328 BELL SYSTEM TECHNICAL JOURNAL 



Thus, in the neighborhood of their respective critical frequencies, the 

 attenuations of the two waves are functionally the same; ultimately, 

 however, while the attenuation of the fundamental £-wave increases 

 asf-, the attenuation of the fundamental //-wave decreases as f-^'^; 

 a remarkable property peculiar to this type of wave alone. 



By extending the preceding treatment to the harmonic waves, it is 

 found after some rather laborious analysis that for all the component 

 £- waves, 



/>/n. (51) 



Vi - (fn/ir 



Care must be taken, of course, to choose the correct critical frequency 

 (/„ = /„„,) for the particular component wave under consideration. 



For all the //-waves (including the fundamental //-wave) it is found 

 that 



«0 / ,r r,r^^, , {fljr'Y 



Here n is the order of the geometric harmonic wave (//n-wave) and 

 r' is the root of Jn'iy) corresponding to the particular component wave 

 under consideration. 



The foregoing formulates the attenuation due to dissipation in the 

 sheath alone. If we suppose that the dielectric has a very small but 

 finite conductivity a\, then there must be added to the attenuation, 

 for all types of waves, a term 



{^^) 



Vl - {fnlf? 



To a first order approximation the dissipation has no effect on the 

 phase velocity, which is simply v' . 



Comparative values of attenuation are shown on the accompanying 

 drawing for the fundamental and for the first harmonic E- and //- 

 waves. This is the attenuation due to the loss in the conductor only. 

 That due to the dielectric loss, the term given by (53), must be added. 

 In many instances, we cannot say how large this term will be, for the 

 losses in many dielectrics at the high frequencies involved herein are 

 not known with any certainty at present. Such approximate calcu- 

 lations as we have made, however, ha^'e shown them to be very large 

 except in the case of air. 



