200 Mr Cookson, The Oscillations of a Fluid 



The Oscillations of a Fluid in an Annular Trough. By 

 B. Cookson, B.A., Trinity College. 



[Bead 6 May 1901.] 



The wave motion of a fluid in a circular trough can be dis- 

 cussed by the use of Bessel's Functions of the first kind. It may 

 be found in Lamb's Hydi^odynamics, Art. 187: but the case of an 

 annular basin is passed over with the remark that it is " easily 

 treated, theoretically, with the help of Bessel's Functions of the 

 second kind." The analysis has therefore been carried out and 

 applied to a particular case. The case is that of a trough contain- 

 ing mercury into which another annulus fits : this second annulus 

 is floated by the mercury and on it is mounted an astronomical 

 telescope, the combination forming a floating Zenith-Telescope. 

 It is of interest to compare the period of a free wave of the fluid 

 in the trough with the observed period of oscillation of the floating 

 instrument and to know the character of the contour lines in the 

 case of the free wave. 



The fluid is assumed to start from rest and to be frictionless : 

 the motion is irrotational and the velocity potential <f) satisfies 

 Laplace's equation 



V 2 c/) = (1). 



The motion is further imagined to be so small that the squares 

 of the volocities may be neglected, so that the dynamical equa- 

 tion is 



* P +3- L * < 2 > 



Here z is measured vertically upwards from the free surface 

 when the liquid is in equilibrium. These two equations determine 

 the small oscillations. 



