314 BELL SYSTEM TECHNICAL JOURNAL 



conductor and hence no circuit, this concept breaks down. There is 

 another way, however, in which the characteristic impedance may be 

 defined, and by aid of which it remains a useful concept in hollow 

 cylinder transmission. Writing K = Kr + iKi as the complex 

 expression for the characteristic impedance, then it may be shown that 



K = W + i2u^{f - U), 



where W is the mean power transmitted, T is the mean stored magnetic 

 energy, and U the mean stored electric energy, corresponding to an unit 

 current. Now in the hollow conducting cylinder, for, say the £o-wave, 

 we can calculate 



W + t2co(r - U) 



for an unit axial current, and call this the characteristic impedance. 

 Again for the ilo-wave we can calculate this quantity for an unit 

 circulating current per unit length and designate it as the characteristic 

 impedance. In addition, somewhat similar conventions apply to the 

 harmonic waves. 



One of the chief uses of the foregoing concept of characteristic 

 impedance is in the calculation of_the attenuation in the dissipative 

 system. For, if corresponding to W we calculate the mean dissipation 

 Q per unit length, then the attenuation a is given by 



a = Q/2W. 



All actual systems are of course dissipative and consequently the 

 wave is attenuated. If the hollow conducting cylinder were to be 

 employed in practice for hyper-frequency wave transmission the 

 securing of low and desirable attenuation characteristics would 

 probably be the controlling consideration. 



The attenuation in the free transmission range is due to (1) dissipa- 

 tion in the cylinder or sheath and (2) dissipation in the internal di- 

 electric medium. The former is inherent and can be reduced only by 

 employing a sheath of high conductivity and by properly designing the 

 dimensions of the system. As regards the dielectric loss, this may be 

 substantially eliminated by employing air as the dielectric medium. 

 The use of a dielectric medium of high specific inductive capacity has 

 the advantage of substantially reducing the critical frequency; on the 

 other hand it inevitably introduces heavy losses and thus sharply 

 increases the attenuation. The analysis of Section III brings out the 

 remarkable fact that for the fundamental if-wave the attenuation 

 decreases with increasing frequency ; for all the other types it increases. 



