Ogilvie 



3cp 1 



k= 1 



6 , 



k = l ^ \ OXj / 



6 



- E ak(t) {(-V 3^ + V(p„ -v) 02k(^) ■ 



6 ^t 



k= 1 ''- 00 



k = 1 -^ - 00 



i^(r) 



Vr) 



^ + -V 



_3_ 

 3t 



3x, 



3x, 



+ Vcp„ • V 



(x, t - t) dr 



(x, t t) dr. 



The first line on the right-hand side represents a steady pressure. The second, 

 third, and fourth lines, respectively, depend on the acceleration, velocity, and 

 displacement in the six modes of motion. The last two lines are convolution in- 

 tegrals, involving the whole past history of the velocity and displacement in the 

 six modes. 



The computation of force and moment components has been relegated to 

 Appendix A, because it is rather tedious and does not add much perspicuity to 

 the result. There are two problems which may be mentioned here: 



1. The pressure must be evaluated on the instantaneous position of the ship 

 hull and not on the mean position. Similarly, the instantaneous extent of the 

 wetted hull surface must be calculated and used as the domain over which the 

 pressure is integrated. 



2. Since we wish to use the force and moment resvilts to write down equa- 

 tions of motion, we must express these quantities in terms of an inertial refer- 

 ence frame. The geometry of the ship is most easily described in a reference 

 frame attached to the ship, but this is accelerating and therefore it is not ac- 

 ceptable. The procedure adopted in Appendix A is to calculate the force and 

 moment components with respect to the moving axes and then to use standard 

 transformations to express them in the steadily translating Newtonian system. 



The six components of force and moment are written out in Eqs. (Al) - (AlO) 

 of the Appendix A. The form of these components is as follows: 



32 



