MEMBRANEiS AND PLATES 147 



Since f is a periodic function of 0, of period 2-Tr, it can be 

 expanded (for any particular value of r) in a series of sines and 

 cosines of multiples of 0, thus 



f = R Q + R l cos 6 + $ sin + ... 



+ .RsCoss0-f &sms0+ ...-, ...(9) 



by Fourier's theorem; and this formula will apply to the 

 whole membrane if the coefficients be regarded as functions 

 of r. Moreover on substitution in (8) it appears that each 

 term must satisfy the equation separately. Thus we have 

 a typical .solution 



.cos(nt+e), ............ (10) 



provided + i ! + *_ fi. _0. ...(11) 



2 z 



r 



The solution of this, which is finite for r = 0, can be found in 

 the form of an ascending series. In the accepted notation we 

 have R 8 = A 8 J 8 (kr), where the function /, is defined by 



This is known as the Bessel's Function of the sth order, of the 

 first kind. As in the case of (2) there is a second solution 

 which becomes infinite for r = 0, but in the case of the complete 

 circular membrane this of course is inadmissible. We have then 

 the normal modes 



=AJ s (kr)coss0.cos(nt + e), ......... (13) 



where k is determined by 



J 9 (ka)=0 ...................... (14) 



Similarly, taking a term 8 g sin sO from (9) we should have been 

 led to the modes 



=BJ 8 (kr)sms0.cos(nt + e), ......... (15) 



with the same determination of k. Owing to the equality of 

 periods the normal modes are to some extent indeterminate. 

 Thus, for any admissible value of k, we may combine (13) 

 and (15) in arbitrary proportions, and obtain 



?= CJ, (kr) cos (80 + a) . cos (nt + e) ....... (16) 



102 



