122 KENNELL Y-T A YLOR— EXPLORATIONS OVER [April 22, 



a„ = a phase-angle measured around the diaphragm (radians), 

 (0 = 27r;? = angular velocity of vibrating motion (radians/sec), 

 w= frequency of diaphragm vibration (cycles/sec), 

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

 ^ = a time-phase determined by the epoch (seconds), 

 .a = radius of the diaphragm (cm.). 



For the fundamental mode of motion, n=o; or there must be no 

 nodal diameters. Consequently (i) reduces to: 



Wo = P{Joikr) + \Jo{ikr) ] cos {wt -f- e) cm. (2) 



Here the amplitude of vibration at any point Wo, ceases to be a func- 

 tion of 6, and depends only on Bessel functions of r. Since we 

 shall consider only the fundamental mode of vibration in what fol- 

 lows, the subscript will be unnecessary, and we may substitute w 

 for Wo. 



Continuing Lord Rayleigh's method of demonstration, if a flat 

 circular diaphragm is clamped at its edge between a pair of flat 

 circular rings, then, referring to (2), we have w vanishing at r^=a, 

 the clamping radius, and since there is to be no bending or slope of 

 the diaphragm at the clamped boundary, we have also (dzu/dr) =0 

 at r = a. 



Entering (2) with w = o, we have: 



Jo(ka) . 



X = - numeric. (3) 



Jo{ika) 



Also dififerentiating (2) with respect to r, for r = a, we obtain: 



whence 



I div 



- "1- = Jo'ika) -{- i\Jo'{ika) = o numeric, (4) 



Joika) . 



X = - ~j~,y^^ numeric. (5) 



Combining (3) and (5) we obtain: 



Jo(ka) Joika) Ji(ka) 



Joiika) iJo{ika) iJx{ika) 



numeric. (6) 



