Recently, [^6] used a quasi^steady treatment and calculated the 

 hydrodynamic properties at different angles of heel. His treatment may be 

 useful at very low frequency range. 



In the dynamic problem, two additional complications arise: 



1) The effect of non-linearity of rolling motion is not confined 

 to the equation of motion of this mode alone, but also makes the coupled sway- 

 roll-yaw equations non-linear. 



2) The existence of the position-dependent added inertia gives 

 rise to the existence of additional velocity-dependent terms which may take 

 both positive and negative values. Some of these problems have been considered 

 already by [24], [21]. 



However, if it is assumed that the frequency of ship motions is suf- 

 ficiently small (which means that the slope of the body generated waves will 

 also be small) and that the slope of the Incident waves is fairly small, then 

 it may be valid to linearize the free-surface conditions even for large body 

 displacements. There are some occasions when the oscillation frequency is 

 low, e.g., ship motions in following and quartering seas, roll motions in 

 beam seas, pitching and heaving in long head waves. 



Chapman [5] is developing such a method (JSR vol. 23, No. 1 

 also) . 



[ 3 ] has developed a three dimensional numerical method for predict- 

 ing ship motions which solves the complete three-dimensional hydrodynamics 

 problem and satisfies correctly all forward speed effects. 



The hydrodynamics problem is solved by distributing three-dimensional 

 oscillating (Kelvin) sources (which satisfy the linearized free-surface boundary 

 condtion) on the wetted hull surface. The strength of these singularities is 

 obtained by solving the hull boundary condition. It is assumed that ship 

 motions are small enou gh that the hulP boundary condition can be satisfied at 



the mean position of the hull, 



76 



