76 Samuel Karp, Jack Kotik, and Jerome Lurye 
forms defined by a certain class of polynomials. The wave resistance is then a function of 
the polynomial coefficients, and minima can be found by simple computations (see Ref. 5). 
The other type of investigation has been concerned with minimization within the class of 
vertical struts of infinite depth having fixed length, fixed volume-per-unit depth, and an 
otherwise arbitrary form. The latter problem, with which this paper is concerned, has been 
considered by a number of authors. Their results have been unsatisfactory for one or more 
of the following reasons: 
1. The problem was found to have no solution at all. 
2. The shapes obtained were infinitely wide at bow and stern. 
3. Shapes (which in some cases have negative width) were obtained by solving an in- 
tegral equation numerically without regard for the fact that solutions of the equation are 
known to be (in general) singular. 
The cause ofthese difficulties is a nonuniformity in the accuracy of the perturbation 
procedure used to linearize the problem. As in the case of thin air foils (see Ref. 6) for 
fixed position along the air foil the results are arbitrarily accurate for sufficiently small 
thickness. However, the convergence is not always uniform with respect to position, the 
difficulty occurring at the ends of the foil. 
In this paper the problem is formulated with dipole density instead of strut shape as the 
unknown function. The dipole density satisfies one of the integral equations studied by 
previous workers, and is generally singular at the ends of the interval in which it is defined. 
The associated shape, defined by the closed streamlines in the flow generated by the di- 
poles, is approximately proportional to the dipole density (as in the usual theory) except at 
the ends. The shape has the required length and volume-per-unit depth, to first order in the 
perturbation parameter, and finite positive width. Hence, although refinements can still be 
made in the method of calculating the shape the basic problem is regarded as solved. 
In Section 2 we formulate the problem and derive the integral equation for the dipole 
density. In Section 3 we discuss the optimum shape for large Froude number. In Section 4 
we describe the numerical solution of the integral equation. In Section 5 we determine the 
shape of the optimum form. In Section 6 we compare the shape and wave drag of various 
forms. 
2. FORMULATION AND THE INTEGRAL EQUATIONS 
We reproduce here the usual theory of the thin strut of minimum wave resistance in order 
to motivate the formulation given later in this section. Throughout the discussion, the fluid 
is assumed to be incompressible and inviscid and the flow to be irrotational. Referring to 
Fig. 1, we introduce a right-handed rectangular coordinate system (x, y, z). The x-z plane 
(y = 0) represents the upper surface of the fluid, the acceleration of gravity is downwards 
and the flow velocity is in the +x-direction. 
It is convenient also to introduce the dimensionless coordinates x,y,z defined by 
x = Lx, y = Ly, z= Lz (1) 
where L is the length of the strut’s horizontal section in the x-direction. 
