2 Lamb, Velocity of Sound in a Tube. 



applying the general equations of elasticity to this case is 

 indicated in § 3. In order to diminish to some extent the 

 complexity of the analysis, attention has been directed 

 mainly to the case where the external radius is infinite. 

 This has, moreover, a special interest as forming the 

 opposite extreme to the circumstances considered in 

 the previous part of the paper. Even here the final 

 equation for determining the wave-velocity is of a some- 

 what unmanageable character ; but it leads to a very 

 simple result when the wave-length is very great compared 

 with the circumference of the tube. It is to be remarked, 

 indeed, that the more definite results obtained throughout 

 this paper are, as a rule, of the nature of limiting forms 

 which are approximated to more closely the greater the 

 wave-length. In some of the experiments on the subject, 

 the ratio of the wave-length to the circumference of the 

 tube cannot be said to be very great ; a correction on this 

 account, when the ratio is moderately large, might be 

 investigated without much difficulty. 



I. Let the axis of x be taken along the axis of the 

 tube, whose thickness (//) will for the present be assumed 

 to be small compared with the radius {a). In the defor- 

 mations to be considered, the displacement of any point 

 of the wall will be in a plane through the axis, and if u 

 and w denote its components in the directions of the 

 generating line and of the radius, respectively, these 

 quantities will be functions of x and / only. The linear 

 extensions in the tangent plane, along and perpendicular 

 to the generating line, will be dujdx and wja, respectively. 

 Hence if a denote Poisson's ratio (of lateral contraction to 

 longitudinal extension in a bar of the same material), the 

 tensions called into play in these two directions wdll be 



* See Proc. Lo7id. Math. Soc, t. xxL, pp. 137, 138 (1890.) 



