476 kennelly and affel. 



Appendix II. 



Elementary Theory of Equivalent Mass. 



Let it be assumed that a flat diaphragm is clamped tightly aroimd a 

 circular edge of radius a cm., without stretch or tension, that it exe- 

 cutes very small sinusoidal vibrations, perpendicularly to its plane, 

 in the fundamental mode of motion, i. e., without either nodal circles 

 or nodal diameters, and that these vibrations may be considered as 

 having a certain amplitude which is a function of the radial distance 

 from the center, and which does not vary appreciably in azimuth 

 around the diaphragm. 



Then let 



X,. = the vibration amplitude at radius /• cm. 



.ro = max. " " " center cm. 



Xr = velocity " radius r cm/sec. 



r = radius of a given point on the diaphragm cm. 



a = " " the diaphragm clamping circle cm. 



M = total mass of the diaphragm within the clamping circle gm. 



p' = superficial density of the diaphragm gm/cm.^ 



m = equivalent mass of the diaphragm gm. 



Wr = kinetic energy of an annulus ergs 



W = total " " the diaphragm. ergs 



Then in any steady state of vibration, the kinetic energy of motion 

 of any elementary annulus of radius r and width dr in the diaphragm, 

 will be 



(l\\\ = ■•Zwrp {xry-dr ergs (51) 



and the total kinetic energy will be 



W = r Wr = I ■ 2tvp' ■ f XvrY ■ r ■ dr ergs (52) 



Also if we define the equivalent mass of the diaphragm as such a 

 mass as, moving with the vibrational velocity at the center of the 

 diaphragm, possesses the same kinetic energy as the whole dia- 

 phragm in its distributed amplitudes, we have 



W 



= f (-^-o)- = \-2irp'-£{xr)--r-dr ergs (58) 



