Nonsinusoidal Oscillations of a Cylinder in a Free Surface 



FgCz) = Wg(x,y) +iW*(x,y) 



a e'^ y j(K'lxl-^) ^ .. j(K'lxl-iQ) 



. a^e'J^ r e"^'E,(iK'z) , e"'''E,(-lK'z) 1 

 -J^r— L K' + Kg "^ K' - Kg J 



+1 r H (e)e-"'«^'-^^ E,(-iK (z-e)) de + 21 Th (e)e-"'^^-^^ de 



+ (j-i) P hg(e)e""'«^'"^^d|, (39) 



where ± signs correspond to x $ and hg(x) =0 in -b<x<b. 



4. 2 Solutions for y^ and y^ 



We will now show how to use the solutions obtained in the pre- 

 ceeding section to find <p, and <p , We introduce a new harmonic 

 function G. defined by 



G^(x,y) = ^^- W^ (40) 



in the domain of y < except for the portion occupied by the cylinder. 

 Here the subscript k can be either 5 or 6 unless specified as one or 

 the other. The boundary-value problems posed in Sections 3,1 and 

 3.2 can be written in terms of G. as 



G^y(x,0) - K^G^= 0, (41) 



GkK'yo)f'K) - GKy= n.,(^o'yo) - (W,,(x,.y,)f(x,) - W,y), (42) 



or in terms of the harmonic conjugates of G,^, denote by G,^ , the 

 above boundary condition can be written as 



^5(^0' Ye) = - J I (^lyt^o'^o) ^ *2y) " ^5 = ^^(x^'yo) ^^^a) 



and 



G6(-o'yo) = J 7 <*.yK'yo) - \) - K " Bg(x^'yo)- t^^b) 



Furthermore Gj^ is even in x, G|^ -* as y -* - 00, and G^ 



919 



