338 BELL SYSTEM TECHNICAL JOURNAL 



imaginary terms in (5.1-13) occur in the coefl&cients of xq and arise from the 

 imaginary quantity Viqc. By making the substitution 



the set (5.1-13) may be reduced to 



3 



(KiiTtCtanh 7ic)i^i + S U;»7»^ tanhyiC + {KV~'^)ijVjQc]uj = (kF"^)^ 



(5.1-15) 



in which the coefficients are all real. It should be noticed, however, that 

 nothing is gained by making the substitution (5.1-14) when the frequency 

 is so high that other modes in addition to the dominant are propagated. 



The equation for y corresponding to (5.1-15) may be obtained by re- 

 placing tanh by coth and u by v where now 



1 -h Vi Cl 10 



Incidentally, if we set 7 = 1 in (5.1-14) and (5.1-16) and substitute in 

 the expressions (5.1-4) for ^to we may show that, since ui and vi are real, 



I gto r + I gro P = 1 (5.1-17) 



Equation (5.1-17) may be obtained at once from the fact that the energy 

 of the waves leaving the bend must equal the energy of the incident wave. 

 It may also be shown that gTo vanishes when uiVic^Fio = 1. 



When the above numbers are set in the three equations obtained from 

 (5.1-15) we get 



-8.382 ui -1.748 m -1.122 u^ = .9492 

 .055 ui +2.780 U2 +1.688 W3 = -.2608 

 .053 ui -3.105 U2 +7.191 U3 = .1427 



from which 



ui= - .0940, xi = .1400 + i.0113 



The equations for vi obtained by substituting coth for tanh are 



.2354 vi - .3732 V2 + .3861 vs = .9492 

 .0717 vi + 3.1417 V2 + 1.924 vz = - .2608 

 .0534 vi —3.1143 V2 + 7.211 V3 = .1427 



from which 



vi = 3.930, yi = -.1045 + i.9803 



