124 KENNELLY-TAYLOR— EXPLORATIONS OVER [April 22, 



This is a transcendental equation involving Bessel's Functions of 

 the zeroth and first orders. It is capable of being satisfied, by trial, 

 with an indefinitely great number of roots, each corresponding to a 

 possible mode of vibration with nodal circles. Fig. lA indicates 

 graphically the method of determining the successive roots of (6). 

 The points of intersection of the lower curve with the successive 

 descending branches, indicate the values of y = kr which satisfy 

 (6). In order to have the fundamental mode of vibration, there 

 must be no nodal circles, which means that the first and lowest root 

 for ka must be taken in (6). This root is at ^0 = 3.196. . . . Plac- 

 ing this value for ^a in (3) we have: 



-/o(3.i96) -0.3197 , • / N 



X = ,,.'-' I' = - -——^ = + 0.05571 numeric. (7) 



Re-entering (2) with this value of A, we have for the fundamental 

 mode of vibration of the circular diaphragm: 



zi^m^. = P{Jo(kr)-^o.oS57^Joiikr)} cm. (8) 



In Fig. iB, the abscissas correspond both to kr, where ^=1.21 cm.-^ 

 and to r in cm., the relation being as already pointed out that at the 

 boundary r=a = 2.62 cm. and At =3.196. The ordinates are the 

 numerical values of Bessel's functions as taken from Tables. They 

 also represent vibratory amplitudes of the diaphragm, taking the 

 maximum amplitude at the center (r = o) in microns, correspond- 

 ing to the heavy curve. The upper faint curve shows the graph 

 'of the first Bessel function Jo{kr) ; while the lower faint curve 

 ishows the corresponding graph of A times the second Bessel func- 

 tion, or 0.05571/0 (t^^')- Adding these two graphs, as called for by 

 (2), we obtain the heavy curve, which represents the theoretical 

 amplitude of vibration along any radius of this particular dia- 

 phragm, assuming such a scale that 1.056 corresponds to the maxi- 

 mum or central amplitude. The small circles near this curve show 

 the amplitudes observed with the aid of the vibration explorer. 



