HYPER-FREQUENCY TRANSMISSION 317 



is zero everywhere, which will be called generically the Til-wave and (2) 

 a wave for which E^ is zero everywhere, which will be designated 

 generically as the i7-wave. (If the cylinder is dissipative, however, the 

 E- and //-waves can exist alone only for the case of circular symmetry. 

 In other words, unless d/d9 = 0, neither the E^ nor the //^ component 

 of the field can be identically zero. This will be discussed further in 

 Section III.) 



Assuming first a non-dissipative system, it will be seen that when //j 

 is zero everywhere, 



E, and Ee ^ /n(Xp) j . " . 

 [ sm nd 



Thus the possible £-waves are determined by the boundary equation 



Jn{\a) = 0, (5) 



where 



X2 = ^2 _^ 0)7^2. 



This has an infinite number of real roots in X determining an infinite 

 number of possible waves. Only a finite number, m, of these waves 

 will be unattenuated, however, for, if X is to be real and y pure imagin- 

 ary, the frequency must be so high that 



OO/V > \nm, (6) 



where Xn,„a is the mth root of /„(Xa) = 0. It is therefore convenient to 

 designate as the £„m-wave that component of the £-wave for which 



„ J. ,. , f cos n( 



\ sm ni 

 Thus if 



An, m+1 > W^ ^ X„TO. 



the components En, m+i, En, m+2, • • • of the £-wave will all be attenu- 

 ated but En\, F.ni, • • ■, Enm wiU bc unattenuated. There will also be 

 only a finite number « + 1 of the components iioi, En, • • • Eni, for the 

 frequency must be at least sufficiently high so that 



oo/v > X„i, 



where X„ia is the lowest root (excluding zero) of Jni^a) = 0, in order to 

 transmit the component -E„i of the E-wave without attenuation. 

 Hence the jE-wave consists of a doubly terminating series of possible 

 components; for each of the finite number k -\- 1 possible values of n 

 there will be w„ possible values of \a or a total of 



mo + mi + m2 + • • • + Wk 



possible modes of propagation. 



