OPTIMUM SHAPES OF BODIES IN FREE SURFACE 
FLOWS 
Thee WE 
Caltfornta Instttute of Technology 
Pasadena, Caltfornta, U.S.A. 
and 
Arthur K. Whitney 
Palo Alto Research Laboratory, Lockheed Atreraft Corp. 
Pato’ Alto, “Calt fornia, “UsS.A: 
ABSTRACT 
The general problem of optimum shapes arising in 
a wide variety of free-surface flows can be charac- 
terized mathematically byanew class of variation- 
al problems in which the Euler equation is a set of 
dual integral equations which are generally nonli- 
near, and singular, of the Cauchy type. Several ap- 
proximate methods are discussed, including linear- 
ization of the integral equations, the Rayleigh-Ritz 
method, and the thin-wing type theory. These me- 
thods are applied here to consider the following 
physical problems : 
(i) The optimum shape of a two-dimensional plate 
planing on the water surface, producing the maxi- 
mum hydrodynamic lift ; 
(ii) The two-dimensional body profile of minimum 
pressure drag in symmetric cavity flows ; 
(iii) The cavitating hydrofoil having the minimum 
drag for prescribed lift. 
Approximate solutions of these problems are dis- 
cussedundera set ofadditional isoperimetric cons- 
traints and some physically desirable end condi- 
tions. 
i Be 
