lo Lamb, Velocity of Sound in a Tube. 



Hence we may put 



n = zcja .... (34), 



where ^ is a root of 



B 



,f ap, Jjz) \ B 

 V hp zj:{z)]-pc,^ 



(35), 



Jq denoting the ordinary Bessel's Function of zero order. 

 If the fraction a^J/tg were small, one root of this would be 



approximately ; this root determining the frequency of the 

 radial vibrations of the wall as modified by the presence of 

 the fluid. The remaining roots are in the neighbourhood 

 of those of 



and correspond to the various radial vibrations of a 

 cylindrical mass of elastic fluid, as modified by the want 

 of rigidity in the boundary. It is easy to continue the 

 approximation, and so to estimate the extent of the modi- 

 fication in the several fundamental types of radial vibra- 

 tion, but it is hardly worth while to write down the results, 

 as in cases of any interest the fraction a^JIiQ would not 

 be small. The only plan would then be to solve (35) by 

 iheans of the tables of Bessel's Functions. 



2. We have so far investigated only the case of fluid 

 internal to the tube. When, as in Wertheim's experiments, 

 the tube is immersed in liquid, we shall have in place of (6) 



(36), 



where/' denotes the external pressure. The solution of 

 (17) appropriate to the external space is 



ii = DKlvr) . . . (37), 





('- 



B\ iaB 



\u^ w- 

 p / map 



--0 





\ 



icr 

 ma 



B 



~u + 

 9 



('•- 



iB \ 



—5—,— mi-- 

 m^a^ p J 





m 



-p' 



