ELECTRIC CIRCUIT THEORY 359 



Setting G = for simplich>- and substituting in (284) we get, after 

 easy simplifications, 





^^(o=^+^2:™r4^^"(-^(v)^-3 -'• ^-^) 



/TWTTX^ p^ 



If we write 



mir 



Mm 



5 



(286) can be written as 



Ax(t) = -^-\ ^ sin — (M,«i'/-.v)+sni — {fx„,vt+x) '-. (280 



sR s ^-^ m-K IS s ) 



Mw — 

 s 



This type of solution is often referred to as a wave solution and the 

 component terms of the series regarded as travelling waves. As a 

 matter of fact it is a solution in terms of normal or characteristic 

 vibrations, each of which is to be regarded as instantaneously pro- 

 duced at time / = 0. The solution in terms of true waves has been 

 fully discussed in the preceding. 



Formula (287) is practically useless for computation on account of 

 the slow convergence of the series (the series are only conditionally 

 convergent), and cannot be interpreted to bring out the existence of 

 the actual direct and reflected waves and the physical character of 

 the phenomena it formulates. In fact, as stated above, this form of 

 solution is useful only in connection with the non-inductive cable. 



In the cases considered above we have taken the simplest possi- 

 ble terminations — these of short circuits in which case the roots of 

 Z{p) are easily evaluated. If, however, the line is closed by arbi- 

 trary impedances, the case is quite different, and the location of the 

 roots becomes, except for simple impedances, and then only in the 

 case of the non-inductive cable, practically impossible. While, there- 

 fore, the expansion theorem solution can be formally written down, 

 its actual numerical evaluation is a practical impossibility, except 

 in a few cases. For this reason it will not be considered further here. 



The physically artificial character of the expansion solution, as 

 applied to transmission lines, may be seen from the following con- 

 siderations. When a wave is sent into the line, for a finite time 

 equal to the time of the propagation of the line, it is independent of the 

 character of the distant termination. Yet in the expansion solution 

 every term in\ol\es and is dependent upon the impedance constitut- 



