A SOLUTION OF THE MINIMUM WAVE RESISTANCE PROBLEM 
R. Timman and G. Vossers 
Netherlands Ship Model Basin 
INTRODUCTION 
The problem of the determination of ship hulls with minimum wave resistance has been 
the subject of numerous investigations. The usual way of approach, based on a classical 
paper by Weinblum (1930) is to consider a ship hull in the form of an infinitely deep cylinder 
and to solve the variational problem of minimizing the resistance integral with prescribed 
horizontal cross section of the ship. In Weinblum’s original paper a Ritz method is used; 
the waterline is represented by a polynomial with unknown coefficients which are determined 
from a minimum problem for a function of a finite number of variables. Computation of the 
Weinblum-functions (1955) facilitates the procedure. 
The method, however, is open to some criticism, since it is not a priori clear, that it 
yields a good approximation to the variational problem. 
It is known that Pavlenko (1934) reduced the variational problem to the solution of an 
integral equation of the first kind. It is known that its solution has certain singularities, 
which cannot properly be represented by a polynomial. 
In this paper the problem is reduced to an integral equation of the second kind, obtained 
by the following consideration. The resistance integral in the thin ship approximation is 
known to be a quadratic functional of the hull function. If the additional condition is also 
expressed by a quadratic functional, the problem is simply equivalent to a principal axis 
problem and the solution exists, as is known from the general theory of quadratic functionals. 
Essentially, it is assumed that the cross sections of the ship are similar in shape, which 
(with a slight modification) produces the required result. This means, that the ship consid- 
ered has a finite draft. 
In order to obtain a simple integral equation, the influence of the bottom in the evalua- 
tion of Michell’s integral is neglected, although a more correct treatment lies within the 
power of numerical methods. 
MICHELL’S FORMULA FOR THE RESISTANCE INTEGRAL 
We introduce a coordinate system with the x axis along the axis of the ship, the y axis 
athwartships, and the z axis downward. The ship hull is represented by a function 
y= 2x, Zz) ; IES BIS & BID A QO Koes T 
67 
