Nonsinusoidal Oscillations of a Cylinder in a Free Surface 



the kinematic and dynamic boundary conditions on the free surface 

 can be given respectively by 



$,(x,Y(x,t),t)Y^(x,t) - c^y + Y, = 0, (10) 



and 



$^(x,Y(x,t),t) +gY +i(€>,^ +$y) = 0, (11) 



where a constant atmospheric pressure and an absence of surface 

 tension on the free surface have been assumed. Taking the substantial 

 derivative of Eq. (11) and eliminating Y(x,t) by using Eq. (10), 

 we obtain 



$^^(X,Y(X,t),t) +g^y +2$,$^, +2$y<|>^y 



Let the equation of the cylinder surface at its rest position 

 be given by 



S{xo.yo) = f^) - Yo = (13) 



where f(xJ represents an implicit functional relation between x^ 

 and Yo through the parameters X^ and a. Then the equation of the 

 oscillating surface can be written as 



S{Xo,yo + y(t)) = f(x^) + h^(sin (r,t + sin Cgt) - y = 0. (14) 



The kinematic condition to be satisfied on the cylinder surface is 



St 



V^(Xo,yo + y(t) ,t) • n = Vn = - -^ 



(15) 



where n is the unit normal vector on the cylinder surface and points 

 into the"fluid and Vn is the nornicd component of the cylinder- 

 surface velocity. Since 



_ VS (f'(Xn),-l) 



ii-"]Vst Tvil ' 



Eq, (15) becomes 



909 



