Minimum Wave Resistance for Dipole Distributions 85 
distribution which has (and hence produces a shape o(x,y;V/2L) which has) the least 
wave resistance.* 
Since the minimizing density is infinite at the endpoints x = +1/2, the stagnation points 
of the flow defined by Eq. (27) lie outside the interval —-1/2 < x<1/2. Hence the minimiz- 
ing shape o(x,y; V/2L) actually extends beyond the limits of this interval. However, as 
V/2L ~ 0 the stagnation points approach +1/2, as shown in Sections 3 and 5. 
Finally, we remark that any solution of Eq. (30) is an even function of x, so that if free 
surface effects are disregarded the minimizing struts may be said to be symmetrical fore and 
aft. The above discussion summarizes our formulation of the problem of the strut of minimum 
wave resistance. 
3. THE MINIMIZING SHAPE FOR LARGE FROUDE NUMBER 
When the Froude number is large, it is possible to derive an algebraic expression for 
the limiting form assumed by the minimizing shape at great depths. In this section we indi- 
cate briefly how this can be done. 
For large Froude number the parameter F is small, so that the kernel in the integral 
Eq. (30) can be approximated in the usual way by the formula 
~ 2 1 7 t 
Y,(F|x-x'|) = — log (57 F| x-x |) (31) 
where log y’ = y = Euler’s constant = 0.577... . 
Thus, the integral equation becomes 
1/2 
2 | &(x') log ce y'Flx-x'|) dx' =). (32) 
-1/2 
The solution to this equation is an elementary function. Specifically, we have [15] 
(x) = oh 
2 (33) 
where A is a constant that depends on A and F, i.e., it depends on the side condition Eq. 
(26) and on the Froude number, f. Naturally, when A = A(F) is chosen so that g(x) satisfies 
Eq. (26) then A = constant independent of F. Wehausen [3] regards the singularity at 
|x| = 1/2 as depriving the solution of physical reality. This criticism is valid in the context 
*We recall, that, to first order in V/2L, Eq. (26) fixes the volume-per-unit draft. Thus within the limi- 
tations of first-order theory, the minimization is still being performed among all struts having a given 
carrying capacity per unit draft. 
