Understanding and Prediction of Ship Motions 



things. The non-existence of a closed body can be rationalized easily. However, 

 the manner in which the streamlines cross the linearized free surface curve ap- 

 pears to be most unreasonable. At the least, it suggests that much more study 

 must be devoted to these streamlines before we jump to far-reaching conclu- 

 sions about how best to improve satisfaction of the body boundary conditions. 

 Perhaps it is more important to satisfy the free surface condition exactly. Tuck 

 has made force calculations which suggest that this may be the case. 



It must be emphasized that the strange behavior of the streamlines depicted 

 in Fig. 6 has nothing to do with nonlinear ities, except inasmuch as we are neg- 

 lecting them. The streamlines shown are those which result from solution of 

 the first order (linearized) problem. We usually accept the idea that solutions 

 of linearized free surface problems may be physically invalid just because the 

 problems are linearized, that is, because we have omitted and/or simplified 

 some terms in the boundary conditions. This example demonstrates that the 

 linearized solution can be meaningless because it is internally physically con- 

 tradictory. 



Regardless of these problems, we can formulate a self-consistent math- 

 ematical theory for the motions of a thin ship, and the theory will include the 

 Michell-Havelock wave resistance theory as a special case. This was first 

 done in a general way by Peters and Stoker (1954). Their work is quite well 

 known in our field, especially since most of it was reproduced by Stoker (1957) 

 in his monograph on wave problems. Only a very brief discussion of it will be 

 presented here. 



Peters and Stoker first formulate the exact nonlinear problem of a ship 

 performing arbitrary motions in a nonviscous, incompressible fluid with a free 

 surface not able to sustain surface tension. They then assume that all variables 

 can be expanded in perturbation series in powers of /3, a small parameter which 

 may be considered as the beam/length ratio. Some quantities must be allowed 

 to have a zero-order term; in particular, ship speed is assumed to have the 

 expansion: 



S(t) = S^(t) + /3Si(t) + yS^s^Ct) + ... . 



However, most of the variables are assumed to represent small disturbances, 

 and so their expansions start with terms linear in /3. For example, the dis- 

 turbance potential is written: 



(I)(x,y,z,t) = /3cpj(x,y, z, t) + /S^ cp2 (x, y, z, t ) + ... . 



The free surface elevation, the motion variables, the thrust, etc., all have 

 similar expansions. 



These expansions are all substituted into the various conditions and equa- 

 tions, and the terms are all arranged according to powers of /3. Before solving 

 any boundary value problems, Peters and Stoker make a number of observa- 

 tions. For example, ds^/dt = 0, which means that s(t) represents a steady 

 forward speed with perturbations superposed on it, the perturbations being of 

 order /3. The usual conditions for hydrostatic equilibrium are also obtained. 



51 



