of the fluid motion. Because of the periodic motion of the body, average 

 flux across the control surface must be zero. Therefore, 



II 



(U n +n»V<(>) dS = 



(241) 



where the bar means the time-averaged during one period. 



The average force in the x-direction yields the resistance to the forward 



motion of the ship. 



D = F = 



J P n x dS + p J J 



tj* (U n +n»V<j>) dS 



3x X - 



(242) 



Taking cylindrical coordinates R and 8, 



x = R cos 8, y = R sin 



(243) 



making use of the pressure equation 



P-Pr 



9t 2 



L (V<f>) 2 - U H - gz 



3x 



(244) 



and designating the elevation of the free surface by £, we obtain 



D = p J 



2tt C 



R d8 

 



3(f) 3<j> I 34 1 ,„,,2 



3^ 3R- ht + 2 (V(J5) +g2 



dz (245) 



Because of the periodic motion, we can put 



2tt 



R d6 

 



| (If*) 



cos 6 dz = 



