BELL SYSTEM TECHNICAL JOURNAL 



pedance ^ equal to Zo per unit area. At one end, 0, let the velocity 

 be uniform over the whole cross-section and equal to ile*"^ At a 

 distance / from let the tube be terminated by the material which is 

 to be investigated, and the acoustic impedance of which may be rep- 



j J J ■ ^ ^ J ' • 



'>>>>>>' I > I 



Fig. 1 



resented by Z2 = i?2 + i^t per unit area. Under these conditions, 

 the pressure, p, at any point in the tube at a distance x from O, is by 

 analogy with the electrical transmission line 



y- 



Zi cosh PI + Zo sinh PI , „ • u r> 1 v •! 



. p, . „ — ■ . p, cosh Px - smh Px Zq.^ 



Zo cosh PI + Z2 smh PI J 



(1) 



If there is no attenuation along the tube, we get, on dropping the time 

 factor, 



Z2 cos /3/ + iR sin /3/ 



P = Rh 



[ 



R cos jS/ + *Z2 sin ^l 



cos (8.r — i SI 



n i3x , 



(2) 



where R = rp, the product of the velocity of propagation along the 

 tube and the density of the medium, and 



/3 = 



27r/ 



Equation (2) indicates numerous possible ways of determining 

 Z2, e.g., from the values of ^1 and of p at any point in the tube; from 

 the pressures for two values of either x or /, if ii is constant; from the 

 pressures at any point in the tube for the unknown and for a known 

 value of Z2; from the magnitude of /? as a function of either x or /. 

 However, we shall confine our discussion to three methods, which 

 appear to be most practicable. 



^ The term acoustic impedance as here used may be defined as the ratio of pressure 

 to volume velocity; the characteristic impedance is this impedance if the tube were 

 of infinite length. 



^ J. A. Fleming, "Propagation of Electric Currents in Telephone and Telegra[)h 

 Conductors," page 98; 3d Ed. 



