problem the Neumann-Kelvin problem. Several numerical works have been 

 carried out so far. Since the smallness of the disturbance velocity can- 

 not be assumed apriori, other conditions are needed in order to realize it. 

 The simplest case is the thin ship. If the beam of the ship is very small, 

 the inner region, in which the potential <j> is defined, shrinks to a narrow 

 slit and, consequently, 9<j> /9n becomes higher order with respect to the 

 beam-to-length ratio of the ship. Therefore, a(Q) is determined by 

 (1/4tt) 3<j>/9n. The line integral becomes third order and can be omitted. 

 If we write the equation of the hull surface as 



y = f(x,z) sgn y (23) 



the velocity potential is reduced to 



<j>(P) = - ^ [ [ G(P,Q) f x (x',z') dx'dz' (24) 



S 



where S is the center plane of the ship and f = 9f/9x. This is identical 

 c x 



with Michell's potential. 



Wave Resistance at Low Speed 



Most existing theories of ship waves and wave resistance are based on 

 the linearization of the flow field by a small parameter which specifies 

 the slenderness of the ship hull. Since ship hulls, in practice, are 

 neither so slender nor thin enough to secure the validity of the linearized 

 theory, the agreement between the theoretical prediction and the experi- 

 mental results is, in general, not satisfactory. The disturbance ve- 

 locities are not small enough to make their square negligible everywhere 

 on the free surface. Since the inclusion of the nonlinear terms in the 

 free surface condition makes the boundary value problem intractable, some 

 simplification other than the linearization by the beam-to-length ratio 

 as a small parameter is needed to formulate the wave resistance of 

 practical hull forms, expecially full-form ships. 



15 



