Newman and Tuck 



theory results from retaining only those contributions to the velocity potential 

 and forces acting on the body which are of leading order in e, and higher order 

 approximations follow by systematically including the next-higher -order con- 

 tributions, etc. However, our problem is complicated by the fact that a slender 

 ship, oscillating in six degrees of freedom, will produce hydrodynamic disturb- 

 ances in the various modes and encounter hydrodynamic, hydrostatic, and iner- 

 tial forces in each mode, which are of different orders in e. It is clear, for 

 example, that the surging or rolling oscillations of a slender ship will not gen- 

 erate disturbances of the same order as pitching or heaving modes. Moreover, 

 even within one given mode, say heave, certain types of forces will dominate 

 others; for example the hydrostatic restoring force will be of the same order as 

 the waterplane area, or 0(e), while the inertial force will be of the same order 

 as the ship's displaced volume, or 0(6^). As a result many of the accepted 

 components to the total forces and moments acting on the ship are higher order, 

 and do not appear in the first order theory for each mode. This situation is 

 illustrated in Table 2, which shows the order of magnitude, for each mode of 

 oscillation, of various physical quantities. These include the normal fluid ve- 

 locity B0g/3n on the ship's surface induced by its oscillations and by the incident 

 wave system, the corresponding body velocity potential 4>^ representing the dis- 

 turbance of the fluid by the ship, the hydrodynamic body force Fg due to this 

 disturbance, the hydrodynamic force Fpj^ due to the pressure field of the undis- 

 turbed incident wave system (the "Froude-Krylov" force), the hydrostatic re- 

 storing force Fjjs, and the inertial force Fj due to the body's own mass or 

 moment of inertia. For each mode the forces of leading order are underlined, 

 and the first order eqxiation of motion is written symbolically in the last column. 

 Conceptually this table can be derived most easily for uncoupled motions, but in 

 fact the inclusion of coupling between modes does not affect the order of magni- 

 tude in each case (assuming that the origin is taken at the center of gravity). 



Table 2 

 Relative Orders of Magnitude for each Degree of Freedom 



To illustrate how the entries of Table 2 are obtained, consider the case of 

 surge. The normal velocity on the ship's surface is proportional to the direction 

 cosine in the longitudinal direction, which is 0(e) for a slender body. The 



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