Lee 



^X^^O'Vo ■*" y(*)'^)^'W - ^y= - ho(or, COS 0-,t + CTg COS (Tgt) . (16) 



In terms of the stream function which Is the harmonic conjugate of 

 $ the boundary condition on the body is 



84j _ 8J _ _ dy(t) 8x 

 9s 3n dt 9s 



where s is the arc length of the cylinder contour in the counter- 

 clockwise direction. Thus we have 



To complete the specification of the boundary-value problem, 

 following conditions should be also given; the symmetry of the flow 

 about the y-axis implies that 



$(x,y,t) = $(-x,y,t) , 



the zero normal component of the fluid velocity on a rigid surface 

 at the infinitely deep horizontal bottom is described by 



^y(x,-oo,t) = 0, 



and the solution should represent outgoing plane waves as |x| -♦ oo. 



III. PERTURBATION EXPANSION 



Assume a frequency- response system in which the relation 

 between the input X and the output Y is given by 



Y = AX + BX^ 



where the input X is given by 



X = e-J"-' + e'J^a' 



ajid A and B are constarits; it can then be shown that the frequency 

 components involved in th output Y are cr| , CTg, 2(r, , ZcTg* cr, + Cg. 

 (T, - o-g, and a "d.c," sh?ft. Therefore, we make a perturbation 

 expansion of the complex velocity potential 



910 



