Variational problems in the theory of hydroelastic waves

P. I. Plotnikov, J. F. Toland

Research output: Contribution to journalArticle

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 languageEnglish
Article number20170343
Pages (from-to)1-12
Number of pages12
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume376
Issue number2129
Early online date20 Aug 2018
DOIs
Publication statusPublished - 28 Sep 2018

Fingerprint

elastic sheets
Variational Problem
Gravitational potential energy
Periodic Wave
Potential energy
Mean Curvature
Energy
gravitational fields
Stretching
Existence of Solutions
Critical point
critical point
Gravity
Gravitation
Directly proportional
potential energy
curvature
Boundary conditions
boundary conditions
gravitation

Keywords

  • Hydroelastic waves
  • Theory of shells
  • Willmore functional

ASJC Scopus subject areas

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

Cite this

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

In: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 376, No. 2129, 20170343, 28.09.2018, p. 1-12.

Research output: Contribution to journalArticle

@article{94597e468b104be9bfc7d0c8d92d5119,
title = "Variational problems in the theory of hydroelastic waves",
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'.",
keywords = "Hydroelastic waves, Theory of shells, Willmore functional",
author = "Plotnikov, {P. I.} and Toland, {J. F.}",
note = "{\circledC} 2018 The Authors.",
year = "2018",
month = "9",
day = "28",
doi = "10.1098/rsta.2017.0343",
language = "English",
volume = "376",
pages = "1--12",
journal = "Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences",
issn = "1364-503X",
publisher = "Royal Society of London",
number = "2129",

}

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 -