Data for Ships of Minimum Resistance 



be the case and that there be no question of finagling with data in order to "im- 

 prove" it. 



The authors cannot agree with Dr. Pien's statement that the "relative re- 

 sistance qualities are just opposite to the theoretical predictions." For both 

 models the residuary resistance is greater than the Michell resistance at the 

 design Froude number 0.316. This is, in fact the usual occurrence at this 

 Froude number for hulls with these prismatics. The only thing which really 

 seems out of the ordinary is the very low ratio of Michell to residuary resist- 

 ance for the symmetric model, but here there are no similar experiments to 

 compare with. However, it is because of the extremely low Michell resistance 

 in this case that we called attention to the necessity of considering the possi- 

 bility that the second-order term overpowers the first-order term. This neces- 

 sity does not seem so pressing for the asymmetrical model. An accurate 

 assessment of the effect of viscosity is also correspondingly more important 

 for the symmetric model. 



Although Dr. Pien might appear to have deprecated the importance of the 

 second-order approximation, he is, in fact, also proposing that we take it into 

 account. He states, ". . . we know the linearized ship-surface condition is not 

 accurate for the beam value . . . ." Indeed, we know much more, for we know that 

 the linearized approximation is not accurate for either the hull shape or the 

 wave surface. In a paper by one of the authors presented at the Ann Arbor 

 Symposium in 1963 it was shown that the more important error is associated 

 with this phenomenon in a related situation. Dr. Pien's argument that, in cases 

 where the first-order resistance is very low, it is legitimate to use the linearized 

 free-surface condition together with the exact body boundary condition is tempt- 

 ing, but assumes that low wave resistance is associated with small surface dis- 

 turbance everywhere . However, it is still possible that the local disturbance is 

 substantial and it is just in this locality where the inaccuracy is most important. 



With regard to further experiments, it is our own opinion that the influence 

 of viscosity should be clarified before any attempts to improve the approxima- 

 tion are made, and, in particular, that one should have a more reliable experi- 

 mental determination of the wave resistance. This also appears to be the import 

 of Prof. Ward's remarks. It is a pleasure to add that he has later volunteered 

 to undertake an investigation of the wave resistance of the symmetric model ac- 

 cording to his method. 



Prof. Winblum states that he has preferred polynomial representations for 

 hulls partly on mathematical grounds because of Weierstrass's Theorem. We 

 hope he will take it as good-natured malice if we point out that there are two 

 Weierstrass approximation theorems. One states the uniform approximability 

 of continuous functions on closed intervals by polynomials, the other by trigo- 

 nometric sums. Thus there is no mathematical reason for preferring polyno- 

 mials. The advantage of the trigonometric sums lies in the orthogonality of the 

 expansion functions, which results in smaller coefficients for higher harmonics. 

 The same can, of course, be achieved with Legendre or Chebyshev polynomials 

 (Prof. Maruo worked out the details for the latter during a visit to Berkeley), 

 but now the numerical computation becomes somewhat more complicated. 



1063 



