126 KENNELLY-TAYLOR— EXPLORATIONS OVER [April 22, 



Appendix II. 



Elementary Theory of the Steady Vibration Amplitude of a Dia- 

 phragm Vibrating in its Fundamental Mode, as a 

 Function of the Impressed Frequency. 



Let w= the vibration amplitude at the center of the diaphragm^ 



(cm.Z), 



Wr = the vibration amplitude at the radius r (cm. Z), 



w=the vibration velocity at the center of the diaphragm 



(cm./sec. Z), 

 zif = the vibration acceleration at the center of the diaphragm 



(cm./sec.^Z ), 

 r=frictional resistance to motion of the diaphragm, re- 

 ferred to the equivalent mass, see below (dynes/cm. 

 per sec. Z ), 

 ? = elapsed time from a given epoch (seconds), 

 .?= elastic force of the diaphragm per cm. of displacement, 

 referred to the equivalent mass (dynes per cm.Z), 

 / = -F^^"'':^ impressed simple harmonic moving force on the dia- 

 phragm tending to produce displacement w, and 

 measured in the direction of w, referred to the equiv- 

 alent mass (dynes Z), 



i=V~^^, 

 co==27r» = the angular velocity of a simple harmonic motion of 



frequency n (radians/sec), 

 m = equivalent mass of the diaphragm, defined by the con- 

 dition that the energy of motion of this mass with 

 the velocity w at the center, is equal to the actual 

 energy of the diaphragm with its distributed mass 

 and velocities, according to the equation : 



— (w)^ = I riwr^dr ergs, (i) 



2 2 J^ 



where p' = superficial density of the diaphragm (gm./cm.^), 



m = -2 — I {wrYrdr gm., (2) 



'^max t/o 



8 The sign £ after a unit indicates a " complex quantity." 



