136 DYNAMICAL THEOKY OF SOUND 



necessarily periodic functions of 6, the period being 2?r, and 

 can accordingly be expanded by Fourier's theorem in series of 

 sines and cosines of multiples of 6 ; moreover it is easily 

 proved that the terms of any given rank in the expansion must 

 satisfy the equations separately. We find, in fact, that a 

 sufficient assumption for our purpose is 



u = A cos s6 . cos (nt + e), v = B sin sB . cos (nt + e), (7) 

 where 5 is integral or zero. This leads to 



(8) 



- 2 )5 = 0, 



/ 

 where {3 = ri*a?p/E. (9) 



Hence 



Since tc/a is small, the sum of the roots of this quadratic in ft is 

 s 2 + 1, approximately, whilst the product s 2 (s 2 I) 2 /c 2 /a 2 is small. 

 The two roots are therefore 



<?2 / Q 2 _ 1 \2 ,.2 



0-*+i, *-'-^J, ............ (ID 



approximately. 



On reference to (8) we see that the former root makes 

 B = sA nearly. The corresponding modes are closely analogous 

 to the longitudinal modes of a straight bar, the potential 

 energy being mainly due to the extension ; and the frequencies, 

 which are given by 



2 = (* 2 + l) / . .................. (12) 



are, for similar dimensions, of the like order of magnitude. The 

 case s = is that of purely radial vibrations. 



The vibrations corresponding to the second root are more 

 important. We then have, from (8), A + sB = 0, nearly ; thus 



A 



u = A cos s& . cos (nt + e), v = -- sin s& . cos (nt + e), (13) 



s 



with n'--. ...(14) 



2 4 



