80 Samuel Karp, Jack Kotik, and Jerome Lurye 
solution of Eq. (13), we conclude finally that Eq. (12) has no solution at all.* Thus, the 
seeming|ly natural way in which the problem of the strut of minimum wave resistance has 
been formulated turns out to be inadequate. We will show that the functions ¢(x) satisfying 
the conditions 1, 2, and 3 comprise a class which is too severely restricted for the minimiz- 
ing C(x) to be found within it. 
The problem can also be approached as follows. Since we require C(+1/2) = 0, two in- 
tegrations by parts in Eq. (7) lead to 
/2 1/2 
1 
CS 4F? | | U(x) U(x!) Yo(Flx-x"|) dx'dx, Qe) 
= / 22 
and the corresponding integral equation is 
1/2 
| C(x") Y,(F|x-x'|) dx' = a on - 
-1/2 
Nl = 
1 
| a7) 
where @ is determined so that Eq. (11) holds. As mentioned above it is shown in Ref. 10 
that the solutions of Eq. (17) are generally infinite at x = +1/2, so that the derivation of 
Eq. (16) is suspect and the solution physically unacceptable. Nevertheless Pavlenko [12] 
has solved Eq. (17) numerically by replacing it by a set of algebraic equations and imposing 
C(+1/2) = 0. According to Wehausen, for Froude numbers f < 0.325 he finds negative ordi- 
nates near the ends. For higher Froude numbers his solutions are fairly good except at the 
ends, as we shall see. Shen and Kern [13] solved Eq. (17) numerically by assuming that 
¢(x) was a polynomial with three undetermined coefficients. Their results are discussed in 
Section 6. 
We now show that Eqs. (16) and (17), suitably interpreted, are still the correct equations. 
Let (27,2; V/2L) be the potential for the flow generated by a strut advancing into still 
water. Assuming the strut to be fixed and the free stream directed along the positive x axis 
with velocity c, we have cx as the potential of the free stream.t Then 
Alva aa V V Og a as a 
U (8.9.8: =| = OE WX,Y,Z) + cx (18) 
where Vf/2L is the “perturbation potential” expressing the amount by which the strut dis- 
turbs the free stream. The well-known [3, p. 113] linearized boundary conditions on ¢ for 
a thin strut are 
(19) 
at the mean free surface y = 0 and 
*The exceptional case k = 0 in Eq. (12) leads to the nontrivial solution ¢ = constant, but this fails to 
satisfy the condition C(t 1/2) = 0 unless @ =0. Sretenskii [11] has concluded that there are no square- 
integrable solutions, but his reasoning has been criticized by Wehausen [3]. 
tThe flow velocity, v, is given by 0 = grad W. 
