70 BELl. SYSTEM TRCHXICAL JOURN.-IL 



It follows at onrc from tlu' i)re'ce(ling and Theorem \'II that 

 7(/)=0for/<r 



If the coefticienls ^u /S-. . . . are evaluated, a simple matter of elemen- 

 tary algebra, the foregoing expansion in the retarded time t — r will 

 be found to agree with the solution (211) when a is put equal to p. 



We shall now discuss the outstanding features of the propagation 

 phenomena in the light of equations (210) and (211) for the current 

 and ^•oltage. We observe, first, that we have a true finite velocity of 

 propagation v = l/\/'LC. No matter what the form of impressed 

 e.m.f. at the beginning of the line (.\- = 0), its effect does not reach the 

 point X of the line until a time t = x/v has elapsed. Consequently 

 v^x/t is the velocity with which the wave is propagated. This is a 

 strict consequence of the distributed inductance and capacity of the 

 line and depends only on them, since v = 1/\/LC. It will be recalled 

 that in the case of the cable, where the inductance is ignored, no 

 finite velocity of propagation exists. 



The question of velocity of propagation of the w^a\-e has been the 

 subject of considerable confusion and misinterpretation when dealing 

 with the steady-state phenomena. It seems worth while to briefly 

 touch on this in passing. 



As has been pointed out in preceding chapters, the symbolic or com- 

 plex steady-state formula is gotten fromjhe operational equation by 

 replacing the symbol p by iu where i = \/ -\ and oj/2ir is the frequency. 

 If this is done in the operational equation (203) for the voltage, the 

 symbolic formula is 



Y = e -— V(»co+p)2 o'-g-:ut^ 



If the expression \/ (iu -\- p)- - a- is separated into its real and im- 

 aginary parts we get an expression of the form 



where (w2+'ff^^V + VP+^^^7F+^^' 



^=\ 2^^ 



and 



a = p/^v. 



Now if we keep the expression /-/S ^ constant, that is, if we move 



along the line with velocity dx/dt = v'^, the phase of the wave will 

 remain constant. This is interpreted often as meaning that the 



