6 Lamb, Velocity of Sound zjt a Tube. 



tube, (II) the longitudinal vibrations of the tube-wall as 

 modified (very slightly) by the presence of the fluid, and 

 (III) the radial vibrations of the system. 



In the cases (I) and (II), c^ is comparable either with 

 c^ or with BJQ, and for sufficiently long waves va will be 

 a small quantity. The formula (23) then reduces to 



^=^;^ 7e/ . . . (24), 



approximately. If we make this substitution in (6), and 

 then eliminate the ratio tcjw, we obtain 



If we substitute the value of v^ from (18), this may be 

 put in the form 





This is a cubic in c^, but we are only concerned with the 

 two smaller roots. 



For very long waves, one root is very great, and the 

 remaining pair are given by the quadratic 



where k has been written for q^c}. In the case of a gas 

 enclosed in a metal or glass tube, the right hand side of 

 (27) may be neglected, owing to the excessive smallness 

 of the ratio k/^.* The roots of (27) are then c^- and Ejq, 

 the mutual influence of the vibrations of the fluid and of 

 the tube being quite insensible. 



In the case of a liquid, however, the fraction 2aKlhB 

 may well have an appreciable, and even a considerable 

 value, and the alteration of the wave-velocity in the fluid 



* Its value for air and glass is about 2*34 x lo-^. 



