406 BELL SYSTEM TECHNICAL JOURNAL 



whereby Z may be calculated in terms of Zo and two ratios of pressures 

 at the junction. 



Referring now to Fig. 1 it will be seen that the current through the 

 resistance and mutual primary is proportional to the pressure at the 

 junction. The drop in voltage across the secondary and the resistance 

 is equal in magnitude and opposite in phase to a voltage proportional 

 to E, when no current passes through the head-phones. If k signifies 

 the circuit constant and if 2 be the impedance value of the resistance 

 and the mutual inductance, then ez = kE; and the above equation 

 becomes 



-^- 1 

 Z _ z-i _ Zi — Z2 _ (ri — r^) + jco(nii — mz) 



Zq 23 _ 23 - S2 (rs — Yi) +jco{m3 — mi) ' 



22 



where r is the resistance component of 2 and in is the mutual inductance. 



The reactance of a closed tube of uniform bore whose length is one- 

 eighth the wave length of sound for the measuring frequency is chosen 

 as the known impedance. If dissipation in the tube be neglected, the 

 impedance is readily calculated ^ to be a pure negative reactance of 41 

 mechanical ohms per square centimeter ^ at a temperature of 20° C. 

 This value is chosen because it is of the same order of magnitude as 

 most acoustic impedances. By mechanical ohms per square centi- 

 meter is meant the complex ratio of pressure to the linear velocity of the 

 air. The justification for assuming negligible dissipation will be ap- 

 parent when measurements made on a closed tube, several wave- 

 lengths long, are described. 



In making measurements, the three impedance values necessary for 

 balance are read for the three impedance conditions in the 2-1-3 or 

 2-3-1 order. Afterwards, as a check to ensure that the circuit constant 

 has not changed during the measurement, condition 2 is measured 

 again. This series of four measurements is repeated for each fre- 

 quency. 



Application 



Fig. 4 shows the results of measuring the reactance of a closed tube. 

 The tube was 2.4 inches long and 0.7 inch in diameter. The com- 

 parison impedance was the calculated reactance of this same tube in 

 the one-eighth wave-length condition, assuming no dissipation. The 

 impedance was also calculated,^ taking into account viscosity and 



^See I. B. Crandall's "Theory of Vibrating Systems and Sound," p. 104. 

 ' See definitions 8007 and 8011 in "Standardization Report of I. R. E." in "Year 

 Book of I. R. E.," 1931. 



* See Rayleigh, "Theory of Sound," Vol. II, pp 318 and 325. 



