REFLECTION THEORY — PROPAGATION BEYOND THE HORIZON 635 



suggested the unusual idealized antenna patterns shown in Fig. 7, which 

 are described in the next section. 



CALCULATION OF THE RECEIVED POWER 



Substituting (2), (3) and (4) in (1), one obtains for the received power, 

 P« = PrMaAuAA'' f p'' dV (5) 



where 



M ^ 2000b'Ki^N (6) 



To integrate over the volume common to actual antenna patterns 

 would be difficult. We have, as mentioned before, replaced the actual 

 patterns with the idealized patterns shown in perspective in Fig. 7 and 

 in plane projection in Fig. 8. The patterns (Fig. 7) are bounded by side 

 planes of the large wedge and by surfaces of cones with axis TR. The com- 

 mon volume is indicated by broken lines and is well defined. Since the 

 grazing angle A is constant for the incremental cylindrical volumes dVi 

 and dVo shown in Fig. 8, it is easy to integrate over the common volume 

 V and we obtain 



(7) 



1+^-^^. -^1^^ ^1 (8) 



(■ * ? 



The function f(a/d) is plotted in Fig. 9. 



The gain of the idealized antennas is G^ = 87r/Q;jS(o! + 26) and the ef- 

 fective area is 



2X^ 



^ ^ a^{a + 2d) ^^^ 



The area of a cross section of the antenna pattern is bounded by two 

 straight sides, ra, and two curved sides rd^ and r{d + a)|3. The aspect 

 ratio is defined as the ratio of the sum of the lengths of the curved sides 

 to the sum of the lengths of the straight sides. It is equal to one when 



Substituting (10) in (9) gives the effective area of the idealized antenna 



