10 BELL SYSTEM TECHNICAL JOURNAL 



of the coupled oscillations. It can be shown by combining and transforming 

 equations 1.1-14, 15, 19, that the coupled damping coefficients are 



6i + 62 ir , \\\a . , IV^a 



Oa — —. ; — ^7^ — Ol?77 r (>2 



1-i-W Wtot.X IFtotal 



^ 8iW -i- 82 ^ 81 Wib 82 1^26 



' 1 -\-W IFtotal W^total 



The damping constants of two coupled resonances are found by combining 

 the uncoupled damping constants in the same proportion as the energies 

 oscillating in the two resonators. 

 The coupled frequencies are 



03a = 



0)b = 



1 - W 



(J^2 — Wul 



and 



1 - W 



1.2 Forced traveling waves in coupled transmission lines (Fig. lA). 

 The two lines are coupled according to the four equations 



Tci = Ziii + Zmi2 



Til = y\e\ + jmei 



Ve<i = 22J2 + Zmii 



Ti2 = ^2^2 + ymei 



which may be compared to the corresponding equations of section 1.1- 

 There is a dimensional difference because in transmission lines the series 

 impedances 2 are measured in ohm/meter and the shunt reactances y in 

 mho/meter. T is the propagation constant of the wave traveling in the 

 -{-s direction. If a sinusoidal signal with the radian frequency co is impressed 

 upon the input of the lines, the coupled waves have the solution 



ex = Eiae'"'-^''' + £i6e^"'"'''^ 1.2-1 



62 = £2ae^'"'"'°' + £26e^"'-'*' 1.2-2 



For zero coupling one finds, in analogy to Section 1.1 



rr • T-T jut — Fj* 



€20 = 1S.2I2 — .c.2oe 

 ■ it 



K, = 



