I9IS.] SURFACES OF TELEPHONIC DIAPHRAGMS. 127 



since the velocities w and Wr being assumed simply harmonic, are 

 respectively proportional to their maximum displacements Wmax and 



Wr. 



Then on the assumptions that the diaphragm vibrates like its 

 equivalent mass collected at the center, with its observed central 

 velocity, with an elastic opposing force sw on this mass, propor- 

 tional to the displacement, and with a resisting force rw on this 

 mass proportional to the velocity, then the equation of motion of 

 the diaphragm in terms of equivalent mass will be® 



sw -\-T'w-\- mw^f = Fe'"'' dynes Z . (3) 



The solution of this equation, in terms of velocity w, and the 

 steady state, is known to be 



/ / / ^w cm 



■( s\ r 



-\- ix z z sec. 



Z, (4) 



where x is the "mechanical reactance," and z is the complex 

 " mechanical impedance," by analogy to alternating electric current 

 theory. Both x and z have the same dimensions as r. 



The mechanical impedance relations are indicated in Fig. WA 

 at the left-hand side. OX and OY being rectangular coordinates, 

 the " mechanical resistance " r in dynes per unit velocity, is meas- 

 ured along OX, and is assumed to remain constant at all frequencies. 

 As the frequency n is increased (and with it the vibratory angular 

 velocity oj) from zero to infinity, the reactance .i'=(mw — V<^) 

 varies from — co to -|- co along the line yXy' . The mechanical im- 

 pedance z which is the vector sum of r and ix, will be represented 

 by a complex quantity, or plane vector Op, the extremity of which 

 remains on the line yXy' . At the particular or resonant value of 

 (0, for which ww — .y/(o = o, the reactance vanishes, and the im- 

 pedance z coincides with the resistance r. As shown in the figure, 

 p lies above OX, corresponding to a value of w somewhat greater 

 than the critical or resonant value. 



9 See Bibliography No. 8. 



