ASYMPTOTIC D I POLE RADIATION FORMULAS 



667 



<2i 



2^2^ COS 9. 



ki" cos d, + ^i-V^s' - ki^ sin2 ^, 



gikiRi 



R, 



(11) 



This ()i behaves as one would rightfully expect a true asymptotic 

 formula to behave. It is 2(exp ikiR2)/R2 at ^2 = «= and l(exp ikiRi)! 

 R2 at ^2 = ^1- 



In so far as ^2 is considerably larger than ki and the expansion in 

 powers of / is valid Q2 is negligible in comparison with QiJ Perhaps 

 the easiest way of seeing this is to note that equation (10) might just 

 as well have been obtained directly from V instead of from Qi. 



Substituting equation (11) in equation (4) we get 



whence 



^ ^ k2' cos 0. - h^k2^ - k{^ sin^ ^_. ^..,(^.-^, ) e^'-"^ ^ ^j^) 



£/ = -ico( n/ + ;fei-2^n. 



^2^ cos dz + ki-\k2^ — jfei^ sin^ dz 



a^ 



i?i 



where 



- ioi sin2 ^,(1 + i?igi^i2«cos9.)_^^ (13) 



i?l = 



^2^ COS 02 — ki-^ki^ — ki^ sin^ 62 



k2^ cos ^z + ^iV^2^ — ^1^ sin^ dz 

 Substituting equation (11) in equation (2) and adding £2°° we get 



Ez'' ~ - iw cos d:, cos ^^(1 - i?igi*l2acos9.-)^ 



(14) 



i?i is the coefficient of reflection for a plane wave polarized in the 

 plane of incidence. 



The horizontal fields of a horizontal dipole are 





aj' \ dx dy dz 



and 



= — ico n^ 



dx~ 



^ Riemann-Weber's " Differentialglcichungen dcr Physik," p. 556. 



