COMPUTING THE MODIFIED INDEX OF REFRACTION. M 



75 



Example IV 



Mixing ratio and temperature given. Simplified method 

 satisfactory for h < 500 m.** 



' Columns for h, t, and w are the assumed data. These data are the same 

 as in Esample III. 



^ F is read from Table 8, as before. 



' u 18 read from Table 10. 



^ Au is read from Table 11. Au is then to be multiplied by A/lOO to give 

 the column hAu/100. 2 is the average centigrade temperature from 

 ground to h. 



''The column (n — 1)10^ is given by F -• u ~\- hAu/100. If the average 

 temperature t is negative, this column is F — u — {hAu/lOO). 



^ Mc is obtained, as before, from Table 7. M is the sum of (n— 1)10^ 

 and Mc- 



"This method is not accurate above 500 ra. It will be noted, by com- 

 paring the results here with those of Example III, that there are occasional 

 differences of 0.1 M unit^. This is due to rounding off and is not significant. 



6.5.3 Procedure Used in Setting up Tables 



Tables 3 to 7 — Eelative Humidity and 

 Temperatuke Given 



Equation 2 may be written 



where 



M.= 





with 



My, = fe, 

 Mc = Ch, 





(3) 



(4) 



(5) 

 (6) 



(7) 



Tables have been prepared which give the quantities 

 1/d (dry term), My, (wet term), and M^ (curvature 

 term) separately. 



Dry Term. Since equation (4) contains the pressure 

 p, which is not usually measured as a function of 

 height, it is necessary to eliminate direct considera- 

 tion of ;;. For this purpose a simplification of the 

 elaborate formula used in the Smithsonian tables for 

 calculating height as a function of pressure is suffi- 

 cient. This simplification neglects very small pressure 

 effects caused by humidity variations and the change 

 of the acceleration of gi-avity with height. This prob- 

 lem is discussed more fully in the section on pressure 



versus height, below. The pressure may then be written 



(8) 



p = poe-"'"'^, 



where 



and 



Po = barometric pressure at sea level, 

 £ = natural logarithmic base, 

 OS = 0.034163, 



T=ll'n^'^dh 



(9) 



T is thus the average temperature from fe = to the 



height h. Strictly, T should be calculated from equa- 

 tion (9). However, it turns out that p is rather insen- 

 sitive to T, and that, except perhaps where high accu- 

 racy is desired, it is sufficient to replace equation (9) 

 with 



T = T 



AT 



(10) 



where AT = T^ — T and T„ is the temperature at the 

 surface. 



By substituting from equations (8) and (10), equa- 

 tion (4) assumes the form 



For all practical cases, 

 and hence. 



Jt«' 



,, Afi, -{ahlT)U-{ATI2T)] 



Ma = -fj^ e 



= -t' 



Even at heights as great as 10* m it can be seen that 

 the second of these exponentials can be replaced by the 

 first two terms of its expansion. Therefore 



Apo.-"*"' , Av,ah -"'^1'^ ^ 



M,= ^^e + 



T ' ' 2T' 



or M, = H{T,h) + G{T,h) AT, (11) 



H{T,h) 



where "'''^ i,\ — "" 



ahIT 



and 



Avn -ahIT 



G{T,K) =^,cche . (12) 



Table 3 gives the quantity H{T,h) as a function of 

 h and t, where 



t = T - 273 



is the standard centigrade temperature. In this table, 

 it is assumed that p^ = 1,000 mb. 



