112 Samuel Karp, Jack Kotik, and Jerome Lurye 
essentially indebted to Prof. Inui, because he has dealt with the problem at stake in a con- 
sequent and efficient if approximate manner. His severe remark that Wigley and I have ham- 
pered the progress in wave resistance research by neglecting the distinction between hull 
form and distribution is perhaps too hard since our conclusions are primarily based on the 
sectional area curve and this, fortunately, is less affected by the difficulties mentioned 
than that of the actual ship form. 
I am dwelling at some length on this subject as a warning example: one should not 
proceed too long in a well-established groove of thought and therefore lose the connection 
with facts. 
Prof. Timman and Mr. Vossers have, as far as I understand, rigorously remained within 
the concept of the Michell ship. We are looking forward for explicit results. Prof. Karp and 
colleagues have followed rather the way indicated by Havelock, distinguishing between 
body form and distribution. Although the results apply to a rather restricted case of a 
deeply submerged strut, they are in principle extremely interesting. The optimum cross 
sections plotted correspond to some extent to those which have been derived by approximate 
methods, except, however, that they all are more exaggerated and show a rounded nose which 
could not be obtained by the low degree polynomials used before. It would be interesting to 
have similar cross sections for Froude numbers below 0.38. 
This lower range appears to have important practical consequences. Using my earlier 
work my collaborator Kracht has investigated the influence of the bulbous bow (and stern). 
Contrary to experimental results and earlier findings by Mr. Wigley and myself he has estab- 
lished that moderate bulbs may have a beneficial influence on wave resistance even at low 
Froude numbers. We are continuing with these investigations. To get an independent check, 
wholly submerged bodies of revolution moving in the vicinity of the free surface have been 
considered. I have already treated this problem several times earlier. However, while in 
the case of the Michell surface ship it is plausible from physical reasoning to assume zero 
end ordinates for the distributions, there is no need to introduce this restriction for bodies 
of revolution. Therefore, together with Dr. Eggers and Mr. Sharma, I have investigated sin- 
gularities systems which include continuous line doublet distributions, concentrated sources, 
and, following Wigley, doublets located on the axis of the body. 
The plots in Fig. D1 show the optimum longitudinal distribution calculated for a con- 
stant area coefficient 9 = 0.60 (except for plot (g)), three depths of immersion ratios 2f/L, 
and two Froude numbers. 
The symbol <2, 4,6,8 D> indicates the powers of the polynomial terms; D stands for 
concentrated dipoles at the ends of the axis with a doublet moment ap. 
Computations have been made under three assumptions: 
1. The end ordinates of the dipole distribution, (+1) = 0. 
2. End ordinates —7(+1) are not prescribed but ap = 0; a result 7(+1) > 0 means a con- 
centrated source at the ends. 
3. The dipole moment a, + 0. In this case the area of the circle at the ends of the 
axis represents the dipole moment to the scale of the distribution curve. 
Figures D1(a)-(c) show the optimum doublet distributions for y, = 1/2F* = 6, F = 0.289. 
These distributions coincide sometimes nicely for different 2f/L (Fig. D1(a)), sometimes 
