antenna height is discussed further 

 in Section IV. The error that can result from the use of this artifice does 

 not exceed ±3 db and occurs where the actual antenna height is ap- 

 proximately equal to the minimum effective antenna height. 



The sine function in (4) shows that the received field intensity oscil- 

 lates around the free space value as the antenna heights are increased. 

 The first maximum occurs when the difference between the direct and 

 ground reflected waves is a half wavelength. The signal maxima have a 

 magnitude 1 + | i? | and the signal minima have a magnitude of 1 — \R\. 



Frequently the amount of clearance (or obstruction) is described in 

 terms of Fresnel zones. All points from which a wave could be reflected 



