Slender Body Theory for an Oscillating Ship 



The case that both parameters are of order unity is also treated here. Al- 

 though the same difficulties in the equations of motion can be expected as in the 

 corresponding problem with Froude number equal to zero it is nevertheless 

 worthwhile to carry out the calculations in order to get some insight in the range 

 of validity of the theory. 



FORMULATION OF THE PROBLEM 



In the coordinate system used in the following the Xj, yj plane coincides 

 with the free surface. The origin moves with the ship speed v in the same di- 

 rection as the ship and the z ^ axis is taken positive in upward direction. 



The hull surface in equilibrium position is assumed to be of the form 



Vl = f i(Xi, Zj) sgn yj . (2.1) 



As an additional condition the bow and the stern have the shape of sharp wedges. 

 Between B^ and S^ the bottom of the ship is flat. 



The length of the ship is L, the beam B and the draft T. A cross section 

 contour is denoted by C(xj) . The bow contour and the stern contour are denoted 

 respectively by r^ and r^. Only heaving and pitching motions of the ship are 

 considered, which are harmonic in time with angular speed w. The same pro- 

 cedure can be followed for swaying and yawing motions. 



In the inviscid fluid a velocity potential exists defined by 



(D(Xj,yj,Zj,t) = -Vxj + 0(Xj,yj,Zj,t) , 



0(Xj,yj,Zj,t) must satisfy the Laplace equation 



Ac;6 = , 



the linearized free surface condition for z ^ = o 



^tt + V^c^,^,^ - 2V0tx, + g<^z, = 



and the boundary condition on the hull 



169 



(2.2) 



(2.3) 



(2.4) 



