at a fixed point. The velocity potential of the fluid motion in this case 

 is expressed by Ux + <{) as before. Consider a vertical circular cylinder 

 of large radius with its axis through the origin of the coordinates. Tak- 

 ing the immersed part of the surface of the ship S, the cylindrical 

 surface Z, and the portion of the free surface Z , inside the surface Z, as 

 the momentum control surfaces, we apply the momentum principle to the 

 fluid contained in the space bounded by these surfaces and consider the 

 fact that there is no flux across S and Z and the pressure is constant 

 on Z Q . Then the rate of change of momentum M of the fluid enclosed by 

 these surfaces is 



-^ = pndS+ pndS + p V <J>(n«V$) dS (238) 



+ (gravitational force) 



where n is the unit normal to each surface directing inward to the fluid 

 under consideration, and we have put 



$ = Ux + cf> (239) 



The force acting on the ship is the integral pressure on S. Taking the x- 

 component, the force in the x-direction is given by 



F x = - ) | P n„dS 

 S 



dM 

 dt 



p n x dS + p (u+ || ) (U n+n.V<j>) dS (240) 



where subscript x means the x-component of each vector. 



If the time average is taken, dM /dt vanishes because of the periodicity 



87 



