Motion and Resistance of a Low-Wat enplane Catamaran 



where (J) e and (p o are even and odd functions in y , respectively, 

 and satisfy the following body boundary conditions 



6 I = A' e ° Z (n cos K y'- ni sin K y') (84) 



enL 3 o 2 o 



c o 



<$> | = - j A' e ° ( n' cos K y' + n sin K y') (85) 



on I _ 2 o 3 o 



c o 



The potentials <P e and (p also independently satisfy the remaining 

 conditions prescribed for (Drj . With this arrangement, we can easi- 

 ly relate <y e to the problem of heaving twin cylinders and relate y 

 to the problem of swaying twin cylinders with the only trivial differen- 

 ce being the magnitudes on the right side of Equations (84) and (85). 



The solution for oscillating twin cylinders can be obtained 

 by using the method of source distribution along the contour C . 

 The expression for the source is given by 



= J_ [/„ { (y- v) z + (z-n 2 } +i J(y+>o 2 + (z- n 2 \ 

 47r L {( y +r,) 2 + (z+ n 2 \~ {(y+nf + ( z + n z } 



+ —^r~ ~h ^r 7- S cos k (y- ?7 ) ± cos k (y+ V ) > dk 







{ cos K Q (v-v)± cos K q (y+ v ) } (86) 



- J e 



where the plus sign corresponds to heave (G ) and the minus sign cor- 

 responds to sway (G ) , and J 1 means Cauchy's principal-value 

 integral. 



Let 



*e = i Q e °e d£ < 87 » 



♦o = J Cr °o °o di < 88 » 



where C R is the integral along the immersed contour of the right- 

 half of the cross section, and Q e and Q are the unknown source 

 strengths. Applying the boundary conditions given by Equations (84) 

 and (85) to Equations (8 7) and (88), respectively, and solving the 

 integral equations for Q e and Q , we obtain the solutions of V 



537 



