18 KENNELLY AND KUROKAWA. 



ance offered to this frequency by an indefinitely great length of the 

 tube, and 8a is the "position angle" at the sending end A of the tube 

 as the length of the same is varied. The value deduced for z^ is 

 187 Y 5?2, or A Z on Figure 9. This point Z is thus the inferred 

 inner terminus of the spiral E D C B if the tube were prolonged indefi- 

 nitely. The hyperbolic angle 5^ appears to be 



5 A = {L (0.000706 + j 0.107) + (0.0974 + i 0^^ } hyps Z (21) 



In the electric-conductor analogy, if the tube were stopped with a 

 perfectly reflecting air-tight plug, the position angle at the point of 



• IT 



stoppage would be j - hyperbolic radians or _/ 1; i.e., one imaginary 



quadrant, ^° or 90°. This imperfection of the plug apparently alters 

 this to 0.0974 + j 0.97 . 



The* dotted spiral in Figure 9 indicates the computed locus accord- 

 ing to (21). The agreement between the heavy spiral of observations 

 and the dotted spiral of formula (21) is seen to be fairly satisfactory. 

 At the time when the obserA^ations were made, the theory of the 

 analogy between the impedance of an acoustic-tube conductor and 

 that of an electric-line conductor had not been reached. It has been 

 pointed out, however, by Fleming ^^ and Fitzgerald ^^ that an analogy 

 exists between the waves along air tubes and electric waves. 



According to this analogy, therefore, the plugged tube of varied 

 length offered an acoustic impedance similar to the electric impedance 

 of a uniform alternating-current line, of varied length, put to ground 

 in each case at the distant end through a certain leak, or high resist- 

 ance. At each quarter wave of tube lengthening, the impedance 

 of the tube crosses the axis of z^ and reaches alternately a high and a 

 low value. Thus at C = 37.7. cm., the impedance is a maximum: 



Z3„ = 187 Y 5?2 tanh { (0.0266 +.? 4.03) + (0.0974+^0.97)} 

 = 187 V 5?2 tanh (0.124 + j 5) 

 = 187 Y 5?2 coth 0.124 = 187 \ 5?2 X 8.106 = 1517 \ 5?2, 



acoustic absohms Z 



The quarter wave length is 9.35 cm.; so that at L = 0.30, 19.0, 37.7, 

 56.4 cm., the acoustic impedances should be maxima, and should fall 

 on the axis of z^ at E, D, C and B respectively; while at L = 9.65, 



10 Bibliography 11 and 1.3. 

 H Bibliography 12. 

 12 Bibliography 2. 



