TR.4NSMISSIOX CII.IR.ICI ERISTICS OF If.trEFH.TF.RS 577 



\W (S) aiul (9) 



rosh (.•H-/B)=cosh .1 cos B + i sinh .1 sin B=\+2['+i2V. (\l) 



whence 



rosh .1 cos B=\+2U, 

 .ni.l (12) 



sinh .1 sin B=2V. 



The solution of this pair of sinuihaneoiis i'<niations leads to separate 

 relations for .1 and B, 



l+2U\- , / 2V \- 

 and 



\ cosh A/ \sinh A/ 



c-^BY-i^y^'- 



\ cos B J vsin B/ 



As is well known from (13) equal attenuation constant loci are repre- 

 sented in the U, V plane by confocal ellipses with foci at U= — l, 

 V = and U = 0, V = 0, thus having symmetry about the [/-axis. 

 The locus for A =0, the limiting case, is a straight line between the foci 

 and it corresponds to the transmitting band in a non-dissipative wave- 

 filler. Similarly from (14) e{|ual phase constant loci are represented 

 by confocal hyperlx)las which have the same foci as above and are 

 orthogonal to the equal attenuation constant ellipses. It will be 

 assumed that the phase constant, B, lies between — r and +5r, which 

 amounts to neglecting multiples of 2w. Then from (12) B has the 

 same sign as V, so that loci in the upper half of the plane correspond 

 to a positive phase constant while those in the lower half correspond 

 to a negative one. 



It is possible, howe\er, to represent all ilii.-. in just the u()|)er li.iil 

 of the plane using coordinates 6' and \V[. Put 



1' = <-;F1, (15) 



where <= ±1, the sign being that of V. The attenuation constant is 

 independent of the sign of I', i.e., of c. But for the phase constant 

 we get from (12) 



MnrB = ^^^, (16) 



smh A 

 and 



O^cB^+ir. 



Thus, as here considered, the product cB, where c= i 1 has the sign of 

 V. is always positive with a value less than or equal to ir. 



