FUNDAMENTALS OF PROPAGATION 



19:} 



discontinuously by the amount </>, which is the phase 

 angle of the reflection coefficient. For horizontal 

 polarization, to again take the simplest case, the 

 phase shift <t> at reflection is practically 180°, or 

 x radians. (For vertical polarization, see Figure 4, <t> 

 is more complicated.) Adding the phase change ^, 

 corresponding to difference in path length, gives the 

 complete phase change 5 in the form 



8=^+4> = 2hil3— + tt (for horizontal polarization) . (3) 



A 



Maximum values of the earth gain factor G occur 

 when 5 is an integral multiple of 2ir ; minimum values, 

 for odd integral values of ir. The corresponding 

 values of the angle of elevation /3 are given by 



earth, the refraction of the atmosphere, and diffrac- 

 tion into the region below the line of sight. 



P 4/n \m = 0, 2, 4, • 



(maxima) 

 (minima) 



(for horizontal polarization.) 

 If the reflection coefficient F of the surface is 

 assumed to be unity (see Figure 4) the plane earth 

 gain factor G, from equation (2) , reduces to 



,2, 



G = 2 cos 



which fluctuates between the limits of 2 and zero. 



The coverage diagram drawn for propagation over 

 a perfectly conducting plane on horizontal polariza- 

 tion is illustrated in Figure 5. As an example, consider 



FLAT EARTH 



Figure 5. Simplified coverage diagram. 



/=200 mc, A =1.5 m, Ax = 30.5 m. The values of $ 

 for the first three lobe maxima are 0.68°, 1.37°, and 

 2.05°, and the maximum ranges are twice the free 

 space values. The angles at which the minima occur 

 lie half way between. The scale of vertical distances 

 is greatly exaggerated compared with the horizontal 

 scale. Coverage diagrams for the same frequency and 

 transmitter height, but taking account of the earth's 

 curvature, are shown in Figure 24. 



Coverage diagrams for more complicated situa- 

 tions must take into account, in addition to the 

 factors already mentioned, the curvature of the 



_ HORi_zqNTAL_, h ' _ .f^'^ifirr^ 



3j__ ~ " " Gfi 0"N0-p^ F 



Figure 6. Use of equivalent ground plane. 



When the ground is sloping, the above construction 

 may be modified as indicated in Figure 6. For any 

 specified lobe, determine approximately the part of 

 the ground where reflection takes place. Draw a 

 tangent to the ground in this region and determine 

 the perpendicular projection of the antenna site on 

 this plane ("equivalent ground"). Use the equivalent 

 height thus determined in equation (3), and let the 

 angle /3 refer to the plane of the equivalent ground. 

 This procedure is also required when the transmitter 

 and receiver or target are of comparable height so 

 that the reflection point is not near the transmitter. 



When the transmitter is set up near a coast, the 

 lobe pattern over the ocean will undergo periodic 

 variations caused by the tides. Since, in equation (3), 

 /3 is multiplied by hi, it follows that the lobes will be 

 low at high tide and high at low tide. This phenom- 

 enon may become very important for heightfinding 

 sets. 



A more complicated case occurs if ground reflec- 

 tion is not complete. Then p is less than| unity, and 4> 

 differs from 180°. In this event the lobes have max- 



Hi 



FLAT EARTH 



r~ 



Figure 7. Coverage diagram for incomplete reflection. 



ima which are less than twice the free space field and 

 minima which never reach zero. The angular posi- 

 tions of the lobes are changed somewhat, but the 

 most noticeable change is found on the lower side of 

 the first lobe. It is likely to lie at a lower elevation 

 and reaches the ground at some distance from the 

 transmitter (compare Figures 5 and 7). 



