ACOUSTIC IMPEDANCE. 29 



Appendix. 



On flu AtKthxjic.'f hcficccii Souiid-Wave Trftn.'nin'.s.sion (ilong a I' nifonn 

 Acititstic TiiJjc Conductor, and Electromagnciir-Wave Tnnisntission 

 (doiKj (I f'nU'oDit Klcdric J.inc Conductor . 



\Ve may assume that when a telephone l■eee^^•er, actuated as an 

 acoustic generator, sings a pure tone steacHly into a long smooth 

 straight tube of S sc{. cm. internal cross section, arranged as in Figure 

 2, the sound waves travelling along the tube are plane waves, and that 

 the effects on them of friction at the walls of the tube may be neglected. 

 P2ven if the actual conditions do not conform closely with these 

 postulates, the results thus arrived at may safely be regarded as first 

 approximations. 



The distance of any cross-section of the tube, measured along the 

 axis, from the generator diaphragm may be expressed as 1 cm. The 

 clamping-circle diameter of the receiver diaphragm may first be sup- 

 posed to be equal to the internal diameter of the tube. The very 

 small A'ibratory displacement of the air, in a sound wave, from its 

 normal quiescent position, measured parallel to the axis, in the direc- 

 tion of increasing 1, may be denoted by x cm. Since x, in the assumed 

 case of a pure tone and simple harmonic \ibration, may be taken as 

 the projection, at time t seconds, on the tube axis, of the rotating 

 plane vector 



•r = .r,„ e^"' instantaneous cm. Z (22) 



where o) is the impressed angular velocity of rotation, and .r,„ is the 

 maximum cyclic vibratory displacement, the vibratory velocity of a 

 transverse layer of air in the tube, also measured along the axis, and 

 in tlic direction of increasing 1, is 



dx . . . . inst. kines or , ..,. 



dt ■ cm. sec. Z 



The velocity x is also a planeA'ector quantity, capable of being dealt 

 with by the rules of complex arithmetic. Likewise, the vibratory 

 acceleration will be 



^"•'' .. . . . . , , , inst. kines sec. ' .,,. 



-^^ = X = jo:x„.e'-' = jo^x = - co^r„,e^-= - c.-.r ,,,. ^.,,,_ ,^^^_,^ (24) 



