Nonsinusoidal Oscillations of a Cylinder in a Free Surface 



H(z,t) = $(x,y,t) + ii|j(x,y,t) 



in terms of a perturbation parameter e = h^/b in the following 

 fashion: 



= e(^^'^ + 14.^'^ ) + €^4)^'^ + i^'^') + e^(<^^^^ + ic^^^M + ... (17) 



Each velocity potential and stream function given above is further 

 expanded as 



$<" = (j),{x,y,t) + 4)2(x,y,t) 



-j<r,t -jCTjf 



= ^i(x,y)e + <p^x,Y)e "^ , {i7a) 



^ = <l>3(x,y,t) +<|)^(x,y,t) +4)5(x.y,t) +<f>g{x,y,t) +(t>^(x,y,t) 



-J2<r,t -i^o-gt -j(<r, + o-2)t 



= ^3(x,y)e + <p^e +^5© 



+ ^ge"^^'''"*"2^V<p^(x,y) (i7b) 



4i^"= 4>,(x,y,t) +4j2(x,y,t) 



= >I',(x,y)e"^'''^ + *2(x,y)e"^''2^, (17^) 



i|j<2'= 4;3(x,y,t) +i|;^(x,y,t) + i|j5(x,y,t) + 4ig(x,y,t) + ^^(x,y) 



-j2o-,t -j2<r,t -j(cr, + 0-5)1 



= *3(x,y)e '+^46 ^ + %e ' ^ 



+ ^ge ' 2 +^7(x,y) {17d) 



etc. , where 



for k= 1,2,.. .,6, and j = V^. In these expressions , only the real 



911 



