Understanding and Prediction of Ship Motions 



(See p. 14 of Stoker (1957).) Here, n is an outward unit normal vector, and v^ is 

 the velocity component of the surface normal to itself. The surface may be a 

 physical surface, always containing the same fluid particles, in which case 

 (D^ = v^, it may be a mathematical surface following an arbitrary prescribed 

 law, or it may be a combination of real and mathematical boundaries. The same 

 formula may be interpreted as the energy flux rate across a non-closed surface. 

 However, one must be careful to note that a positive flux rate is to be taken in 

 the direction opposite to the standard normal vector. 



On the ship hull, which is a physical surface, we have ^^ = v^, and so the 

 average rate at which the ship does work on the water is: 



s 



As a control surface far away, we take a vertical right circular cylinder 

 extending from the free surface far down into the water, capped on the bottom by 

 a flat horizontal surface. This is a fixed (mathematical) surface on which v^ - 0. 

 Furthermore, we assume that all disturbances vanish sufficiently rapidly with 

 increasing depth so that there is no contribution at all from the deep horizontal 

 surface. Then the average rate at which energy passes outward through the con- 

 trol surface is: 



W = - p<D. (f„ dS 



where l denotes the cylindrical control surface. (Physically it is clear that no 

 energy can pass through the free surface. Mathematically this statement follows 

 from the fact that the free surface is both a physical surface, so that 0^ - v^, 

 and a zero-pressure surface.) 



We now have three expressions for w. Actually, we do not need the second 

 one, for the desired resvilt comes from equating the first and third: 



6 

 i = 1 k = 1 



E ^i'^k I ['^'V-k-c-i^jsinCei^- ep + wb-kCOsCe^- e.) . ^-pj cD^ <D^ dS . 



(20) 



We have here one eqviation relating all of the hydrodynamic coefficients in 

 the eqviations of motion with an integral of the velocity potential far away (at 

 "infinity," for talking purposes). The problem still remains of separating as 

 far as possible the various coefficients in (20). At the beginning of this deriva- 

 tion, it was assumed that the ship was forced to oscillate in an arbitrary mode 

 or combination of modes by an external force system. Because of this arbi- 

 trariness we can separate a number of special cases of (20), by selecting the 

 amplitudes a.^ and the phases e . in appropriate ways. 



First, let us assume that only one particular a. is non-zero. If o is the 

 velocity potential for such motion, then 



47 



