Nonsinusoidal Osail'lations of a Cylinder in a Free Surface 



mapped according to Eq. (2) 

 b 

 



F = 2^ dxo[€|p,(x^,y„)e'^''i^ + Pg^'^'"'' } 



^( -j2o-,t -J2o-2^ , -J(o-, + o-2^^ 



+ e^jpje +p^e + p^e 



+ p^e-^^-a)VpJ] + 0(e^). (62) 



If we let 



F = c(f ,e-^"'^ + f^e'^"^^ 



9 -j2<y,t -j2o--t -j(<r,+(T,)t 



+ e^(f,e ' + f^e ^ + f.e ' ^ 



3 4 5 



+ fge-^^"'-"2^^f,) +0(e^), (63) 



then we find that 

 b 



f j = 2 \ Pi(xo(Xo,Q^),yo(\o,Qr)) dx 



Jo 



= 2^* Pi(Xo(\o,a),yo(\o,a))T(c^) da for i=l,2,....7 (64) 



where the expressions for Pj are given in Eqs. (55) through (58) 

 and 



00 



T(c.) = - a|x^sina + y l^Illil^2Jl±L sin (2n + 1) c. | . 



n=0 ^0 



If we let 



Yj = tan' (Rej fj/lmj fj) for i = 1,2,... ,6, 

 we can show that 



fje ' = j|f j|e 'e ' 



= Ifjl sin (cojt + Yj) for 1=1, 2,. ..,6 (66) 



925 



