''-^,- • ■ ■ (9). 



4 Lamb, Velocity of Sound in a Tube, 



a quadratic in c^. If, as is assumed throughout, the wave- 

 length be large compared with the circumference of the tube, 

 m^a^ will be a small fraction, and the roots of (7) will be 

 approximately 



.2^(,_^2)^^^ . . . (8), 



P P 



where E denotes Young's modulus, and 



respectively. In the case of (8) we have, by the former 

 of equations (6), 



w/u= - lama . . . (10), 



This ratio is therefore small, and the vibrations are mainly 

 longitudinal. The wave-velocity given by (8) is, in fact, 

 that of longitudinal vibrations in a bar. If we continue 

 the approximation, we find, in place of (8), 



^2 = (i-A2V)^ . . . (11). 

 P 



On the other hand, the solution (9) makes 



ulw= - iama . . . (12), 



so that the vibrations are chiefly radial. The corresponding 

 type of motion is, indeed, best described as a radial 

 vibration, with a very gradual variation of phase as we 

 pass along the tube ; and the result is best expressed in 

 terms of the " speed." Thus if we put n = 7nc, so that n/27r 

 measures the frequency of the vibrations, we have 



.^=1 ., . . . (13)* 



The next approximation gives 



n^={i+a^m^a^)^^ . . . (14). 



* The results (8) and (9) are given by Love, Theory of Elasticily,. 

 t. ii., p. 259. 



