Ogilvie 



There are probably several reasons for this long delay, and a quick inspec- 

 tion of these reasons will suggest something about the nature of slender body 

 theory. One important problem arises immediately which distinguishes the 

 slender body approach from thin ship analysis. If we introduce the slenderness 

 parameter into the formulation of the boundary value problem, the effects of the 

 free surface are generally lost and we are left with an infinite fluid problem. 

 This is clearly quite unsatisfactory, because we seek primarily a description of 

 just those phenomena which result from the presence of the free surface. Fur- 

 thermore, such a formulation turns out to be equivalent to a set of two-dimen- 

 sional problems, and the effects of interactions between various cross sections 

 are apparently quite ambiguous. These problems can be resolved by a refor- 

 mulation which is altogether different from the thin ship approach, but then the 

 final formulas depend on the geometry of the ship in an elementary way — which 

 is offensive to the naval architect, for it implies that the complicated geometry 

 of a hull is of little importance for ship motions. 



These difficulties all demand that our attempts to apply slender body theory 

 in ship problems be done with great care, by a systematic procedure. Using a 

 perturbation analysis, we can answer to all of these problems, even the last, 

 for, by being systematic, we can (in principle) proceed to higher approximations 

 which involve more and more details of the hull geometry. 



Before proceeding to the logical development of slender body theory, we 

 should note that Grim (1957, 1960) anticipated much that would later come out of 

 the theory. He pointed out that there were apparently two general approaches to 

 representing the ship in studies of ship motions: (1) The ship can be represented 

 by a set of three-dimensional singularities which clearly predict three-dimen- 

 sional effects but which are loosely connected to ship geometry. (2) The exact 

 shape of the ship in each cross section can be generated as if that cross section 

 were part of an infinitely long body of uniform shape, with no account taken of 

 three-dimensional effects (either interactions or forward speed effects). In 

 order to combine the advantages of both, he proposed to solve the potential prob- 

 lem corresponding to the second approach, representing the potential as a two- 

 dimensional multipole expansion about a line in the centerplane, and then at each 

 section to replace the two-dimensional singularities by three-dimensional sin- 

 gularities of the same strength. The resulting potential would then be used to 

 calculate pressure and force. In other words, he proposed to use strip theory 

 only to find singularities for representing the ship and then to use truly three- 

 dimensional potential functions to represent the flow. Grim (1960) published the 

 details of the analysis and some calculations, all for the case of zero forward 

 speed. He also stated that the theory had been worked out for forward speed 

 cases as well, but apparently he has not yet published that. 



There is considerable similarity between Grim's procedure and the rules 

 for calculation which follow from slender body theory. However, the two are 

 not identical, and it is obscure as to what meaning should be attributed to the 

 differences. It will appear that slender body theory actually gives simpler re- 

 sults than Grim's, and one may say that the slender body results fall into the 

 category which Grim criticized for not providing a realistic representation of 

 the ship geometry. But the systematic approach is logical if the assumptions 



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