14 Lamb, Velocity of Sound in a Tube. 



For sufficiently long waves (53) reduces to 



Prr---^^r . . . (54); 

 and the conditions to be satisfied for r = a are therefore 



( 



and 



2Ltw2\ 2u dc . ,,0 o 2ta7n dw 



2iin dl ,^„ „,^y , ^. 



The conditions to be satisfied at the second surface may 

 be obtained in a similar manner. 



If we write 



m^-H^ = i]^, 7?t^-k^ = e . . . (57), 

 the differential equations to be satisfied by the functions 

 ^ and ^ take the forms 



^'^ I ^? „. , ^, 



d?^'7dr-''^' = ^ ' • • (58), 



d^X I '^Y o 



^. + p^-4\ = o . . . (59). 



If we aim only at a determination of the wave-velocity in 

 the fluid, as modified by the elasticity of the tube, <: will be 

 less than c^, and a fortiori less than the velocity of free 

 waves, whether of expansion or of distortion, in an infinite 

 elastic solid. Hence the quantities v, rj, Z, will be real. 



The case where the internal and external radii are both 

 finite leads to some rather complicated equations. The 

 problem is a little simplified if we suppose the external 

 radius to be infinite, i.e., we consider a tunnel bored in an 

 infinite solid. This case is also of some interest as forming 

 the other extreme to the state of things considered in § i 



