Vibrations of a Rectangular Sheet of Rotating Liquid. 297 



In (18) )3i2 is supposed to be of not higher order of 

 small quantities than ls , ^ 2 s- F° r example, we are not at 

 liberty to put |3 12 = 0. 



In the above we have considered the modification intro- 

 duced by the j3's into a vibration which when undisturbed is 

 one of two with equal frequencies. If the type of vibration 

 under consideration be one of those whose frequency is not 

 repeated, the original formulae (6), (7) undergo no essential 

 modification. 



In the following paper some of the principles of the present 

 are applied to a hydrodynamical example. 



XXVIII. On the Vibrations of a Rectangular Sheet of 

 Rotating Liquid. By Lord Rayleigh, OJL, F.R.S.* 



THE problem of the free vibrations of a rotating sheet of 

 gravitating liquid of small uniform depth has been 

 solved in the case where the boundary is circular f. When 

 the boundary is rectangular, the difficulty of a complete 

 solution is much greater ; but I have thought that it would 

 be of interest to obtain a partial solution, applicable when the 

 angular velocity of rotation is small. 



If f be the elevation, u, v the component velocities of the 

 relative motion at any point, the equations of free vibration, 

 w r hen these quantities are proportional to 



e wt . are 



(i) 



io-u — 2nv= — g dtydx, 

 icrv + 2nu = —g d%jdy, 

 and 



<n d% *-m 



dx? + dy 2 + gh >- U > :' • ■ ' W 



in which n denotes the angular velocity of rotation, h the 

 depth of the water (as rotating), and g the acceleration of 

 gravity. The boundary walls will be supposed to be situated 

 at x= ±^7r, y— +y lt 



When n is evanescent, one of the principal vibrations is 

 represented by 



u. = cos %, v = 0; (3) 



and f is proportional to sin <#, so that 



a 2 =gh (4) 



This determines the frequency when ;* = 0. And since by 



* Communicated by the Author. 



f Kelvin, Phil. Mag. Aug. 1880; Lamb, 'Hydrodynamics,' §§ 200, 

 202, 203. 



