OPTICAL I'HOI'EUTIES <)!•' THE EARTH S SURFACE V\l> \TM()SI»HERE 



35 



dn 

 dh 



= -0.039 ■ L0 ,; per meter . 



(II) 



Refraction at the boundary of two media is fa- 

 miliar from optics and is expressed l>y Snell's law: 



/(, cos ai = ii-> COS as , (12) 



where Bi and rii are the refractive indices of the two 

 media and «i and a» the angle between the boundary 

 and the direction of the ray in the first and second 

 media respectively. In the atmosphere the refrac- 

 tive index is a continuous function of height, and the 

 sudden change of direction at a boundary is then 

 replaced by a curvature of the rays. Equation (12) 

 can be written 



n cos a = n cos a , (13) 



where n and a are now continuous functions of the 

 height and the subscript designates a reference level. 

 The above formulas refer to a plane earth. If the 

 earth's curvature is taken into account so that the 

 planes relative to which the angle a is measured are 

 replaced by spheres about the earth's center, for- 

 mula (13) must be modified; and the mathematical 

 analysis shows 44 ' 2 that it is replaced by 



nr cos a = n r cos a a (14) 



where r is the distance from the center of the earth 

 to the level considered. 



If now we set r = r (1 + h/r ) where h = r — r 

 and h/r is a small quantity and, furthermore, if we 

 note that with a linear gradient of n 



dn 



n = n 



dh 



h 



(15) 



we obtain on substituting into (14) and neglecting 

 small quantities of the second order 



1 + 



+ Th) h 



cos a = cos a 



(16) 



It results from this equation that a linear gradient 

 of refractive index has the same effect on refraction 

 as the curvature of the earth, l/r . By introducing 

 an effective earth's radius it is possible to eliminate 

 the refraction term entirely and to treat the atmos- 

 phere as if it were homogeneous. This device was first 

 introduced by Schelleng, Burrows, and Ferrell, 24 and 

 has since been generally accepted. Some German 

 writers have introduced a quadratic function to 

 represent the variation of refractive index with height 

 in the atmosphere, 443 the coefficients of the quadratic 

 terms being characteristic of the air mass or type of 



atmosphere involved. This lias the advantage of per- 

 mitting a close fit with observed refractive index 

 curves up to heights of 6 to 8 km. It seems, however, 

 that the advantage of the greater analytical simplic- 

 ity of the linear refractive index curves far outweighs 

 the increased accuracy of the quadratic form, and the 

 latter has therefore not found acceptance in this 

 country and Great Britain. 



It is customary to designate the effective, or modi- 

 fied earth radius by ka where k is a numerical con- 

 stant and a replaces r used above and represents 

 the mean radius of the earth. Hence 



1 dn = 1_ 



a dh ka ' 



(17) 



and by comparison with equation (11) it follows that 



k = 



(18) 



since dn/dh = - 0.039 • 10 - 6 = - l/4a. The earth's 

 radius a — 6.37 • 10 6 meters. 



In view of this result coverage diagrams of radar 

 and radio communication sets are commonly drawn 

 with a % earth's radius. In such a diagram the rays, 

 which are curved in a "true" geometric representa- 

 tion, appear as straight lines. 



The value k = % does not, of course, represent a 

 universal law. It is merely an expression of the fact 

 that the rate of decrease of the refractive index with 

 height has, in the middle geographical latitudes, a 

 certain average value. In arctic climates k as a rule 

 is somewhat smaller, lying between % and %, while 

 in tropical climates fc is somewhat larger, between 

 % and %. In temperate and tropical climates, the 

 main factor determining the magnitude of k is the 

 humidity gradient in the lower atmosphere. In 

 Figure 5 is shown a nomogram from which the ap- 

 propriate value of \/k can be read directly as function 

 of the gradient of relative humidity and air tempera- 

 ature. The table has been computed under the as- 

 sumption that the temperature gradient has the 

 "standard" value of —0.65 C per 100 m, but the 

 value of k is relatively insensitive to variations in 

 the temperature gradient. 



Usually the value of k 



is referred to as the 



standard case, but this term is also used to designate 

 more generally an atmosphere with a linear refrac- 

 tive index distribution where k might differ somewhat 

 from %. Experience shows that the atmospheric 

 conditions under which the refractive index is a 

 linear function of height are quite common, but this 



