Nonsinusoidal Oscillations of a Cylinder in a Free Surface 



Infinite series in Eq. (B-20) is truncated to a finite series to obtain 

 an approximate solution by a matrix inversion. After finding some 

 finite number^ of b^^ and c^ and using these coefficients in Eq. 

 (B-19) we find the values of Q and q. 



APPENDIX C 



Solution for the Problem of Sinusoidal Pressure 

 Distribution on a Free Surface 



We seek a solution for the following boundary- value problem: 



vV,{x,y) =0 in y < 0, 



W (x,0) - KW| = Ae^*^'*' (C-1) 



where K = to^g* A is a real constant, and K' = co' /g# K Further- 

 more we require that 



W,y(x,-oo) = 0, 



W, ~ Be'^'yeJ*^''*' as |x|— oo 



where B is a complex constant, and 



W,(x,y) = W, {-x,y). 

 If we let 



w, = W,^ +jW,^, 

 we can easily show that 



W,^y(x.O) - KW,^ = A cos K'x, (C-Z) 



W,jy(x,0) - KW,g = A sinK'|x|. (C-3) 



^The exact number is deternained in the sense of "an approximate in 

 the mean" for the function on the right-hand side of Eq. (B-20). by 

 the series on the left-hand side. 



947 



