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diaphragm and cj, is the i-th frequency of the free diaphragn (all modes of 

 vibration are radially symmetric). The matrix ^c\ depends on "^ only 

 (except for zeros of J"^ ), therefore J^ the roots of Z^ depend only on "X , 

 and consequently the same for the coefficients C^- • 



It is of particular interest to calculate the maximum central 

 deflection 



^H^ = 2:(o,-e^') , (17) 



where "6^ is the smallest positive value of t, satisfying 



2-z(o.rir)=:0 . (18) 



tyt 



It happens that Zh, is given by the relatively simple formula 



(19) 

 in which "t^ can be obtained by solving Eq, (18) by successive approximation. 



The permanent plastic deflection of the diaphragm, as has been 

 previously shown, will differ from aiC^jt^^) by a small amount, usually 

 less than one percent. It can be shown that the center point of the dia- 

 phragm reaches its maximum deflection first. As the rate of strain changes 

 sign at the center an unloading wave travels out toward the center with 

 acoustical velocity. The diaphragm passes in a very short time from the 

 plastic to the elastic state during the unloading and continues to vibrate 

 elastically until the residual kinetic energy is radiated. Since the elas- 

 tic recovery is small the diaphragm takes a permanent set in very nearly 



32 



