8 Lamb, Velocity of Sound in a Tube. 



As a numerical example, let us take the case of water 

 in a glass tube whose thickness is one-tenth of the radius. 

 Assuming 



^'= 2"22 X io'°, ^=6-03x10^^ ^ = 6*46x10" 



in C.G.S. units*, we find that the velocity of longitudinal 

 waves in the tube-wall must lie between 



{E\.^Y and i-o35(^/p)*, 



and that the wave-velocity in the liquid will lie between 



•744^, and 770^,. 



If we actually solve the quadratic (27) we find, with 

 the same data, that the accurate value of the former 

 velocity is 



rois{Elp)\ 



and that of the latter 



•758^0. 



To avoid a possible misconception it may be well to 

 point out explicitly that the forced vibrations of the tube- 

 wall, due to waves of expansion in the contained liquid, 

 are, under the present conditions, mainly longitudinal. 

 This follows from the former of equations (6), which shews 

 that the ratio wju is of the order 7na. An increase of 

 pressure in any part of the tube tends (it is true) to produce 

 in the first instance a radial enlargement, but this in turn 

 tends to produce a longitudinal contraction ; and, owing 

 to the length of the waves, and to the relatively great 

 velocity of wave-propagation in the solid, this latter effect 

 is cumulative. 



If we had assumed that the strains in the tube have 

 at each instant the statical values corresponding to the 



* The numbers refer to a specimen of flint glass whose elastic constants 

 were determined by Everett (see his Units and Physical Constants). 



