ON THE PROBLEM OF MINIMUM WAVE RESISTANCE 
FOR STRUTS AND STRUT-LIKE DIPOLE DISTRIBUTIONS 
Samuel Karp,* Jack Kotik, and Jerome Lurye 
Technical Research Group, Inc. 
Syosset, New York 
In this paper we consider the problem of minimizing the wave resistance 
of a strut of fixed length and volume-per-unit depth at a given Froude 
number. The work of others has shown that a satisfactory solution of 
the problem within the framework of the classical linearized theory is 
not possible. In this paper we show that by regarding the dipole dis- 
tribution rather than the form as the unknown function, and by correcting 
the classical thin-ship relation between the dipole distribution and the 
form, a satisfactory solution can be found. The universal minimum curve 
of Cw vs f is shown, as well as Cy vs f for a number ofoptimum and non- 
optimum forms. Among the problems left unanswered in this paper are 
the influence of three-dimensional effects and other corrections to the 
linearized theory. 
1, INTRODUCTION 
The linearized theory of ship wave resistance has been developed by a series of stu- 
dents beginning with Michell [1]. Accounts of this theory together with extensive bibli- 
ographies will be found in Refs. 2-4. Based on the assumption that the fluid is inviscid and 
that the ship is thin enough to generate only waves of small amplitude, the theory imposes 
the linearized free surface condition on the velocity potential that characterizes the flow. 
By means of these assumptions, the problem is made mathematically tractable and expres- 
sions for the wave resistance are derived. The expressions have the form of integrals involv- 
ing either the functions that define the shape of the hull or the functions that define the dis- 
tribution of sources and sinks by which the hull is generated. Consequently, if we wish to 
find the hull of minimum wave resistance within the linear approximation, we have to solve 
a problem in the calculus of variations. Specifically, a quadratic functional (the integral 
representing wave resistance) is to be minimized subject to a suitable side condition. The 
need for imposing some constraint on the minimization process becomes apparent upon ob- 
serving, for example, that a hull of zero beam has zero wave resistance. 
Previous studies have been of two types. One type depends on the fact that Michell’s 
integral for the wave resistance can be evaluated in terms of tabulated functions for ship 
Note: This work was supported by the Office of Naval Research under Contract Nonr-2427(00). 
*Institute of Mathematical Sciences, N.Y.U., New York, New York. Prof. Karp’s contribution was 
made in his capacity as consultant to TRG, Inc. 
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