186 Mr. Nikhilranjan Sen on a Type of 



cylinder. This consideration gives 



a^/a*+cB^/an=o, (i) 



where cf> is the velocity-potential and c the velocity of wave- 

 propagation. There is also the condition 



ct> = constant, (2) 



which should also be satisfied on the surface of discontinuity. 

 We shall now proceed to the solution of the problem. 



3. 



Let us first suppose the shell to be of finite thickness 

 bounded by the two surfaces r = a and r = b, (a<b), the 

 interior of the shell being filled with a gas of density p u 

 while the shell itself is surrounded by a gaseous atmosphere 

 of density p. The motion of the gases inside and outside the 

 shell should satisfy the wave equation ' 



^t=< 2 V 2 </>, (3) 



while the equation of motion of: the shell (supposing the 

 vibration to be radial) is * 



u being the radial displacement and 8 the density of the 

 material of the shell. If: <pi and <f> be the velocity-potentials 

 at the internal and external space, the surface conditions are 



'du/'dt— —"d^ifdr at r — a 



'du/'dt=—'d(j)l'dr at r=b 



fr = pi"d<pif5t at r = a i 



fr = — />c)</>/cK at r-=bj 



where fr~ (X-f- 2p,)'du/'dr + 2\u/r. 



If u is proportional to e tkct , the equation of motion of the 

 shell is reduced to 



~d 2 u 2 ~du 2u 



~dr 2 r "dr r 



where tf = k 2 c 2 8/{\4-2fL)==kc*/v 2 , v =[(\ + 2ji)/8f 



so that v is the velocity of propagation of the wave of 

 dilatation within the material of the shell t- If we call the 



* The notations used here are those of Love in his ' Theory of Elas- 

 ticity,' 2nd ed. 



t Love's ' Theory of Elasticity/ 2nd ed. p. 282. 



k • • • (5) 



r-:;:w ^+^ = ' • • • • ( 6 >' 



