No abstract

The paper presents a method of computing periodic water waves based on solving an integral equation by means of discretization and automatically finding the mesh on which the functions to be found are approximated by the best way. The power of the method to describe ‘bad functions’ well makes it possible to reproduce all the main results of asymptotic theory for the almost-highest waves (Longuet-Higgins & Fox, 1977, 1978, 1996) by a direct numerical simulation. The method is able to compute two full periods of the oscillations of wave properties for all wave height-to-length ratios. The end of the second period corresponds to the wave steepness that achieves 99.99997% of the limiting value. So, the validity of the asymptotic formulae by Longuet-Higgins & Fox is proved for the steep waves of any finite depth. The refined value of the maximum slope of the free-surfaces is found to be 30.3787°.

We study progressive gravity waves at the interface between two unbounded fluids of different densities. The main concern is to find almost limiting configurations for the so-called overhanging waves. The latter were first computed by Meiron & Saffman (J. Fluid Mech., vol. 129, 1983, pp. 213–218). By means of the Hopf lemma, we rigorously prove that, if $\unicode[STIX]{x1D703}$ is the angle between the tangent line to the interfacial curve and the horizontal direction, then $-\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<\unicode[STIX]{x03C0}$. This inequality allows us to put forward a criterion of proximity of the interface to the limiting configuration, namely, the angle $|\unicode[STIX]{x1D703}|_{max}$ must be close to $\unicode[STIX]{x03C0}$ but may not exceed $\unicode[STIX]{x03C0}$. We develop a new numerical method of computing interfacial waves based on the representation of a piecewise-analytic function to be found in such a manner that only the shape of the interface is unknown. All other hydrodynamic quantities can be expressed analytically in terms of functions describing this shape. Using this method, we compute almost limiting configurations of interfacial waves with $|\unicode[STIX]{x1D703}|_{max}>179.98^{\circ }$. Analysing the results of computations, we introduce two new concepts: an inner crest, and an inner solution near the inner crest. These concepts allow us to make a well-grounded prediction for the shapes of limiting interfacial configurations and confirm Saffman & Yuen’s (J. Fluid Mech., vol. 123, 1982, pp. 459–476) conjecture that the waves are geometrically limited.

The paper presents an exact analytical solution to the problem of finding the optimum profile of a two-dimensional plate which planes on a water surface without spray formation and maximizes the lift force. The lift is maximized under the only isoperimetric constraint of fixed total arclength of the plate. The exact solution is compared with approximate analytical and numerical results by Wu & Whitney (1972). The shape of the optimum plate turns out to be technically unrealizable because of small, tightly wound spirals near the end points. It was shown numerically that cutting off small segments near the end points leads on the one hand to insignificant change in the lift force and on the other hand to a non-separating boundary layer along the remaining part of the optimum plate.