PKOCEEDINGS 



OF THE 



Camkitrgt ^j^ilaisapl^kal ^^omtg. 



On the Oscillations of Superposed Fluids. By H. J. Priestley, 

 M.A. [Communicated by W. Welsh, M.A.] 



[Received 28 October 1909.] 



[Read 8 November 1909.] 



In his classical research "On the Theory of Oscillatory Waves" 

 Stokes* has shewn that the first order period equation for waves 

 at the free surface of a liquid is modified when we take into 

 consideration the third order terms. The present paper is under- 

 taken with the object of finding the corresponding modification 

 for waves at the common surface of two fluids ; and ascertaining 

 the effect of the small terms on the stability when the upper 

 fluid has a stream velocity. It is found that the range of wave 

 lengths for which oscillations are possible is greater than that 

 given by the first order equation. When the upper fluid has no 

 stream velocity the period given by this equation is too long 

 in the case of propagated waves and too short in the case of 

 standing waves. 



For the propagated waves the method adopted is a modification 

 of that used by Stokes in his Supplementf to the paper quoted 

 above. This method is found unsuitable for the standing waves, 

 and it is necessary to use direct approximations from the ordinary 

 equations of motion. In each case the origin is taken in the 

 undisturbed surface, with the axis of y vertically upwards. The 

 sign given to the velocity potential is that adopted by Professor 



Lamb, so that the velocity along the x axis is — ^ , 



* Math, and Pliijs. Papers, Tome i. pp. 197 et seq. 

 t Ibid. pp. 317 et seq. 



VOL. XV. PT. IV. 20 



