242 BELL SYSTEM TECHNICAL JOURNAL 



If r„ is imaginary (an active mode) the power flow is real, while if r„ is 

 real (a passive mode) the power flow is imaginary (reactive or "wattless" 

 power). 



The spatial distribution functions 7r„ and the corresponding propagation 

 constants r„ are a means for si)ecifying the electrical properties of a physical 

 structure, just as are the physical dimensions which describe the physical 

 structure and determine the various 7r„'s and r„'s. In fact, if we know the 

 various 7r„'s and r„'s, we can determine the response of the structure to an 

 impressed current without direct reference to the physical dimensions. 



In terms of the 7r„'s and r„'s, we can represent any unforced disturbance 

 in the circuit in the form 



Y.^n{x, y)[Ane-^"' + ^„/"n (6.21) 



n 



Here An is the complex amplitude of the wave of the ni\i spatial distribu- 

 tion traveling to the right, and Bn the complex amplitude of the wave of 

 the same spatial distribution traveling to the left. 



It is of interest to consider the power flow in terms of the amplitude, An 

 or Bn . We can obtain the power flow P by integrating the Poynting vector 

 over the part of the .v, y plane within the conducting boundaries 



(6.22) 

 P -\ll (^-^* - E^y^*) dx dy 



By expressing the fields in terms of the stream function, we obtain 



---"'^wK-y+fex 



dx dy (6.23) 



We can transform this by integrating by parts (essentially Green's 

 theorem). Thus 



I — — dx — TTn — — / TT,, - ^ dx (6.24) 



Jxi ox ax dx xi Jxi dx' 



Here Xi and x^ , the limits of integration, lie on the conducting boundaries 

 where 7r„ = 0, and hence the first term on the right is zero. Doing the same 

 for the second term in (6.23), we obtain 



