Ogilvie 



Suppose that the ship oscillates sinusoidally in the j -th mode. Let the po- 

 tential be the real part of 



VjrpjCXi,X2,X3) e''"* , 



where vj is a real amplitude. Then cpj satisfies the usual free surface boundary 

 condition, a radiation condition, and a condition on the hull, 



^ = f/Xj,x,,X3) on S, (18a) 



where f ■ depends only on the geometry of the hull and on the mode being con- 

 sidered. We may look on the quantities f j as modal weighting functions. For 

 example, f j = cos (n, ij). If the ship surges, the fluid disturbance due to an 

 element of the hull surface is proportional to this direction cosine. Haskind's 

 formula arises because this same quantity, f j, plays a role in the inverse prob- 

 lem: If there is an external disturbance to the fluid, the surge-force contribu- 

 tion of the pressvure on this element will again be proportional to f j . (See 

 Chertock (1962).) 



To see how this works out, we must consider the potential function for the 

 diffraction problem. Let 



cp^(Xj,X2,X3) e"^ , cpj(Xj,X2,X3) e "^ 



be the functions of which the real parts are the potentials, respectively, for the 

 incident wave and the diffracted wave. (The ship is fixed in this problem.) The 

 functions cPq and p^ satisfy the same free surface condition as cpj , and cp^ satis- 

 fies the radiation condition as well, (p^ is known everywhere, but it clearly 

 does not satisfy the radiation condition, since it represents a wave which is in- 

 cident on the ship. These two potentials yield normal velocity components on the 

 hull surface which are equal and opposite, since the hull is not moving; that is, 



3n 3n 



on S . (18b) 



dS 

 J 



The force in the j-th mode on an element of the surface of the hull is just 

 proportional to f . , so that we may write for the generalized force 



X, = - Jpf, 

 where p is the hydrodynamic pressure: 



P = "-^ gT ^^ {C*o + ^d^ e^"''! = -Re |ia,'p(cp^ + Cpj) e""^*! . 



We substitute this expression into the previous equation, and we also make use 

 of (18a): 



X. = Relicope'"^ J (cPo + cpj) 1^ "^^ J 



44 



