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LVI. On the Vibrations of a Cylindrical Vessel containing 

 Liquid. By Lord Rayleigh, F.R.S., Professor of Expe- 

 rimental Physics in the University of Cambridge* . 



THE problem of a uniform cylinder vibrating in two dimen- 

 sions is considered in my book on the Theory of Sound, 

 § 233. If the displacements at any point a, 6 of the circum- 

 ference be Sr, aS6, then for a single component 



8r = aA n co$n0j 80= — w _1 A w sin nQ. . . (1) 



If d be the thickness, and cr the volume-density of the ma- 

 terial, the kinetic energy of the motion for a length z measured 

 parallel to the axis is 



T=i^daa\l + n^){^) (2) 



The corresponding potential energy is 



v=, £v- i ) 2A »> 



in which B is a constant depending upon the material and 

 upon the thickness. As a function of thickness B oc d 3 ; so 

 that w r e may write B = B <i 3 , in which B depends upon the 

 material only. Thus 



V=E ¥ 1 ^- 1 ) 2A » ( 3 > 



If the cylinder be empty, these expressions suffice to determine 

 the periods of vibration. Thus, if A n oc cos p t, 



2 _ B cP (n 2 -!) 2 

 p o~ aaf l + n~ 2? W 



showing that for a given material the frequency is proportional 

 to the thickness and inversely as the square of the radius. 



If the cylinder contain frictionless fluid, the motion of the 

 fluid wall depend upon a velocity-potential </> which satisfies 

 the equation 



(d? 1 d 1 d 2 t d 1 , A , A /KX 



W + r-oh* + 7 2 W 2 + a& +K r = > ' * * ( 5 > 

 in which 



K=a / ~ 1 p, (6) 



a! being the velocity of propagation of sound within the fluid. 

 If the fluid can be treated as incompressible, we may put k = 0. 

 For the present purpose we will retain k, but we will assume 



* Communicated by the Author. 



