Variational problems in the theory of hydroelastic waves

• Published in: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
• Main Authors: Plotnikov, P. I., Toland, J. F.
• Other Authors: Simons Foundation, EPSRC, Ministry of Education and Science of the Russian Federation
• Format: Article in Journal/Newspaper
• Language: English
• Published: The Royal Society 2018
• Subjects: Sea ice
• Online Access: http://dx.doi.org/10.1098/rsta.2017.0343; https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2017.0343; https://royalsocietypublishing.org/doi/full-xml/10.1098/rsta.2017.0343

This paper outlines a mathematical approach to steady periodic waves which propagate with constant velocity and without change of form on the surface of a three-dimensional expanse of fluid which is at rest at infinite depth and moving irrotationally under gravity, bounded above by a frictionless elastic sheet. The elastic sheet is supposed to have gravitational potential energy, bending energy proportional to the square integral of its mean curvature (its Willmore functional), and stretching energy determined by the position of its particles relative to a reference configuration. The equations and boundary conditions governing the wave shape are derived by formulating the problem, in the language of geometry of surfaces, as one for critical points of a natural Lagrangian, and a proof of the existence of solutions is sketched. This article is part of the theme issue ‘Modelling of sea-ice phenomena’.