August  2014, 34(8): 3095-3107. doi: 10.3934/dcds.2014.34.3095

Particle trajectories in extreme Stokes waves over infinite depth

1. 

School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8

Received  July 2013 Revised  September 2013 Published  January 2014

We investigate the velocity field of fluid particles in an extreme water wave over infinite depth. It is shown that the trajectories of particles within the fluid and along the free surface do not form closed paths over the course of one period, but rather undergo a positive drift in the direction of wave propagation. In addition it is shown that the wave crest cannot form a stagnation point despite the velocity of the fluid particles being zero there.
Citation: Tony Lyons. Particle trajectories in extreme Stokes waves over infinite depth. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3095-3107. doi: 10.3934/dcds.2014.34.3095
References:
[1]

C. J. Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form,, Acta Math., 148 (1982), 193.  doi: 10.1007/BF02392728.  Google Scholar

[2]

C. J. Amick and J. F. Toland, On periodic water-waves and their convergence to solitary waves in the long wave limit,, Philos. Trans. R. Soc. Lond. A, 303 (1981), 633.  doi: 10.1098/rsta.1981.0231.  Google Scholar

[3]

B. Buffoni and J. F. Toland, Analytic Theory of Global Bifurcation. An Introduction,, Princeton University Press, (2003).   Google Scholar

[4]

R. B. Burckel, An Introduction to Classical Complex Analysis,, New York-London Acadenic Press, (1979).   Google Scholar

[5]

A. Constantin, On deep water wave motion,, J. Phys. A, 34 (2001), 1405.  doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[6]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[7]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, CBMS-NSF Regional Conference in Applied Mathematics 81. 2011, (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[8]

A. Constantin, Particle trajectories in extreme Stokes waves,, IMA J. Appl. Math., 77 (2012), 293.  doi: 10.1093/imamat/hxs033.  Google Scholar

[9]

A. Constantin, Mean velocities in a Stokes wave,, Arch. Ration. Mech. Anal., 207 (2013), 907.  doi: 10.1007/s00205-012-0584-6.  Google Scholar

[10]

A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves,, Nonl. Anal.-Real World Appl., 9 (2008), 1336.  doi: 10.1016/j.nonrwa.2007.03.003.  Google Scholar

[11]

A. Constantin and J. Escher, Symmetry of deep-water waves with vorticity,, Eur. J. App. Math., 15 (2004), 755.  doi: 10.1017/S0956792504005777.  Google Scholar

[12]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 423.  doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[13]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[14]

A. Constantin and W. Strauss, Pressure beneath a Stokes Wave,, Comm. Pure Appl. Math., 63 (2010), 533.  doi: 10.1002/cpa.20299.  Google Scholar

[15]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation,, Arch. Ration. Mech. Anal., 199 (2011), 33.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[16]

A. Constantin and G. Villari, Particle trajectories in linear water waves,, J. Math. Fluid Mech., 10 (2008), 1.  doi: 10.1007/s00021-005-0214-2.  Google Scholar

[17]

M. Ehrnström, On the streamlines and particle paths of gravitational waves,, Nonlinearity, 21 (2008), 1141.  doi: 10.1088/0951-7715/21/5/012.  Google Scholar

[18]

L. C. Evans, Partial Differential Equations-2nd ed.,, AMS Graduate Studies in Mathematics, (2010).   Google Scholar

[19]

L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems,, Cambridge University Press, (2000).  doi: 10.1017/CBO9780511569203.  Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Berlin: Springer, (2001).   Google Scholar

[21]

D. Henry, The trajectories of particles in deep water Stokes waves,, Int. Math. Res. Not., 2006 (2006), 1.  doi: 10.1155/IMRN/2006/23405.  Google Scholar

[22]

D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves,, Phil. Trans. R. Soc. A, 365 (2007), 2241.  doi: 10.1098/rsta.2007.2005.  Google Scholar

[23]

D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves,, J. Nonlin. Math. Phys., 14 (2007), 1.  doi: 10.2991/jnmp.2007.14.1.1.  Google Scholar

[24]

D. Henry, On the deep-water Stokes wave flow,, Int. Math. Res. Not., 2008 (2008), 1.  doi: 10.1093/imrn/rnn071.  Google Scholar

[25]

D. Henry, Pressure in a deep-water Stokes wave,, J. Math. Fluid. Mech., 13 (2011), 251.  doi: 10.1007/s00021-009-0015-0.  Google Scholar

[26]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge University Press, (1997).  doi: 10.1017/CBO9780511624056.  Google Scholar

[27]

P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form,, Arch. Ration. Mech. Anal., 171 (2004), 349.  doi: 10.1007/s00205-003-0292-3.  Google Scholar

[28]

Ch. Pommerenke, Boundary Behaviour of Conformal Maps,, Berlin: Srpringer, (1992).   Google Scholar

[29]

Ch. Pommerenke., Conformal Maps at the Boundary,, in Handbook of Complex Analysis: Geometric Function Theory, (2002).  doi: 10.1016/S1874-5709(02)80004-X.  Google Scholar

[30]

G. G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change in form,, in Math. and Phys. Papers, (1880), 225.   Google Scholar

[31]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (2006), 1.   Google Scholar

[32]

E. Varvaruca, Some geometric and analytic properties of solutions of Bernoulli free-boundary problems,, Interfaces Free Bound., 9 (2007), 367.  doi: 10.4171/IFB/169.  Google Scholar

[33]

W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation,, New York: Springer, (1989).  doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

C. J. Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form,, Acta Math., 148 (1982), 193.  doi: 10.1007/BF02392728.  Google Scholar

[2]

C. J. Amick and J. F. Toland, On periodic water-waves and their convergence to solitary waves in the long wave limit,, Philos. Trans. R. Soc. Lond. A, 303 (1981), 633.  doi: 10.1098/rsta.1981.0231.  Google Scholar

[3]

B. Buffoni and J. F. Toland, Analytic Theory of Global Bifurcation. An Introduction,, Princeton University Press, (2003).   Google Scholar

[4]

R. B. Burckel, An Introduction to Classical Complex Analysis,, New York-London Acadenic Press, (1979).   Google Scholar

[5]

A. Constantin, On deep water wave motion,, J. Phys. A, 34 (2001), 1405.  doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[6]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[7]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, CBMS-NSF Regional Conference in Applied Mathematics 81. 2011, (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[8]

A. Constantin, Particle trajectories in extreme Stokes waves,, IMA J. Appl. Math., 77 (2012), 293.  doi: 10.1093/imamat/hxs033.  Google Scholar

[9]

A. Constantin, Mean velocities in a Stokes wave,, Arch. Ration. Mech. Anal., 207 (2013), 907.  doi: 10.1007/s00205-012-0584-6.  Google Scholar

[10]

A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves,, Nonl. Anal.-Real World Appl., 9 (2008), 1336.  doi: 10.1016/j.nonrwa.2007.03.003.  Google Scholar

[11]

A. Constantin and J. Escher, Symmetry of deep-water waves with vorticity,, Eur. J. App. Math., 15 (2004), 755.  doi: 10.1017/S0956792504005777.  Google Scholar

[12]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 423.  doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[13]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[14]

A. Constantin and W. Strauss, Pressure beneath a Stokes Wave,, Comm. Pure Appl. Math., 63 (2010), 533.  doi: 10.1002/cpa.20299.  Google Scholar

[15]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation,, Arch. Ration. Mech. Anal., 199 (2011), 33.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[16]

A. Constantin and G. Villari, Particle trajectories in linear water waves,, J. Math. Fluid Mech., 10 (2008), 1.  doi: 10.1007/s00021-005-0214-2.  Google Scholar

[17]

M. Ehrnström, On the streamlines and particle paths of gravitational waves,, Nonlinearity, 21 (2008), 1141.  doi: 10.1088/0951-7715/21/5/012.  Google Scholar

[18]

L. C. Evans, Partial Differential Equations-2nd ed.,, AMS Graduate Studies in Mathematics, (2010).   Google Scholar

[19]

L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems,, Cambridge University Press, (2000).  doi: 10.1017/CBO9780511569203.  Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Berlin: Springer, (2001).   Google Scholar

[21]

D. Henry, The trajectories of particles in deep water Stokes waves,, Int. Math. Res. Not., 2006 (2006), 1.  doi: 10.1155/IMRN/2006/23405.  Google Scholar

[22]

D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves,, Phil. Trans. R. Soc. A, 365 (2007), 2241.  doi: 10.1098/rsta.2007.2005.  Google Scholar

[23]

D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves,, J. Nonlin. Math. Phys., 14 (2007), 1.  doi: 10.2991/jnmp.2007.14.1.1.  Google Scholar

[24]

D. Henry, On the deep-water Stokes wave flow,, Int. Math. Res. Not., 2008 (2008), 1.  doi: 10.1093/imrn/rnn071.  Google Scholar

[25]

D. Henry, Pressure in a deep-water Stokes wave,, J. Math. Fluid. Mech., 13 (2011), 251.  doi: 10.1007/s00021-009-0015-0.  Google Scholar

[26]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge University Press, (1997).  doi: 10.1017/CBO9780511624056.  Google Scholar

[27]

P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form,, Arch. Ration. Mech. Anal., 171 (2004), 349.  doi: 10.1007/s00205-003-0292-3.  Google Scholar

[28]

Ch. Pommerenke, Boundary Behaviour of Conformal Maps,, Berlin: Srpringer, (1992).   Google Scholar

[29]

Ch. Pommerenke., Conformal Maps at the Boundary,, in Handbook of Complex Analysis: Geometric Function Theory, (2002).  doi: 10.1016/S1874-5709(02)80004-X.  Google Scholar

[30]

G. G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change in form,, in Math. and Phys. Papers, (1880), 225.   Google Scholar

[31]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (2006), 1.   Google Scholar

[32]

E. Varvaruca, Some geometric and analytic properties of solutions of Bernoulli free-boundary problems,, Interfaces Free Bound., 9 (2007), 367.  doi: 10.4171/IFB/169.  Google Scholar

[33]

W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation,, New York: Springer, (1989).  doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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