Minimum Wave Resistance for Dipole Distributions 83 
whose potential is 0(x,y,z; V/2L) where 
V V 
0 (xv 2: z = cLx + By Sande (27) 
If we could expand o(x,y; V/2L) in powers of V/2L, we would expect to find 
Z=+o0(x Aides saps é(x) + smaller terms = o, |x Vie itis Bigs (28) 
= ep a) 2L 2L 1 »V> 2L 
analogous to Eq. (24). 
Just as V¢(x)/2L in Eq. (21) is the first order approximation to s(x,y; V/2L), as shown 
in Kq. (24), so Vg(x)/2L = 0, in Eq. (25) is the first-order approximation to o(x,y;V/2L), as 
shown in Eq. (28). However, in contrast to C(x), g(x) need merely be integrable and can 
therefore have integrable singularities, so that the first-order approximation o = Vg/2L is 
not always uniform in x. On the other hand, in spite of the nonuniformity, we expect that the 
integral in Eq. (26) when multiplied by V/2L still gives, to first order, the volume-per-unit 
draft. 
Before we can find the function g(x) in Eq. (25) that minimizes the first-order drag coef- 
ficient of the shape o we need an expression for this coefficient. (Equations (7) and (16) 
apply to the restricted class of shapes, s(x,y; V/2L) but not necessarily to the extended 
class, o(x,y;V/2L).) To obtain the drag coefficient for o, we note that -dG(x,y,z;x',y '0)/dx' 
in Eq. (25) gives the potential of an x-directed dipole located at (x‘,y',0).* Therefore, we 
may interpret p in Eq. (25) as arising from a distribution of such dipoles over the half-strip 
-1/2<x«'< 1/2, y'>0, z'=0, the density of the distribution being —(c/27)(V/2L) g(x’). 
It is possible to calculate the force exerted on any one of the above dipoles by the com- 
bination consisting of the remaining dipoles and the uniform flow. The force on the entire 
distribution, i.e., the wave resistance, is then the product of the density function, —(c/27) 
(V/2L) g(x’), and the force on a single dipole, integrated over the distribution. When the in- 
tegration is carried out and the integral divided by (p9c?/2)(V/2L)2, the drag coefficient of 
the distribution and therefore of the shape o(x,y; V/2L) is 
1/2 1/2 
cy = ~4F? [ J 800 ex") Yo(Flx-x'|) dx'dx. 9) 
1/2 1/2 
This result is proved in Appendix A. 
Comparison of Eq. (29) with Eq. (16) shows that the expression for the wave drag of the 
shape o(x,y; V/2L) has the same form as that for the drag of the shape s(x,y; V/2L) with the 
restricted functions ¢(x) replaced by g(x). Note, however, that Eqs. (7) and (16), which were 
equivalent for the restricted class €(x), are no longer so when € is replaced by g. Indeed, 
if dg/dx is not integrable, such replacement makes Eq. (7) meaningless. Thus, only Eqs. 
*Recall that G is the potential of a sink. 
