GUIDED I'HOl' \G\TIOIN IN NONIIOMOGENEOUS ATMOSPHERE 



245 



E = the electric intensity; 

 h = the height; 



« = the modified dielectric constant; 

 co = the radian frequency; 

 ji = the phase constant in the direction 

 parallel to the stratification. 



We can try to approximate this solution by a curve 

 of the type shown in Figure 1 in which case the 

 problem is to select a proper value for H. The ques- 

 tion may be raised regarding our preference for this 

 particular curve rather than for curve 2 or 3 in 

 Figure 2. We shall return to this point later; for the 

 present we shall merely point out that curve 1 

 occupies a "mean" position among other curves of 

 this type. 



There are two methods for selecting H. In one 

 method H is defined as that value of h for which the 

 coefficient /3 2 — coVe(^) in equation (1) vanishes. 

 This value of h separates the region in which the 

 solution of equation (1) is "more" or less sinusoi- 

 dal" from the region in which the solution is "more 

 or less exponential." This definition leads to one 

 equation connecting H and p. Next, the stratified 

 region < h < H is replaced by a homogeneous 

 region in which the dielectric constant is equal 

 to the average value of e(h) in the interval (0,/i). 

 If we impose the requirement that curve 1 repre- 

 sents the exact field distribution under the new 

 conditions, we obtain the second equation for H and 

 fi. Eliminating f3 and expressing the result in sym- 

 bols approved by the wave propagation committee, 

 we have 



i l 



H M(h) dh - H*M(H) = 



9 X 10 6 

 128 



(2) 



If the lapse rate of M is constant, this equation gives 



dh J 



F = 65X*(-2^f) 



If M{h) is proportional to h 2 , then 



H = 18X 



*W- 



(4) 



If the lapse rate of M is constant, the exact solution 

 may be expressed in terms of Bessel functions. 

 Figure 3 shows the exact and approximate solutions. 

 For this comparison I am indebted to J. E. Freehafer 

 of the Radiation Laboratory. 



The second method is based on the fact that the 



1.0 



0.6 



0.2 



0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 



Ey = (const) U\ [J\(U) + J-\ (U)] Ey = sin p p< ^ 



U = ^U{l-^} h Ey = ^He-P p >^ 



Figure 3. (1) Exact solution normalized to have min- 

 imum value of unity. (2) Approximation. 



solutions of equation (1) minimize and reduce to 

 zero the following function : 



/ = /3 2 / E* dh -w 



/o 



-M« (0) / -7- 



(h) 

 ) 



E*dh 



+ 



I \dh) 



dh . 



(5) 



In deriving this equation we should remember that 

 we are concerned with solutions which vanish at 

 h = and h = oo. Hence, if we wish to approximate 

 this solution by a function of one parameter H, we 

 eliminate H from the following two equations 



°-w = °- 



(6) 



If, for instance, we wish to approximate the field 

 distribution by the master curve in Figure 1, we solve 



dP 9 X 10 5 m 



H ~W~ Hp - ~l2T~ x ' (7) 



where 



I ' - I M sin 2 ~ dh 



" 5 j H Mex v^ir dh - (8) 



By this variational method the numerical coefficient 



