(41) 



and 



-xm 3 + n 3 (42) 



The first two potential terms in parentheses in Equation (40) represent the exciting 

 forces, and the other two potential terms are due to the added-mass and damping 

 forces. Complete lists of added-mass and damping forces are given in Table 1 of 

 Reference 6. Further, the detailed derivations of all equations not derived in this 

 report are given in References 4 and 6. 



SINGULARITY OF THE KERNEL FUNCTION 



The procedure employed here to compute the hydrodynamic forces and the motion 

 begins by solving for G„ and a„ in Equations (14) and (18) through the use of strip 

 theory. The kernel function f (x) of Equation (28) is computed and retained for 

 later use. Next, the three-dimensional source strength q . (x) is computed by solving 

 the Fredholm Integral Equation (27). Once q . (x) is obtained, the sectional inter- 

 action coefficient C.(x) may be computed through Equation (26). 



The most difficult and time-consuming part of numerical computation is the 

 evaluation of the kernel function Equation (28) and the three-dimensional source 

 strength Equation (27) . In a previous publication the present author presented a 

 numerical method for the evaluation of the kernel function, Equation (28), through a 

 small-order procedure in order to avoid the l/|x|-type singularity. However, this 

 method is not generally applicable to the SWATH ship motion problem since the so- 

 lution is not uniform, i.e., the predicted response depends upon the size of the 

 small-order parameter. In order to avoid this singularity, Equation (27) may be 

 integrated by parts to yield 



.+0 . 



q.(x) - i P^ -q.(OF(x-0 



3 \2t\o. J ( J 



L/2 



- L ' 2 L 



k 



+ |q!(OF(x-£)dE; = a + a (j=3,5) (43) 



where q! (x) is the derivative of q . (x) with respect to x; the function F(x) is the 

 integral of Equation (28) with respect to x: 



12 



