Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T20:46:26.020Z Has data issue: false hasContentIssue false
  • English
  • Français

A new equation describing travelling water waves

Published online by Cambridge University Press:  01 February 2013

Katie Oliveras  and
Vishal Vasan
Katie Oliveras*
Affiliation:
Mathematics Department, Seattle University, Seattle, WA 98122-1090, USA
Vishal Vasan
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
*
†Email address for correspondence: oliverak@seattleu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A new single equation for the surface elevation of a travelling water wave in an incompressible, inviscid, irrotational fluid is derived. This new equation is derived without approximation from Euler’s equations, valid for both a one- and two-dimensional travelling-wave surface. We show that this new formulation can be used to efficiently derive higher-order Stokes-wave approximations, and pose that this new formulation provides a useful framework for further investigation of travelling water waves.

JFM classification

Mathematical Foundations: Mathematical Foundations Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 717 , 25 February 2013 , pp. 514 - 522
Copyright
©2013 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ablowitz, M. J., Fokas, A. S. & Musslimani, Z. H. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 562, 313–343.CrossRefGoogle Scholar
Ablowitz, M. J. & Haut, T. S. 2008 Spectral formulation of the two fluid Euler equations with a free interface and long wave reductions.. Analysis and Applications 6, 323–348.CrossRefGoogle Scholar
Babenko, K. I. 1987 Some remarks on the theory of surface waves of finite amplitude. Sov. Math. Dokl. 35, 599–603.Google Scholar
Bridges, T. J., Dias, F. & Menasce, D. 2001 Steady three-dimensional water-wave patterns on a finite-depth fluid. J. Fluid Mech. 436, 145–175.CrossRefGoogle Scholar
Craig, W. & Nicholls, D. P. 2002 Traveling gravity water waves in two and three dimensions. Eur. J. Mech. (B/Fluids) 21, 615–641.CrossRefGoogle Scholar
Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83.CrossRefGoogle Scholar
Deconinck, B. & Oliveras, K. 2011 The instability of periodic surface gravity waves. J. Fluid Mech. 675, 417–430.CrossRefGoogle Scholar
Dyachenko, A. I., Kuznetsov, E. A., Spector, M. D. & Zakharov, V. E. 1996 Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221, 73–79.CrossRefGoogle Scholar
Iooss, G. & Plotnikov, P. I. 2009 Small divisor problem in the theory of three-dimensional water gravity waves. Mem. Am. Math. Soc. 200, 940.Google Scholar
Iooss, G. & Plotnikov, P. I. 2011 Asymmetrical three-dimensional travelling gravity waves. Arch. Rat. Mech. Anal. 200, 789–880.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1978 Some new relations between Stokes’s coefficients in the theory of gravity waves. J. Inst. Maths Applics. 22, 261–273.CrossRefGoogle Scholar
Okamoto, H. & Shoji, M. 2001 The Mathematical Theory of Permanent Progressive Water-Waves. World Scientific Publishing.CrossRefGoogle Scholar
Toland, J. F. 2002 On a pseudo-differential equation for Stokes waves. Arch. Rat. Mech. Anal. 162, 179–189.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194.CrossRefGoogle Scholar