Ship Form of Minimum Wave Resistance 



by Eq. (67) is not the minimum of the asymptotic value of Michell's integral for 

 vanishing draft, as a result of comparison of it and the wave resistance of a 

 slender ship with vertical stem. In the latter case, dA dx or /(/ ) does not van- 

 ish at the ends. If the condition Eq, (58) is discarded, one of the two coefficients 

 in Eq. (62) becomes undetermined unless another side condition is introduced. 

 Then solutions of the integral Eq. (62) give a family of curve of sectional area 

 by which the wave resistance excluding the end effect is minimum for a constant 

 displacement. One may have a doubt since the above indeterminateness seems 

 to contradict with the fact that the minimal solution for Michell's integral of 

 finite draft is unique. By a proper choice of the midship ordinate, k^ can be 

 eliminated. Then the principal part of the wave resistance given by Eq. (67) 

 vanishes. Though difficult is it to identify the true asymptote, the above solu- 

 tion may be regarded as the asymptotic form of the minimal solution with the 

 single condition of constant volume for vanishing draft. Figure 3 shows a com- 

 parison between the curve of sectional area obtained from the above method and 

 the dipole distribution for the optimum infinite strut. The difference between 

 them is small especially at lower Froude numbers. Kotik calculated the opti- 

 mum form of the elementary ship of finite draft at Froude number 0.4, one of 

 which concerned a 4th power vertical section and the other concerned a wall- 

 sided section. His results with respect to a draft- length ratio 0.05 are plotted 

 in Fig. 3 for comparison. They fall between the result for a infinite strut at 

 Froude number 0.397 and that of the aforementioned approximation. When finite 

 value of k2 is retained, a family of solutions with various midship section area 

 is obtained. As mentioned before, there is no minimal solution for dual condi- 

 tion of constant volume and constant midship section. Then the above results 

 seem to correspond to the asymptotic solution for the condition of constant 

 volume and constant moment of inertia. Figures 4-12 give the asymptotic opti- 

 mum curve of sectional area at various speed coefficient y^ = g t/u^ with pris- 

 matic coefficient cp as a parameter. In some of the figures Weinblum's results 

 [14] are given by dotted lines. Difference is not remarkable except the case of 

 7o = 2 where the polynomial representation employed by him seems to lose its 

 accuracy. In Fig. 6, curves of forebody sectional area of the Taylor Standard 

 Series (T.S.S.) are shown for comparison. There is a surprising agreement 

 between the T.S.S. and the theoretically optimum form for medium prismatic 

 coefficient at Froude number 0.25. On examining the chart of residuary resist- 

 ance coefficient, one may find out that T.S.S. shows an excellent behavior at 

 Froude number near 0.25, if the prismatic coefficient is around 0.60 where the 

 best agreement is obtained. It is of some interest to observe that a hump ap- 

 pears at Froude number 0.25, if the prismatic coefficient is reduced to 0.48 or 

 raised to 0.68 where deviation from the optimum curve becomes remarkable. 

 At Froude numbers other than 0.25, T.S.S. does not agree with the optimum 

 curve. Therefore better results than those of T.S.S. can be expected by em- 

 ploying the theoretical curve of sectional area. 



1035 



