Nonsinusoidal Osoillations of a Cylinder in a Free Surface 



where 



. /->00 ny ,. 



M^ = \ .-^ dp + lire " sm K x, (47) 



ko J^ p - K^ ^ -^ k ' ^ ' 



T,,* sin Ima 



M,,_ = - — 



km ^ 



00 



mof , „ I cos ( 2m. - i)a 

 ^ """^M (2m- 1)'^— 



V (2n •*'i)apn^ cos (2m ■*-2n •H)q' ) 



"4 2m+2n!^l \2m*2n+i / 



n=0 



for m > 1 . (48) 



Our task is now to find the unknown coefficients bi,_ and c,,_ and 

 the phase relationship q^^ from the boundary conditions on the body- 

 surface given in Eq. (42). Since the boundary conditions on the body 

 are simpler when written in terms of the stream functions Gy^ than 

 in terms of the potential functions G|^, we use the Eqs. (42a) and 

 (42b) to obtain these unknown quantities. Thus we find that 



00 



V^ % JQk 



I (^m+JSm)Mk„(x,.y,)e =B^(x^.y^). (49) 



m=0 



Since the q|^'s are independent of the points (xQ.yg) on the body, we 

 see by choosing an arbitrary point on the body, say (Xq(\ ,Qr'), 



e'^'^= - BhK>yi) (50) 



2 (bK, + jc,jM*„(Va') 

 m = 



Substitution of Eqs. (50) into (49) yields 



00 



mT6 ^kK'Vi) 



and use of the earlier definition of b|^Q= Q^ and C|^q = In this 

 equation yields 



00 



I 



= BhUn>yn) / 0°° e ° sin px^ ^p + ^^e^^k^'o ^-.^ K,x» - 



921 



