388 Vibrations of a Cylindrical Vessel containing Liquid. 



which a special condition must be satisfied. Hence arises a 

 vertical motion of the surface, which is the proximate cause of 

 the " crispations " usually to be observed under these circum^ 

 stances. In considering this question we may leave the force 

 of gravity out of account, inasmuch as the period of free waves 

 of length comparable with the diameter of the cylinder is much 

 greater than that of the actual motion. 



In accordance with (5), if the fluid be treated as incom- 

 pressible, we may take 



(f> = a n cosp£cosn0r n + 2A K coSj?£cos n6 e~ KZ J„(#r), . (20) 



in which z is measured downwards from the surface, and k, 

 denotes a root of 



J' n («a) = (21) 



The coefficients A K are to be determined by the condition at 

 the surface, which is simply <£=0. Thus for each value of k 



a n ( a r n+1 J n (fcr)dr + A K Cj 2 n (Kr)rdr = 0. . . (22) 

 Jo Jo 



Now (see < Theory of Sound,' §§ 203, 332) 



\ r n+1 J n (fcr)dr= ^- 3 n (jca), 



Jo * 



^Jl(,crydr = W (l- jjfe) JXtca), 



so that 



^ = a „ co.pt cos n6 { r^na^ ^Zi^ia) } ' " ^ 



To calculate the kinetic energy we have to integrate <£ d<f)/dn 

 over the whole boundary of the fluid. Now at the free sur- 

 face </> = 0, and at a great depth the motion becomes two- 

 dimensional. We have therefore only to consider the cylin- 

 drical surface. By supposition J n / (fca) = Q, and thus 



dd> d4> n . , a 



V 1 = -Tj- = nu n a n ~ l cos pt cos n 6. 

 dn dr l 



We get therefore 



l^i/Balno^cosVJ I { 1 " 2n ^ ^2^ } ^s 2 nd dd dz 



*frp&#* cos 2 1 ,*{s-2n2 i? £ 7? }. . . . (24) 

 The value of T is less than if the motion were strictly two- 



