I9I5-] SURFACES OF TELEPHONIC DIAPHRAGMS. ]31 



Appendix III. 

 Elementary Theory of Equivalent Mass. 

 In (2) of Appendix II., the expression for equivalent mass m is 



m = — r — I w,. • r dr gm. (i) 



or m is the mass which, vibrating at the center of the diaphragm 

 with the observed maximum ampltiude Wmax, would have the same 

 kinetic energy as the total distributed kinetic energy of the dia- 

 phragm. 



In order, therefore, to determine the equivalent mass of a dia- 

 phragm, it is necessary to integrate r times the square of the ampli- 

 tude over its surface. Assuming that the vibration follows Ray- 

 leigh's Bessel-function theory as outlined in Appendix L, it should 

 be sufficient to integrate Wr"-r over the surface, mathematically. 

 We are indebted to Dr. Geo. A. Campbell for an indication of the 

 solution of this integral. ^'^ 



In (I) 



w„,ax = i^[/o(o)+A/o(io)]=P(i+A) cm. (2) 



by reference to (8) Appendix I., putting r = o. 

 Also 



Wr = P[J^{kr) -\-XJ^{ikr)] cm. (3) 



+ 2\Jo(kr)Jo(ikr) } rdr (4) 

 = 77^1^2 / JoKkr)r ■ dr -f J \U,\ikr)r • dr 



+ I 2\Jo{kr)Jo{ikr)r • dr 



i-Kp r a 



|'{/o^(M -\- JxK^a)\ 



-f- {J<?{ika) + Ji^ika) 



1-Ii :ili {kJo{ika)Ji(ka) — ikJo{ka)Ji(ika)} , (5) 



10 Bibliography (11), (12), (13). 



