### Abstract

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'.

Original language | English |
---|---|

Article number | 20170343 |

Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 376 |

Issue number | 2129 |

Early online date | 20 Aug 2018 |

DOIs | |

Publication status | Published - 28 Sep 2018 |

### Fingerprint

### Keywords

- Hydroelastic waves
- Theory of shells
- Willmore functional

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)
- Physics and Astronomy(all)

### Cite this

*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*376*(2129), 1-12. [20170343]. https://doi.org/10.1098/rsta.2017.0343

**Variational problems in the theory of hydroelastic waves.** / Plotnikov, P. I.; Toland, J. F.

Research output: Contribution to journal › Article

*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 376, no. 2129, 20170343, pp. 1-12. https://doi.org/10.1098/rsta.2017.0343

}

TY - JOUR

T1 - Variational problems in the theory of hydroelastic waves

AU - Plotnikov, P. I.

AU - Toland, J. F.

N1 - © 2018 The Authors.

PY - 2018/9/28

Y1 - 2018/9/28

N2 - 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'.

AB - 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'.

KW - Hydroelastic waves

KW - Theory of shells

KW - Willmore functional

UR - http://www.scopus.com/inward/record.url?scp=85052489563&partnerID=8YFLogxK

U2 - 10.1098/rsta.2017.0343

DO - 10.1098/rsta.2017.0343

M3 - Article

VL - 376

SP - 1

EP - 12

JO - Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences

SN - 1364-503X

IS - 2129

M1 - 20170343

ER -