68 



CALCULATION OF RADIO GAIN 



Furthermore, the phase lag usually differs from 

 180 degrees, depending upon the frequency and 

 grazing angle. This is especially true for vertical 

 polarization where the reflection coefficient is a 

 critical function of both the grazing angle and fre- 

 quency. The ratio 



R = 



£,• 



pe 



is a complex number and defines the reflection co- 

 efficient R, which has an absolute value p and a 

 phase angle (/>. 



In equation (27), a lagging angle is considered 

 positive. Writing </) = x + 0', equation (27) may 

 be expressed as 



E, 



= - pi 



(28) 



In equation (27), the lag angle <^ is measured with 

 respect to zero-degree phase shift at reflection. For 

 horizontal polarization, (f) varies from 180 degrees to 

 183 degrees, and from 180 degrees to 3 degrees for 

 vertical polarization at 3,000 mc over sea water. 

 In equation (28), lag angle <^' is measured with 

 respect to a 180-degree phase shift (that is, from 

 Ei reversed), and varies from degree to 3 degrees 

 for horizontal polarization and from degree to 

 — 177 degrees for vertical polarization. 

 The resultant field intensity is 



E = E, + pEoe-^^' +*> =Eo + p^oe-^'* + *' + "' (29) 



= i?o(l - pe -^■"), 

 where 



The absolute value of the received field \E\ is 

 given by 



or 



E 



E 



pe 



hj!i 



) (1 - pe-'") 



= Eo J I + p- - 2p cos a 





p)- + 4psm--. 



In equations (31) and (32), n includes all integral 

 values and zero. Equation (30) may be written as 



(30) 



Equation (30) shows that the received field intensity 

 has a maximum of 2Eo when 



P= 1, 



Q = (2n+ l)7r. 



The value of £" is a zero when 



P = 1, 



fi = 2n7r. 



(31) 



(32) 



E 



E, I ^ 



p)- -\- 4psm'--> 



(33) 



where E is the field at distance d from the trans- 

 mitter, and El is the field strength at unit distance. 

 From equation (33) , 



d 



El \ 



n 



py- + 4psin2- 



(34) 



In free space where there is no reflecting earth, 

 p = 0, and 



El 



do = 



E 



(35) 



where rfo is the equivalent free-space distance from 

 the transmitter at which the field strength E would 

 be found. Hence equation (34) may be written in 

 the form 



= ^^^(1- 



p)2 + 4psin2- 



(36) 



5.2.5 



Divergence 



The divergence factor D is introduced to account 

 for the decreased gain produced by the spreading 



FiGtTRE 10. Increased divergence resulting from re- 

 flection by a sphere. 



of a wave reflected from a spherical surface. Re- 

 ferring to Figure 10, 



Eh 

 E2 



_ Irffla _ Ida^^dao _''j Idap _ !j p 



(37) 



