Ogilvie 



of modes, by means of an artificial external system of forces. From knowledge 

 of the force and the ship velocity in each mode, we can calculate the average 

 rate at which the external force system performs work on the ship. Since the 

 ship cannot absorb energy steadily over a long period of time, this energy is 

 then transmitted to the surrounding water, and we calculate the average rate at 

 which work is performed on the water. Finally, we visualize a large fixed 

 mathematical surface far away from the ship which completely encloses the 

 ship. There can be no average rate of accumiilation of energy in the fluid region 

 between the two surfaces, and so the rate of flow of energy out of this control 

 surface must equal the two previous rates of energy flow. 



First, suppose that the forces Fj cos (wt + h ■) are applied to the ship by some 

 external means (there are no incident waves), and let the motions be designated 

 by a.(t) = a- cos (wt + ej). (We suppose further that there is a superimposed 

 steaAy flow past the ship at speed v. Of course, there will be a net drag force, 

 but there will be no work done by the drag force, since the ship has no forward 

 speed in the coordinate system chosen.) Let the eqviations of motion be: 



'• cos (cjt + Sj) 



E |hjk + /^IkC'^)] ^k^t) + b.^(a;) a^(t) + c-^ a^( t ) 

 k = l >■ -" 



The rate at which work is done on the ship by the external forces will then be 



6 



W = Z] ^j<^t) Fj cos (a;t+ S.) 



j = i 



= -cl> y^ /_, '^i °^k ^■'■'^ ("^t + £•) 



j = 1 k= 1 



X J r-a)2(mj^ + Mjk) + c j^l cos (wt + e^) - wb*^ sin (wt + e^) ^ • 



The average value, w, over a whole cycle will be 



6 6 



^ = 1^ E H ""j^-k ■ P^n'jk + Mju) - CjJ sin (e^- e .) + wb*^ cos ( e^ - € .)\ 



; - 1 1, = 1 I -' J 



j = 1 k= 1 



We note that mji^ = mi,j , and so the generalized mass terms cancel each other, 

 due to the presence of the antisymmetric factor sin ( e^^- e ■) . Thus 



6 6 , 



^ = 1"^ Z] E ^j '^k [^V*k-Cjk] sin C^k- ^j^ + '^'^jk ^°s (e^- e.)i . 



j = 1 k= 1 l- ■" 



The rate of increase of energy of the fluid within a closed surface can be 

 written: 



dE 

 dt 



j [P<l>t(^n-Vn) -PVn] ^S 



46 



