Joosen 

 DRIFT FORCE 



First we shall consider a ship at zero forward speed in the presence of in- 

 cident plane progressive waves. 



The fluid is supposed to be incompressible, irrotational, and inviscid; 

 therefore a potential theory can be applied. The x coordinate is taken in the 

 longitudinal direction of the ship, and the z coordinate is taken positive upward. 

 The x-y plane coincides with the position of the undisturbed free surface of the 

 fluid. 



The ship is supposed to be slender, i.e., the beam -to -length ratio e - B/L is 

 small. Moreover, the wave amplitude h and the amplitude of the ship motion a 

 are assumed to be small; consequently a linearized theory can be used. The 

 motion is assumed to be harmonic in time with frequency W27t . The fluid ve- 

 locity can be represented by the gradient of a velocity potential ^. 



The following dimensionless quantities are introduced: 



X = 2x/L , ey = 2y/L , ez = 2z/L , h = 2h/L , a = a/h , 



k = c^2B/2g , (D = gL/2w • h0 e"^'^* . 



In this coordinate system the transverse coordinates are stretched in the ratio 

 e" * ; therefore the longitudinal and transverse dimensions of the ship are of the 

 same order of magnitude. The cross section curves c are given by the equation 

 y= f(x, z) sgny. The frequency parameter is assumed to be of order unity. 



Let the incident wave be characterized by the potential function 



4>Q - exp k (z + i — cos /3 + iy sin /? 



where ji is the angle of wave incidence relative to the x axis. Let (t>^ be the 

 disturbance potential due to the presence of the body. 4>^ must satisfy the La- 

 place equation, the free surface condition, the radiation condition, and a bound- 

 ary condition at the ship's hull. This last condition reads in the slender body 

 approximation 



-rr— - - -^ ikf a (1 - f /) 



av av ^ ■^ 



where v is the direction normal to the ship's hull S. Since the problem is 

 linear 4>^ can be split up in two other potential functions 



01 = 0j + a0^ , 



with 



3x/ 'bv 



638 



