Vibrations in Solid and Hollow Cylinders. 



345 





y,(*m) 



2p 2 A 



J,(«X) 



2p 2 X 



a (^ — i> ) 



m — n \ 2 T , 



Y^aA) 



a 2 (^ 2 -l' 2 ) 

 2x> 2 X 



%_ X 2 



2nfx 

 m— n 



Y,'(6X) 



h — w. X 2 



2n/u oV J a 2 // 



This equation is true irrespective of the relative magnitudes 

 of a and b. It constitutes a frequency equation supplying 

 values of k which apply to the type of vibrations consistent 

 with the surface conditions. If both ends of the rod be fixed 

 there is no restriction to the absolute values of a/l and b/l; 

 but if one or both ends are free, such a restriction is really 

 involved in the fact that unless ia/l and ib/l be both small — 

 i being the order of the harmonic of the fundamental note 

 under consideration— the failure to satisfy exactly the terminal 

 condition ^ 



zr — 



involves an inconsistency which cannot be allowed. 



§ 12. The case of a thick rod fixed at both ends is of little 

 physical interest, and the treatment of (38) in its utmost 

 generality would involve grave mathematical difficulties. 

 I thus limit my attention to the case when ia/l and ib/l are 

 both small. This implies that a\, b\ ayb i and b/n are all small. 

 Thus in dealing with the various Bessel functions we may use 

 the following approximations*, which hold so long as the 

 variable x is small, 



J (#) = 1-^/4, 



^(^ = 1^(1 -<z> 2 /8), 



Y (*) = (1-^/4) log tf + 074, ^ ' ( 39 ) 



:0. 



(38) 



YiW = |(l- 



x 2 / '8) log x— # _1 — w/4tj 



Y/(a ? )=i(l-|^)log^ + ^- 2 + |. J 



Ketaining only the principal terms in (38), we are of course 

 led at once to the first approximation (1). Again, if (1) held 

 exactly w T e should find 



« 2 =p 2 n (3m — n) -f- m (m + n) , 



fju 2 — p 2 =p 2 (m — n) /m ; 

 * Cf. Gray and Mathews' ' Treatise on Bessel Functions,' pp. 11, 22, &c. 



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