260 BELL SYSTEM TECHNICAL JOURNAL 



where A and B are constants and a by analogy with an electric line 

 is the propagation constant of the tube. Substituting (2) in (1), we 

 see that (2) is a solution provided 



2 





/2 



7 



5* ^i 2cop 



(3) 



Now a can be written a = a + ib, where a is the attenuation constant 

 and b the phase constant. If we solve for a and b, assuming 



is a small quantity, we obtain 



We are generally interested in the volume velocity S^ = V, so we can 

 rewrite equation (2) as 



V = i(joSe''^^£A cosh ax + B sinh ax]. (5) 



To determine one constant of equation (5), let x equal zero. Then 



Fx=o = Vi= ioie^'SA 



or 



^ = 



i(j3Se 



We have the additional relation 



A-T^r (6) 



P-P„=_P.,|=^. (7) 



where ^ denotes the excess pressure. Substituting (2) in (7), and 

 differentiating, we have 



p — — Poje'^^'iAa sinh ax + Ba cosh ax). 



Putting X = 0, we have 



^x=o = pi= - Poye'^'iBa) 



or 



B=--^^,. (8) 



