Minimum Wave Resistance for Dipole Distributions 91 
in which the coefficient 0.44 is approximate. The general form (specifically the minus sign) 
of this relation is in agreement with the theorem of G. I. Taylor relating the dipole moment, 
the volume, and the added mass in the x-direction, for a line dipole distribution in a flow 
without free surface. 
4. NUMERICAL SOLUTION OF THE INTEGRAL EQUATION 
Equation (30), with A = —1, was solved by numerical methods as described in Appendix 
B. The solution depends on Froude number and is denoted by g,(x) or g,(x;f). Solutions 
for other values of A are given simply by Ag_(x)/—1. The desired solution g(x) which satis- 
fies Eq. (26) is given by . 
6,(%;5 f) 
a(xsf) = CE (51) 
where 
1/2 
I ,(f) = 6,(x;f) dx. (52) 
-1/2 
I,(f) and g(x;f) have been computed from g,(x;f). As indicated in Appendix B 
h 
6(x) = we ae 
dois 2 (53) 
4 
where h(x) is a bounded function. The behavior of h(+1/2) as a function of f (see Fig. 5) is 
of interest since when h(+1/2) is large (small) the bulge in the shape at bow and stern is 
also large (small), as shown in Section 5. Figure 4 shows g(x;f) for various values of f. 
Figure 5 shows A(x;f) for various values of f. Figures 3, 4, and 5 show that bluntness is a 
predominant feature at high speed, but less so at low speeds. However, the forms appear to 
have bulbs at all values of f yet examined (f> 0.3). Note that (in Fig. 3) A(+1/2) + 7-1 = 
0.3183, as f + 00, in agreement with Eq. (43) et seq. 
The function h(x) (Fig. 5) is of some interest. We have h(x) + 77! as f + 0, as explained 
in Section 3. Hence one can qualitatively define a high-speed range as that in which 
h(x) ~7-1, Figures 4 and 5 suggest that f > 0.6 is a reasonable definition of the high-speed 
range. Below f= 0.6 there is a rapid transition from the dogbone shape characteristic of 
high speed to shapes whose maximum beam is amidship. 
‘ 
Let Chip be the wave resistance coefficient at Froude number f of the form which is 
optimum at Froude number f’ . cf (f), shown in Fig. 6, is then a universal optimum for 
struts. Note that Ch if) is a maximum at f = 0.65. In Fig. 6 we show also C}-%f) and 
Cr: Clif) and C1-°%f) are very close for f > 0.6, which substantiates our identifica- 
tion of f > 0.6 as the high-speed range. The divergence of Ci (f) and C1-°(f) below f = 0.6 
is substantial. chip) and C9-5(f) are close in the range 0.46 < f < 0.6. 
