48 



FACTORS INFLUENCING TR.\NSMISSION 



(8) 



Here the curvature has laeen defined so that it is 

 positive when the ray cur^•es in the same direction 

 as the earth; ^^ith this system the ciu'\'ature of the 

 earth itself is positive. Referring to Figure SB, 

 1 _ f/i/' _ da dcj) 

 p dx d.r d.r 



But d4>/dx = l,'o, and since a is a small angle 

 da da dh _ da _ \ d(or) 

 rf7 " ^dh ' dx " "' dh 2 dh 

 Consequently, by equation (2) 



1 I d(a-) , J_ _ _ i^ 



p " 2 dh a dh ' 



From this, the ciu'vature of the ray is equal to the 

 vertical rate of decrease of the refracti\-e index. 

 Notice that dn/dh is usually negative, so that the 

 true curvature of a ray is usually concave do^\Tiwards. 

 A simple relationship exists between m = p a, 

 the ratio of the radius of curvature of a ray to the 

 radius of the earth, and k. Combining equations (6) 

 and (9) gives 



(9) 





1. 



(10) 



Consider again the special case where du/dh = 

 constant, so that n is a linear function of the height 

 (standard refraction). Consider the plane earth 

 diagram of Figure 6. The angles between corre- 

 sponding curves are the same as in the true diagram. 

 Figure 4. Hence, for the plane earth diagram, 

 equation (8) becomes l/'p' = — da'dx, where p' is 

 the radius of cur\-ature of the ray in the plane earth 

 representation. It is readily found that 



_2_d(o 



2 dh \dh a / (n) 



dh ka 



Since M usually increases with height, the curvature 

 of rays is concave upwards in this diagram. Again, 

 equation (11) shows that when the modified earth's 

 radius ka is introduced (Figure 5), this amount of 

 upward curvature is just canceled and the rays 

 appear as straight hues. 



P 



1 



dh \dh a / 



4.1.6 



Alternate Method 



Instead of taking account of refraction by chang- 

 ing the earth's curvature, another method is some- 

 times more convenient. It mav be shown that the 



ratio of the field E to the free-space field Eo trans- 

 forms in the same way, whether (1) radius a is 

 replaced by ka, or (2) the horizontal distance x is 

 replaced by xk~' ^ and at the same time all elevations 

 h are replaced bj^ hk'^'^. An angle a must then be 

 replaced by ak'^^. Method (2) is usuall3' less con- 

 venient than method (1) because it involves a change 

 of horizontal distance which makes it necessarj- to 

 transform the ratio E/Eo rather than the field itself. 

 In method (1) where only curvatures are changed, 

 tliis difSculty does not appear as the distance x 

 and hence Eo remains unaltered. 



Method (2) may be used to advantage to accoimt 

 for de^'iations of k from the standard value of 

 k = 4 3. Coverage diagrams are usuaOy dra'sTi 

 for this value; the deviations owing to a change in 

 k maj^ then be estimated by multiplying distances, 

 heights, or angles with the appropriate powers of 

 k/(i/S). 



4.1.7 



Computation of Refractive Index 



The following equation gives the dependence of 

 the refractive index on temperature, pressure, and 

 humiditv ; 



(12) 



where T is the absolute temperature, p is the total 

 pressure, and e the water-vapor pressure, both the 

 latter in millibars. 



Introducing .1/ from equation (3), the modified 

 refractive index, for use on a plane earth diagram, 

 is equal to 



4.S00e^ 



M 



T \ T J 



(13) 



where the height h is in meters. If h is in feet, the 

 last term is 0.048/i. 



Tables have been prepared by means of which M 

 can be computed rapidly from meteorological data, 

 namely temperature, humidity, and pressure given 

 as a function of height. For this purpose M is the 

 sum of three terms wiiich are computed independ- 

 ently: 



,1/ = Mi + M.^ + M, . (14) 



The dry term M^ is obtained from Table 1 as a 

 fimction of temperature and height in meters above 

 the ground. (If the pressure at the ground po is 

 substantially different from 1,000 niillibars all 

 values of M^ should be multiplied by po/1,000. In 



