74 



CALCULATION OF RADIO GAIN 



Equation (76) is the plane earth formula. 



Curves showing s as a function of hi/hi and 

 d/dr are given in Figures 19 and 20. These may be 

 used for the direct calculation of f/i = sd within the 

 limits of interpolation. 



5.5.5 



Path Difference 



Referring to Figure 14, the path difference A 

 for a spherical earth is equal to 



A = )■ 



= VfF + {w + wr 



-Vd-+(V-/uT- (77) 

 It is usually sufficient to expand the square roots 

 and neglect powers and products of /h' and /)/ 

 beyond the .second. This gives 



A = r - r. ^ 



2h'h' 

 d 



(78) 



which is the same as the plane earth formula when 

 hi and Jh' are written instead of /u and h. Equation 

 (78) is accurate to within 1 per cent for values of y 

 (the angle at the base of the transmitter) less than 

 about 8 degrees. The error is less than 10 per cent 

 for values of y less than about 24 degrees. When 

 equation (78) is not sufficiently accurate, the follow- 

 ing may be used: 



2h%' ,„„. 



A = r - r, = , > <>y) 



provided 





2 d- 



All the above equations for the path difference 

 depend upon the distance to the reflection point rfi. 

 However, the calculation of rfi may be eliminated 

 by first computing the path difference from the 

 plane earth formula and then subtracting the correc- 



(80) 



If /(2 < hi, interchange h2 and hi on the curves and 



ordinate of Figure 16. 



The maximum value of A (Ap) is 



, hihi 

 A (Ap)„,, = 0..33 X 10-^ -^ 



(81) 



V/ii + V/ia 



If the plane earth correction factor is negligible for 

 the wavelength under consideration, the plane earth 

 formula may be used throughout the whole range 

 within the optical region, not only for the given value 

 of hi/ hi but for all lower values of hi/h with the 

 same hi. 



When hi > > hi, so that the reflection point is 

 much closer to the transmitter than to the receiver, 

 a good approximation to A is ol^tained by replacing 

 hi by hi and hi by hi - d-/2ka in equation (78). 



Figure 16. Plane earth correction factor versus — 

 (Radiation Laboratory.) ^ 



Then 



2h 

 d 



\ 2ka/ 



'h 



^l2ka V/), + V/i. 



which, to the same approximation, means that 



A ^ 2hi tan y. (82) 



In general, equation (82) is an impi-ovcment over 

 the plane earth approximation except close to the 

 transmitter and at low heights where hi is not much 

 greater than hi. 



