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Theory of optimum shapes in free-surface flows. Part 1. Optimum profile of sprayless planing surface

Published online by Cambridge University Press:  29 March 2006

T. Yao-Tsu Wu
California Institute of Technology, Pasadena, California
Arthur K. Whitney
California Institute of Technology, Pasadena, California Present address: Lockheed Palo Alto Research Laboratory, Lockheed Missiles and Space Co., Palo Alto, California.


This paper attempts to determine the optimum profile of a two-dimensional plate that produces the maximum hydrodynamic lift while planing on a water surface, under the condition of no spray formation and no gravitational effect, the latter assumption serving as a good approximation for operations at large Froude numbers. The lift of the sprayless planing surface is maximized under the isoperimetric constraints of fixed chord length and fixed wetted arc-length of the plate. Consideration of the extremization yields, as the Euler equation, a pair of coupled nonlinear singular integral equations of the Cauchy type. These equations are subsequently linearized to facilitate further analysis. The analytical solution of the linearized problem has a branch-type singularity, in both pressure and flow angle, at the two ends of plate. In a special limit, this singularity changes its type, emerging into a logarithmic one, which is the weakest type possible. Guided by this analytic solution of the linearized problem, approximate solutions have been calculated for the nonlinear problem using the Rayleigh-Ritz method and the numerical results compared with the linearized theory.

Research Article
© 1972 Cambridge University Press

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Cumberbatch, E. 1958 Two-dimensional planing at high Froude number. J. Fluid Mech. 4, 466.Google Scholar
Miskhelishvili, N. I. 1953 Singular Integral Equations. Groningen, Holland: Noordhoff.
Rispin, P. P. A. 1967 A singular perturbation method for nonlinear water waves past an obstacle. Ph.D. thesis, California Institute of Technology.
Tricomi, F. G. 1957 Integral Equations. Interscience.
Wehausen, J. V. & Laitone, E. V. 1960 Surface Waves. Handbuch der Physik, vol. 9. Springer.
Whitney, A. K. 1969 Minimum drag profiles in infinite cavity flows. Ph.D. thesis, California Institute of Technology.
Wu, T. Y. 1967 A singular perturbation theory for nonlinear free-surface flow problems. International Shipbuilding Progress, 14, 88.Google Scholar
Wu, T. Y. & Whitney, A. K. 1971 Theory of optimum shapes in free-surface flows. Part 1. California Institute of Technology Rep. E 132 F. 1.Google Scholar