322 BELL SYSTEM TECHNICAL JOURNAL 



So we see that, while the characteristic impedance of the £o-wave 

 approaches a constant at very high frequencies, for the //o-wave we 

 have 



K - 0)2. 



In other words, while the energy transmitted by the £o-wave is inde- 

 pendent of the frequency at sufficiently high values of frequency, that 

 transmitted by the i^o-wave increases as the square of the frequency. 

 For the harmonic E- and iJ-waves, the currents vary as cos tiB 

 around the periphery of the sheath. Hence the total harmonic current 

 is zero over any axial or normal cross-section. For these waves, how- 

 ever, it is possible and convenient to calculate the Complex Poynting 

 V^ector on the basis of the average mean square current intensities, 



w} 2 4^ ^^ ""^ 2^ I 



2,„ ,1 r^Mi^, 



2 4x 



2 



dd, P = a, 



which we may assume for convenience to be of the same value, 1/2, 

 as the mean square currents associated with the fundamental com- 

 ponents. 



On this basis we shall obtain first the characteristic impedance of 

 any harmonic component En of the E-wave, ignoring dissipation. 

 Putting 



/„(Xa) = and \a = r^m, 



the Complex Poynting Vector becomes 



^ a^ rev'/ ccy\A4^_±\BJ^,^ , ,^, ,_^ 



On the basis of the current value which we are assuming 



\AnV-\- \BnV 



Thus 



^^ (/„-i(rn„,))^ = 327rM ^ ) • (27) 



Ku = (iTraycyl^eyll - (fnjf)'. (28) 



Similarly, for the component J/„ of the //-wave, we put 

 Jn(\a) = and \a = r„ 



nm ) 



getting 



^^ = l^^^/^:-(:0'(l^»|'+|^"|^>^=w-X'-(5)')■ '^'' 



