Optimum Shapes of Bodtes in Free Surface Flows 
difficult, and sometimes just impossible, to satisfy. 
Because of these difficulties and the fact that no general 
techniques are known for solving nonlinear singular integral equations, 
development of this new class of variational problems seems to require 
a strong effort, Attempts are made here to present some general re- 
sults of the current study. Some necessary conditions for the existence 
of an optimum solution are derived from a consideration of the first 
and second variations of the functional in question. To solve the re- 
sulting nonlinear, singular integral equation several approximate 
methods are discussed. One method is by linearization of the integral 
equation, giving a final set of dual singular integral equations of the 
Cauchy type. When the variable coefficients of this system of integral 
equations satisfy a certain relationship, this set of dual integral 
equations can be solved analytically in a closed form ; the results of 
this special case provide analytical expressions which can be exten- 
sively investigated to determine the behavior of a solution near the 
end points, Another approximate method is the Rayleigh-Ritz expans- 
ion ; it has the advantages of retaining the nonlinear effects to a certain 
extent, of incorporating the required behavior of the solution near the 
end points into the discretized expansion of the solution, but the method 
is generally not convergent, A third approach depends on a thin wing 
type theory to describe the flow at the very beginning, a variational 
calculation is then made on an approximate expression of the physic- 
al quantities of interest. These mathematical methods will be discuss- 
ed and then applied to three problems described earlier. While the 
results to be presented should be considered as still preliminary, 
since exact solutions to these problems have not yet been found, itis 
hoped that this paper will succeed in stimulating further interest in 
the development of the general theory, and, in turn, aid in the resolu- 
tion of many hydromechanic problems of great importance, 
II. GENERAL MATHEMATICAL THEORY 
To present a unified discussion of the general class of op- 
timum hydromechanical shapes of bodies in plane free-surface flows, 
including the three examples (i) - (iii) depicted in Figure 1, we as- 
sume the flow to be inviscid, irrotational, and incompressible, taking 
as known that the physical plane z=x+iy and the potential plane 
f=¢ +ipf correspond conformally to the upper half of the para- 
metric ¢= & +in plane by the mapping that can be signified sym- 
bolically as 
fee way ase re.. plete, (1) 
PEES 
