i6 Lamb, Velocity of Sojtnd in a Tube. 



on the hypothesis that the strains in the sohd have at each 

 instant the equilibrium values corresponding to the internal 

 pressure. In the case of a cylindrical cavity bored in an 

 infinite solid, the statical value of the displacement at the 

 surface {r = a) due to an internal pressure/ is 



W=pal2y. . . , . (64).* 



Comparing this with (24) we obtain the results (62) and 



(63). 



The corresponding approximation for a tube whose 

 internal and external radii {a and U) are comparable with 

 one another is easily investigated. If the tube be subject to 

 a statical pressure on the interior, the radial displacement 

 {w) at the inner surface is found, on the hypothesis that 

 the tube is free to contract longitudinally, to be 



^= 2^(3\+2;.)l^2_^2) PC^ . • (65). 



Compared with (24), this gives 



c^ r .• . (x+2^K + (3X-f2^)^2 ) 



?='"^\''-/. " (3X + 2;.)(^^-^^) j • (^^)- 

 In the case oi b = a-\-h, where h\a is small, this becomes 

 c^ I 2aK\ . 



in agreement with (31). 



See Love's Elasticity, t. i., p. 226. 



