410 BELL SYSTEM TECHNICAL JOURNAL 



frequency and on the shape of the cyHnder, e.g. whether it is circular or 

 elHptical, and whether it is slender or stubby. The quantity ^—, however, 



A 



depends only upon the mode and shape of the cylinder and has been referred 

 to as the mode-shape {MS) factor. In this formula, 5 refers to skin depth 

 as customarily defined^, and X is wavelength in the dielectric, as given by 



X = - ; both 5 and X are in meters. 



Fniidanieiital Formulas 



s 



Expressions for standing wave patterns and Q- are given in Table I, 



X 



for right rectangular, circular and full coaxial cylinders*. The table is 

 virtually self-explanatory, but a few remarks on mode designation are 

 needed. The mode indices are /, m, n following the notation of Barrow and 

 Mieher.^ In the rectangular prism they denote the number of half-wave- 

 lengths along the coordinate axes. For the other two cases they have an 

 analagous physical significance with / related to the angular coordinate, 

 m to the radial and n to the axial. 



In the elliptical cylinder, a further index is needed to distinguish between 

 modes which differ only in their orientation with respect to the major and 

 minor axes; these paired modes are termed even and odd, and have slightly 

 different resonant frequencies.^" In the circular cylinder they have the 

 same frequency, a condition which is referred to as a degeneracy (in this 

 case, double) ; that is, in the circular cylinder, odd and even modes are dis- 

 tinguishable only by a difference in their orientation within the cylinder 

 with reference to the origin of the angular coordinate. In Table I, the 

 field expressions are given for the even modes; those for the odd modes are 

 obtained by changing cos 16 to sin Id and sin Id to cos Id everywhere. 



The value of TV in the table is based on counting this degeneracy as a 

 single mode; counting even and odd modes as distinct will nearly double the 

 value of N, thus bringing it into agreement with the general equation (1). 

 The distinction between even and odd modes is of limited importance in 

 practical applications, and will not be further mentioned. 



In Table I, the mks system of units is implied. The notation is in gen- 

 eral accordance with that used in prior developments of the subject. For 

 engineering applications, it is advantageous to reduce the results to units 

 in ordinary use and to change the notation wherever this leads to a more 

 obvious association of ideas. For these reasons, in what follows attention 



* The elliptic cylinder (closely allied to the circular c\linder of which it is a generaliza- 

 tion) is omitted as the necessary functions are not widely known or easily available. 



