Minimum Wave Resistance for Dipole Distributions 89 
Finally, 
0 7 646? 4 
sin > = Pee | 1 + 1 + cosh & (48) 
Z 8e cosh 2 (as 
which is a convenient form for computation. It is easily seen that € = 0 corresponds to 
x= 0. In order to get an idea of the range of & we observe that at the stagnation points we 
have y = 0, |x| > 1/2, and hence 0= 7. Solving Eq. (48) with 0 = 7 approximately for €,,,,> 
the value of & at the stagnation point, we find 
1/3 
Geter ae cosh’! Pane (49) 
and 
The midship beam is given to first order by 
2e€ 
z(0) rc} a 5 
Since 2€/z is the beam given by the Michell theory it is interesting to compare z(0) with 
2€/m. Table 1 shows that the relative error is less than 0.008 for € < 0.10.* 
Table 1 
Comparison of z(0) and 2€/7 
0.00318 
0.00637 
0.1273 
0.03183 
0.06366 
The values of x,,,, indicate the extent to which the form fails to have the preassigned 
length. Figure 2 shows the optimum forms for five values of € ranging from 0.005 to 0.100. 
They resemble dogbones. 
*Typical values of € are 0.04 for a destroyer and 0.048 for a tanker, based on the waterplane section 
in each case. 
