Theory of Ships Viewed as Slightly Submerged Bodies: A Simple Explanation and Integral Equation Variants

The classical Neumann–Kelvin (NK) theory of potential flow around a free-surface-piercing ship that steadily advances in calm water or through regular waves is considered. Specifically, this study presents an elementary ‘no-equation interpretation’ of the rigid-waterplane linear flow model and the related modification of the NK theory recently presented by the authors and complements the detailed mathematical analysis given in that earlier study. Specifically, the NN (Neumann–Noblesse) integral equation obtained in that previous study by applying Green’s fundamental identity to an alternative linear flow model called the rigid-waterplane flow model, in which an open free-surface-piercing ship hull is closed by a rigid waterplane slightly submerged under the free surface, is interpreted in light of Saint-Venant’s principle. Briefly, the present study argues that the NK integral equation obtained in the classical NK theory of potential flow around a ship contains a singularity at the ship waterline and that this singularity is removed—in the spirit of the classical Saint-Venant principle—in the rigid-waterplane flow model and the related weakly-singular NN integral equation, which can then be viewed as a ‘regularization’ of the NK integral equation. This study also presents variants of the NN integral equation in which a function defined in terms of the ship hull surface geometry by an integral over the ship waterplane or an integral around the ship waterline is expressed as equivalent integrals over the ship hull surface. Like the NN integral equation given previously, the equivalent variants of the weakly-singular NN integral equation obtained in this study do not involve a waterline integral and hold for a ship that steadily advances in calm water or through regular waves, as well as for an offshore structure or a moored ship in regular waves.