Understanding and Prediction of Ship Motions 



the boundary condition by Timman and Newman (1962) provides the most con- 

 venient procedure for handling this difficulty. Again, see Eq. (10b). 



The potential cpjoi, as found by Newman (1961), points up an interesting 

 difficulty which is still not well understood or appreciated. This potential rep- 

 resents the diffracted flow around the translating restrained ship. It satisfies 

 a straightforward boundary condition on the hull, providing a normal component 

 of velocity which just offsets the corresponding velocity component of the inci- 

 dent wave system. However, its boundary condition on the free surface is 

 unique among the potential problems formulated by Newman. When the series 

 expansions are substituted into the free surface condition and the resulting con- 

 ditions are modified so as to apply on the undisturbed free surface, the poten- 

 tials (Pjoo, cpooi, and cpuo all satisfy a homogeneous condition: 



However, cpioi, must satisfy a nonhomogeneous condition; there is a nonzero 

 right-hand side in the corresponding equation, and this right side contains 

 terms which are essentially products of cpooi and cpjog . This situation is some- 

 what analogous to the problem of satisfying the body boundary condition. There 

 is an interaction between the incident wave system and the steady Kelvin wave 

 system such that an apparent pressure distribution is applied to the free sur- 

 face, and this apparent pressure gives rise to an unexpected addition to the dif- 

 fracted wave. 



This complication with the diffracted wave is an excellent example of the 

 value of systematic perturbation analyses. We could have set up the diffraction 

 problem much more easily. It would have seemed quite reasonable to assume 

 that the usual linearized free surface condition would apply, and so we would 

 have fovind a potential function which satisfied that condition and which also off- 

 set the normal component of the incident wave system on the ship hull. New- 

 man's systematic approach shows that this is not proper. We need another con- 

 tribution to the potential which satisfies a homogeneous condition on the hull and 

 a non-homogeneous condition on the mean free surface. This extra part will be 

 of the same order of magnitude as the potential which we would obtain by the 

 more naive approach. We should note specifically that, since this effect is due 

 to interaction of the incident waves with the steady wave system of the translating 

 ship, this is a problem only when the ship has non-zero forward speed. 



The effects of this difficulty may be quite pervasive. In particular, the Has- 

 kind relations for predicting wave-induced forces (discussed in Chapter IV) 

 were derived only for a ship at zero forward speed, and the extension of these 

 relations to ships with non-zero forward speed will depend on a further satis- 

 factory resolution of the problem discussed here. 



For the thin ship moving through sinusoidal waves, Newman's solution is 

 complete to first order in /3, and he obtained a set of formulas for the coeffi- 

 cients in the force expansions, complete to second order in /3. The expressions 

 are quite xmwieldy, and one can not be very optimistic about being able to use 

 them for practical calculations. However, it is of some interest to point out 



55 



