270 BELL SYSTEM TECHNICAL JOURNAL 



Making use of the above, we can write 



V,Za 



Po = pi -\- 



Si 



Substituting this, the above equation takes the form 



p2 = po cosh r ^ [_Zo sinh F + Z„ cosh F], 



F2 = Fi (cosh F + l^sinh F^ - ^sinh F. 

 \ Zo / -So 



(28) 



EHminating Vi and substituting VtZb/Si for p-i, since here the area 

 remains constant at the two junctions, we have 



F2 = 



The most useful way of writing this equation is 

 2Zo \ / 2Zi, 



Vo = 



2Z5 



Zo -\- Za / \Zo -{- Zb 



X 



(e~n 



1 



1 - e-2r 



Zo Z,i \ I Zo Z 



Zo -\- Za I \Zo -\- Zb 



(29) 



The volume velocity in the termination of the acoustic system, if 

 the filter were not inserted, is obviously po/[_{Za/Si) + (Zb/Si)']. 

 Hence the effect of inserting the filter at any junction is to change the 

 volume velocity of the system by the factor 



Za + Zb 



IZb 



2Zo 



Zo + Za 



2Zb 



Zo + Zb 

 X 



{e-^) 



1 



Zo — Za \ / Zo Z 



Zo -\- Za / \ Zo -\- Zb 



(30) 



A physical interpretation of equation (30) can be obtained in terms 

 of the transmission and reflection factors first introduced by Heaviside.'' 

 Heaviside showed that at a junction, a reflection of a wave takes place 

 if the impedances looking towards the source and away from the source 

 are not equal. He showed that the current reflected on striking a 

 junction, will be the unmodified current in the line multiplied by the 



^ Heaviside "Electromagnetic Theory" Vol. 11, page 79. 



I 



