Minimum Wave Resistance for Dipole Distributions 99. 
900 =2717% [DIPOLE DENSITY] 
ZO = EXACT SHAPE GENERATED BY 30%) 
Fig. 10. Graphs of the shape function for a lens and 
-2n/c times the exact dipole distribution which gen- 
erates it in an infinite fluid. 
(f = 3.162, 2.236, 1.581, 1.292). The forms shown by them are somewhat reminiscent of 
ours. The forms of Pavlenko and of Shen and Kern differ from ours in having finite entrance 
angles. 
In a series of pioneering papers Weinblum has considered the problem of minimizing the 
wave resistance of more realistic, three-dimensional forms. He has defined various families 
of polynomials, imposed constraints on the coefficients by specifying some geometrical 
properties of the form, and used the remaining degrees of freedom to minimize the resistance. 
Again, in spite of these differences in approach, our forms show the same general variation 
with Froude number as his waterplane sections. 
(It should be noted that all wave resistance coefficients of dipole distributions are nor- 
malized with respect to dipole moment (multiplied by —27/c) rather than volume. Hence to 
the extent that volume differs from dipole moment (multiplied by —27/c) a comparison of wave 
resistance coefficients does not provide an exact comparison of resistance values based on 
equal volume. This discrepancy, like the length discrepancy for optimum forms, does not 
affect our theoretical results, which are asymptotically valid for € + 0.) 
7. SUMMARY 
A universal curve of minimum wave resistance for infinite strutlike dipole distributions 
has been found (see Fig. 6). The optimizing distributions have been found, and turn out to 
be infinite at the ends. Despite this last mentioned feature the corresponding shapes, at 
large depths, are determined. The shapes are blunt at the ends, which extend beyond the 
interval on which the dipoles are distributed (see Fig. 8). Attention is called to the dis- 
tinction between shape and dipole distribution, a distinction which is of fundamental impor- 
tance in the present context. Attention is also called to the discrepancy between the wave 
resistances calculated using Michell’s approximate dipole distribution proportional to shape 
and the exact dipole distribution yielding that shape, especially for ships with nonzero en- 
trance angle (see Figs. 9 and 10). Some of the objectionable features of the analysis given 
by previous investigations have been removed. 
