THE MEASUREMENT OF ACOUSTIC IMPEDANCE 



(c) Tube of Variable Length. Pressure Measured at the Source 

 The absolute value of the pressure at the driving end of the tube 

 according to (2) is 



pA = 



i?2' + A"2- + i?- + {Ri + X2- - R') COS 2)8/ 

 + 2X2R sin 2^1 



i?2- + X2- + i?- + {R2' + X2' - R') cos 2/3/ 



- 2X2R sin 2/3/ 



1/2 



i^^i 



and 1^1 1 is a maximum or a minimum when 



2X2R 



tan 2/3/ = 



i?2' + Xs^ - R' 



For the maximum value 2/3/ lies in the first and for the minimum, in 

 the third quadrant. We therefore have 



|/>l|..,aK _ X2' + j?2^ + i?^ + VCX^'^ + i?2^ - R'Y + 4Xo-i?- 

 Xs^ + i?2' + i?2 - V(X2'^ + i?2- - R:')- + 4X2'^i?'^ 



A 



A. 



By analogy from the equations derived in section (b) above, we see 

 that 



2^|AR 



R2 = 



Xo = 



{A -\- I) - {A - 1) cos 2/3/1' 



R(A - 1) sin 2/3/i 

 (^ + 1) - (yl - 1) cos 2/3/1 ' 



^ ^ V.4 - 1 



•VZ + r 



^ = 2/3/i, 

 where /i is the length of the tube when pi has a maximum value. 



Discussion of the Precision of the Methods 



Of the three methods of measuring impedance discussed above, the 

 first is undoubtedly the simplest and most convenient, if an a.c. 

 potentiometer is available. Theoretically, in this case the impedance 

 may be determined with a high degree of precision. However, the 

 method presupposes that the points where the pressures are measured 

 are exactly a quarter of a wave-length apart ; a more detailed analysis 

 shows that, if A is small, variations in this distance will have a large 

 effect on both the ratio of the pressures and their phase difference. 

 It therefore is necessary to keep the temperature of the tube accurately 

 constant or else to determine the distance corresponding to a quarter 



