IMPEDANCE CONCEPT AND APPLICATION 31 



ql being the moment of the doublet. The last equation is obtained 

 from the first two with the aid of Ampere's law. The exact expressions 

 for any medium are 



£/ = ^ — Ai'(o-p) sm <p, Ep+ = -^r^-^Ki(ap) cos >p, 



Zir Zirp 



, icoqla . . Ki((Tp) 



Hz^ =-^^—Ki((rp) sm <p, Z+ = - rj ') \ . 



Zir Ki{<Tp) 



For an internal cylindrical wave, we have 



A 

 E^ = A<jli{(xp) sin (p, Ep~ = /i(o-p) cos ^, 



Hz = — A{g-\- ib}e)Ii{ap) sin <p, Zp~ = 77 



p 



imp) ' 



Close to the doublet this becomes substantially 

 E^- = P sin (p, Ep- = — P cos <p, 



Hz~ = — P(g + iu:e)p sin <p, Zp~ = 



(g + iue)p 



In concluding this set of examples we shall emphasize the fact that 

 the impedance to a wave depends upon the particular manner in which 

 the applied electromotive force is distributed in space, in very much 

 the same way as it depends upon the manner of distribution of this 

 force in time, that is, upon the frequency of the wave. Just as the 

 impedance has a meaning only if the applied electromotive force varies 

 harmonically with a certain well defined frequency,^ there are definite 

 types of applied force distribution in space for which the impedance 

 has a meaning and other types for which it has not. Arbitrary spatial 

 distributions of force may be decomposed into "space harmonics" in 

 a manner analogous to Fourier's frequency analysis of arbitrary time 

 distributions of force. This is just another way of interpreting the 

 well-known method of solving Maxwell's equations with the aid of 

 characteristic wave functions. 



Here is a simple example of the dependence of the impedance to 

 a wave upon the manner of applied force distribution. Consider the 

 wave generated by an infinite electric current filament of radius a 



^ Strictly speaking, the impedance concept is applicable to any impressed force 

 which varies exponentially with time, the exponent being in general a complex 

 number. The only exceptions are the exponents which are either zeros or infinities 

 of the impedance function. Undamped impressed forces constitute merely an im- 

 portant subclass of exponential forces. 



