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    Payne, Peter R. 1986. On the spray sheet thickness of a planning wedge hull. Ocean Engineering, Vol. 13, Issue. 1, p. 1.

    Payne, Peter R. 1982. The spray volume shed by an uncambered planing hull in steady planing. Ocean Engineering, Vol. 9, Issue. 4, p. 373.

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    Wu, T. Yao-Tsu and Whitney, A. K. 1973. Variational Calculus Involving Singular Integral Equations. ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 53, Issue. 11, p. 737.

    Whitney, Arthur K. 1972. Theory of optimum shapes in free-surface flows. Part 2. Minimum drag profiles in infinite cavity flow. Journal of Fluid Mechanics, Vol. 55, Issue. 03, p. 457.


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

  • T. Yao-Tsu Wu (a1) and Arthur K. Whitney (a1) (a2)
  • DOI:
  • Published online: 01 March 2006

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.

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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