92 Samuel Karp, Jack Kotik, and Jerome Lurye 
Fig. 3. Graph of Al +1/2) vs f. Theoptimizing dipole distribution 
has the form -(c/27) h(x)/(1/4 = x2)1/2 = <cg x)/27; see Eq 
(53). Al +1/2) is the coefficient of the edge singularity, andisim- 
portant in determining the shape at the ends. 
As indicated in Appendix B the interval of integration (half a boat-length) was divided 
into six subintervals (seven points). One of the factors affecting the accuracy of the nu- 
merical scheme with seven points is the value of F, since as F increases the number of 
oscillations of Y,(F|x|) increases. We believe that all our numerical results for f < 0.4 
should be regarded as questionable. For some lower values of f the numerical solution of 
the integral equation and the wave resistance coefficient had the wrong sign. Figure 3 
shows h(+1/2) decreasing rapidly for 0.3 < f < 0.4, with intentions of becoming negative. 
Also, a separate computer program was written for calculating C,, for any strutlike dipole 
distribution. The mesh in this program was variable, and it was found that seven points was 
not sufficient to calculate C,, for f= 0.4 with an accuracy of the order of 5 percent whereas 
seven points gave an accuracy of better than 1 percent for f= 0.5. A more powerful com- 
puter program is in preparation for work on the three-dimensional problem, and we expect 
that it will allow us to extend the range of accurate calculation to lower values of f. 
5. DETERMINATION OF THE SHAPE NEAR THE ENDS 
Having found the optimum dipole distribution we are left with the problem of finding the 
form. While digital methods were available and will play a role in future work we preferred 
to determine the shape approximately by analytic means. We expect that z = Vg(x)/2L 
should be a good approximation to the true shape away from the ends, at least for sufficiently 
small V/2L* = € and sufficiently far from the free surface. Figure 7 shows the exact shape 
and the linearized shape g(x) in the case of f=, for €= 0.05. The agreement even away 
from the ends is not as good as one would like” but no attempt will be made to improve this 
aspect of the theory. 
*See also Fig. 10. For additional discussion see Section 6. 
