Ship Waves and Wave Resistance 



noted that Eq. (13) for A^j^^ can be retransformed by partial integration in f and 

 C to a representation by a source distribution over s°*, eliminating the line in- 

 tegral, so that 



0(^) = ^ \l ^{Wx']^ ^ [f4'1 Jci,fd^ '^[J\ '^"'"^ '''^''- ^^^^ 



We should observe that the source density corresponds to the change of internal 

 flow with coordinate rather than to the normal velocity. 



Equation (15) may be compared with Eq. (43) of Ref. 3, in which the influ- 

 ence of trim and sinkage is included. We see that all line integrals presented 

 there can be eliminated under validity of our assumptions. 



Determination of Wave Resistance 



Having thus selected a model for the approximation of the flow, there is a 

 decision to be made for definition of wave resistance to the corresponding de- 

 gree of approximation. Three different approaches may be considered: (a) in- 

 tegrate pressure components over the wetted part of the hull bounded by the 

 calculated wave profile, retaining only terms up to third order, (b) start with 

 expressions for the energy flow through a vertical plane behind the ship, as 

 given in Eq. (8.6), page 460 of Ref. 2, and evaluate these for approximate flow, 

 using the wave contour from this approximate flow, or (c) consider the approxi- 

 mate second-order flow to be physically real in the domain D°, that is, consider 

 a closed surface, part of which is the wetted hull, and from the fact that mo- 

 mentum in the enclosed volume D should not change with time, infer that action 

 of pressure on the hull can be expressed through flow of generalized momentum 

 across the rest of the surface. 



It can be shown easily that for the linearized flow model one and the same 

 expression for the resistance can be derived by either approach (see however 

 the objections raised by Sharma (9)). Up to third order, however, (a) and (c) 

 should only give equivalent expressions R^ and R^, if the boundary condition on 

 the hull is already met exactly by the approximate flow, as otherwise we may 

 have substantial flux of momentum into the ship's interior. The formula for (b) 

 was derived under assumption of a free surface under constant pressure and 

 composed of streamlines. For a second-order flow, it is unrealistic to maintain 

 this assumption. We should therefore expect that resistance R^, calculated by 

 this formula applied to the approximate flow, could, even in a nonmonotonic way 

 depend on the location x^ of the vertical control plane where data are taken. 



But R^, derived by approach (c), should be independent of choice of domain 

 D. We shall select D= D°, the domain bounded above by the undisturbed free 

 surface as described before. Due to conservation of momentum we have for 

 surface integrals enclosing any domain D of the flow (compare Refs. 2 and 7) 



P A(v • v)/2n - (v • n)v} ds = , (16) 



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