Wu and Whttney 
I. INTRODUCTION 
The general problem of optimum shapes of bodies in free- 
surface flows is of practical as well as theoretical interest, In ap- 
plications of naval hydrodynamics these problems often arise when 
attempts are made to improve the hydromechanical efficiency and 
performance of lifting and propulsive devices, or to achieve higher 
speeds of operation of certain vehicles. Some examples of problems 
that fall under this general class are illustrated in Figure 1. The 
first example is to evaluate the optimum profile of a two-dimensional 
plate planing on a water surface without spray formation, and produc- 
ing the maximum hydrodynamic lift under the isoperimetric cons- 
traints of fixed chord length € and fixed wetted arc-length S of the 
plate. The second example depicts the problem of determining the 
shape of a symmetric two-dimensional plate so that the pressure drag 
of this plate in an infinite cavity flow is a minimum, again with fixed 
base-chord & and wetted arc-length S. The third is an example 
concerning the general lifting cavity flow past an optimum hydrofoil 
having the minimum drag for prescribed lift, incidence angle a , 
chord length f and the wetted arc-length S. In these problems the 
gravitational and viscas effects may be neglected as a first appro- 
ximation for operations at high Froude numbers. Physically, there is 
no definite rule for choosing the side constraints and isoperimetric 
conditions, but the existence and the characteristic behavior of the 
solution can depend decisively on what constraints and conditions are 
chosen, Mathematically, it has been observed in a series of recent 
studies that the determination of the optimum hydromechanical shape 
of a body in these free-surface flows invariably results in a new class 
of variational problems, Only a very few special cases from this 
general class of problems have been solved, the optimum -lifting-line 
solution of Prandtl being an outstanding example. 
There are several essential differences between the classic- 
al theory and this new class of variational problems. First of all, the 
unknown argument functions of the functional under extremization are 
related, not by differential equations as in the classical calculus of 
variations, but by a singular integral equation of the Cauchy type. 
Consequently, the ''Euler equation'' which results from the consider- 
ation of the first variation of the functional in this new class is also 
a singular integral equation which is, in general, nonlinear, This is 
in sharp contrast to the Euler differential equation in classical theory. 
Another characteristic feature of these new problems is that while 
regular behavior of solution at the limits of the integral equation may 
be required on physical grounds, the mathematical conditions which 
insure such behavior generally involve functional equations which are 
Fib2 
