94 



CALCULATION OF RADIO GAIN 



and h > -l/l, as described above, except that F^ or 

 F^i is represented by various curves in Figure 32, 

 according to the value of X, and g is modified slightly 

 by a correction factor g'. 



Formula for Sea Water. Vertical Polariza- 

 tion. VHF. 



Equation (160), the general formula for the first 

 mode, in the VHF (1 to 10 meters) range, becomes 



A = [2Ao -li F^.Wi^gg'UH^gg'h, (1(32) 



where H^ is the low antenna height-gain function 

 whose formula is given by equation (152), and 

 gg' = 1 for low antennas. As pointed out above, 

 a h > 4/1 and d > 50/p', equation (162) reduces to 

 equation (161). g', the correction factor for g, is 

 given in Figure 36. For a more extended discussion, 

 see Section 5.7.4. 



5.7.2 



Effect of Changing the Vahie of k 



In the optical region, the effect of a linear gradient 

 of refractive index lias been shown to be equi\'alent 

 to replacing the radius of the earth a by an effective 

 radius ka and then treating the atmosphere as 

 homogeneous (.see Chapter 4). 



In view of the equivalence of the sum of modes to 

 the optical formula in the optical region, it follows 

 that the modes should be changed in the same way 

 in the optical region, i.e., a should be replaced by ka. 

 These same modes supplj'' the solution of the wave 

 equation in the diffraction region, so that in both 

 regions the substitution of ka for a will take care of 

 an atmosphere with a linear variation of refractive 

 index with height for all values of k. 



For given transmitter and receiver antenna heights, 

 hi and h^, the first maximum of the field-strength 

 versus distance curve (see Figure 4) will frequently 

 represent the limit of detection. The first maximum 

 occurs not far from the line of sight which, for a 

 standard atmosphere {k = 4/3) , is given by 



dL = \^2(-\aHhi+ ^lh,). (10.3) 



If 4/3 is changed to k, the more general form is 



diJ = V2fc^ (V/i^ + Vy . (164) 



The first maximum will now be near 



Suppose next that the point at distance (//,' for the 

 given heights was originally well within the diffrac- 

 tive region. The exponential part of the gain due to 

 change of k from 4/3 is eciual to 



or 



(9-l.f)07 (V-UTi- l)s''i. 



The abo^•e formula is obtained by combining equa- 

 tion (149) for Fs, and s is obtained from Figure 31 

 (A- = 4/3). This corresponding gain in decibels is 



20 logioC-Ifi"' (V3f74-l)srfi 



= 20 



- 1.607 X .434 



(Vf-) 



sdL 



= - 14 sd, 



Of-) 



db 



below the free-space value. Since a maximum is now 

 near cl^' as a result of changing 4/3 to k, i.e., the 

 field has a value of 6 db above the free-space value, 

 the gain is approximately the exponential value 



14 sdi 



m~^ 



as a consequence of refraction giving a value A; 

 greater than 4/3. For k = 12, and sd^, = 5, the gain 

 would be about 140 db at some point near rf^' = 3rfi„ 

 at heights /)i and h^ such that 



di' = V2A77 (V/Ti-h V/i7) 



and such that /(2 is weW within the diffraction region. 

 For given transmitter height hi and distance d, the 

 effect of increasing k is to lower the lowest lobe 

 roughly by the amount by which the line-of-sight 

 elevation at the distance d is lowered in changing 

 from 4a/3 to ka. At this distance the height of the 

 line of sight is h^, and 



V/ii = ^= - V/(i (for /,■ = 4/3) 



(165) 



becomes 



V/(/ = 



V2A-a 



- V/h, 



(166) 



so that there is a downward shift of approximately 



8;^V-3^;-2V/.^- 



(167) 



hr. - hr 



2V/,\^^?(l-^|^} 



