430 BELL SYSTEM TECHNICAL JOURNAL 



equations are given; it is only necessary to integrate H^dr over the volume 

 and IPda over the surface and get Q from 



2 / ^'^' 

 Q = I (16) 



j H^da 



with 5 = skin depth, a known constant. Unfortunately the integrations 

 cannot at present be expressed in closed form. A numerical solution can 

 be obtained by a combination of integration in series and of numerical 

 integration. 



The calculations have been made for the ^TE 01 mode with c — 2.0, for 

 which r = 4.154. This value of c corresponds in this case to an ellipticity 

 of about 12%; in a 4" cylinder this would amount to 1/2" difference between 

 largest and smallest diameters. Evaluation* of the integrals yields: 



H-dr = 12.307 k^L + 12.294 klL 

 v 



H"d<7 = 49.228 k^ + 0.1619 kiHL + 6.6847 kiL 



s 



Substituting k] — and kg = — ^ o^^^ obtains, finally 



Q8 = 0.471 D 



1 + 0.1622 nR" 



,1 + 0.0039 «2i?2 ^ 0.1529 n'-R^ 

 For a circular cyhnder, 



'1 + 0.1681 nR" 



Qc8 = 0.5 D 



1 + 0.1681 n'-R 



Comparison of these two formulas for Qd shows that the losses in the end 

 plates {n-R term) are less with respect to the side wall losses in the ellip- 

 tical cylinder. The net loss in Q8, as described by the reduction in the mul- 

 tiplier from 0.5 to 0.471, is thus presumably ascribable to an increase in side 

 wall losses (stored energy assunied held constant). The additional term 

 in n^R in the denominator is responsible for the difference in the attenuation- 

 frequency behavior of elliptical vs circular wave guide as shown by Chu, 

 Fig. 4. Incidentally, these results agree numerically with those of Chu. 



* Numerical integration was by Weddle's rule; intervals of 5° in ^ and 0.1 in x were used. 

 The calculations were made bj^ Miss F. C. Larkej'. 



