Nonsinusoidal Oscillations of a Cylindev in a Fvee Surface 



We expand Pj (i = I,II, . . . , VII) In Taylor series about the mean 

 position of the cylinder (x^.y ) and substitute 



y(t) = hQ(sin o-|t + sin o-gt) 



= eb(sin a-^t + sin o-gt) 



in this expansion. We rearrange the terms in powers of e and in 

 time harmonics to obtain 



P= e|p,(x,,,yo)e ' + p^e ' | 



o( -]2<r,t -j2cr,t -j(<7,+(r-)t 



From the Bernoulli equation. 



+ Pee • ' - +P7(xo,yo)} +0(e^. (54) 



P = - P^t(xo'yo +y(t),t) - Pg(yo + y(t)) - 7 (^x + ^y)' 



we eliminate the static pressure pgyo. expand the right-hand side in 

 accordance with Eq. (17) and equate terms which are of the same 

 power of e and of the same time harmonics. We then find the ex- 

 pressions for Pj (i = 1 , 2, . . . , 7) in terms of velocity potentials <p■^ 

 and their derivatives. The expressions for these Pj's are 



Pj = Jp(o-i^i(xo,yo) - gb) for i = 1,2 (55) 



Pi*2 =- p{-J2°'i^i*2'^-^^iy +^<^i^x +^fy)} f°^ ^=^'2 (56) 



Pj = - P {- j(o-, + <y^<P^{^^.Yo) +7 (<P,x?'2x + ^iy^2y) 



+ ^(«r,<p„+ag^gy)}, (57) 



Pe= - p{-j(o-, - 0-2)?'6(xo.yo) +7(^»x^2x'^*i'ly^2y) 



-l(-.^.y+-a^2y)}» (58) 



Z 



P7 = 7/^ {b(rj?'j2y(xo,yo)-2(^ix^ix + S^iy^iy)} ' '59) 



923 



