# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 193-214. doi: 10.1007/BF02392728. [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), 633-669. doi: 10.1098/rsta.1981.0231. [3] B. Buffoni and J. F. Toland, Analytic Theory of Global Bifurcation. An Introduction, Princeton University Press, Princeton, NJ, 2003. [4] R. B. Burckel, An Introduction to Classical Complex Analysis, New York-London Acadenic Press, Inc., 1979. [5] A. Constantin, On deep water wave motion, J. Phys. A, 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313. [6] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [7] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference in Applied Mathematics 81. 2011, Philadelphia: SIAM. doi: 10.1137/1.9781611971873. [8] A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033. [9] A. Constantin, Mean velocities in a Stokes wave, Arch. Ration. Mech. Anal., 207 (2013), 907-917. doi: 10.1007/s00205-012-0584-6. [10] A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves, Nonl. Anal.-Real World Appl., 9 (2008), 1336-1344. doi: 10.1016/j.nonrwa.2007.03.003. [11] A. Constantin and J. Escher, Symmetry of deep-water waves with vorticity, Eur. J. App. Math., 15 (2004), 755-768. doi: 10.1017/S0956792504005777. [12] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 423-431 (electronic). doi: 10.1090/S0273-0979-07-01159-7. [13] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. [14] A. Constantin and W. Strauss, Pressure beneath a Stokes Wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299. [15] A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation, Arch. Ration. Mech. Anal., 199 (2011), 33-77. doi: 10.1007/s00205-010-0314-x. [16] A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech., 10 (2008), 1-18. doi: 10.1007/s00021-005-0214-2. [17] M. Ehrnström, On the streamlines and particle paths of gravitational waves, Nonlinearity, 21 (2008), 1141-1154. doi: 10.1088/0951-7715/21/5/012. [18] L. C. Evans, Partial Differential Equations-2nd ed., AMS Graduate Studies in Mathematics, 19, 2010. [19] L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge University Press, 2000. doi: 10.1017/CBO9780511569203. [20] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin: Springer, 2001. [21] D. Henry, The trajectories of particles in deep water Stokes waves, Int. Math. Res. Not., 2006 (2006), 1-13. doi: 10.1155/IMRN/2006/23405. [22] D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves, Phil. Trans. R. Soc. A, 365 (2007), 2241-2251 doi: 10.1098/rsta.2007.2005. [23] D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves, J. Nonlin. Math. Phys., 14 (2007), 1-7. doi: 10.2991/jnmp.2007.14.1.1. [24] D. Henry, On the deep-water Stokes wave flow, Int. Math. Res. Not., 2008 (2008), 1-7. doi: 10.1093/imrn/rnn071. [25] D. Henry, Pressure in a deep-water Stokes wave, J. Math. Fluid. Mech., 13 (2011), 251-257. doi: 10.1007/s00021-009-0015-0. [26] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, 1997. doi: 10.1017/CBO9780511624056. [27] P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form, Arch. Ration. Mech. Anal., 171 (2004), 349-416. doi: 10.1007/s00205-003-0292-3. [28] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Berlin: Srpringer, 1992. [29] Ch. Pommerenke., Conformal Maps at the Boundary, in Handbook of Complex Analysis: Geometric Function Theory, 1, Amsterdam: North-Holland, 2002. doi: 10.1016/S1874-5709(02)80004-X. [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, I, Cambridge, 1880, 225-228. [31] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (2006), 1-48. [32] E. Varvaruca, Some geometric and analytic properties of solutions of Bernoulli free-boundary problems, Interfaces Free Bound., 9 (2007), 367-381. doi: 10.4171/IFB/169. [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.

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##### 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), 193-214. doi: 10.1007/BF02392728. [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), 633-669. doi: 10.1098/rsta.1981.0231. [3] B. Buffoni and J. F. Toland, Analytic Theory of Global Bifurcation. An Introduction, Princeton University Press, Princeton, NJ, 2003. [4] R. B. Burckel, An Introduction to Classical Complex Analysis, New York-London Acadenic Press, Inc., 1979. [5] A. Constantin, On deep water wave motion, J. Phys. A, 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313. [6] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [7] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference in Applied Mathematics 81. 2011, Philadelphia: SIAM. doi: 10.1137/1.9781611971873. [8] A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033. [9] A. Constantin, Mean velocities in a Stokes wave, Arch. Ration. Mech. Anal., 207 (2013), 907-917. doi: 10.1007/s00205-012-0584-6. [10] A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves, Nonl. Anal.-Real World Appl., 9 (2008), 1336-1344. doi: 10.1016/j.nonrwa.2007.03.003. [11] A. Constantin and J. Escher, Symmetry of deep-water waves with vorticity, Eur. J. App. Math., 15 (2004), 755-768. doi: 10.1017/S0956792504005777. [12] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. (N.S.), 44 (2007), 423-431 (electronic). doi: 10.1090/S0273-0979-07-01159-7. [13] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. [14] A. Constantin and W. Strauss, Pressure beneath a Stokes Wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299. [15] A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation, Arch. Ration. Mech. Anal., 199 (2011), 33-77. doi: 10.1007/s00205-010-0314-x. [16] A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech., 10 (2008), 1-18. doi: 10.1007/s00021-005-0214-2. [17] M. Ehrnström, On the streamlines and particle paths of gravitational waves, Nonlinearity, 21 (2008), 1141-1154. doi: 10.1088/0951-7715/21/5/012. [18] L. C. Evans, Partial Differential Equations-2nd ed., AMS Graduate Studies in Mathematics, 19, 2010. [19] L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge University Press, 2000. doi: 10.1017/CBO9780511569203. [20] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin: Springer, 2001. [21] D. Henry, The trajectories of particles in deep water Stokes waves, Int. Math. Res. Not., 2006 (2006), 1-13. doi: 10.1155/IMRN/2006/23405. [22] D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves, Phil. Trans. R. Soc. A, 365 (2007), 2241-2251 doi: 10.1098/rsta.2007.2005. [23] D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves, J. Nonlin. Math. Phys., 14 (2007), 1-7. doi: 10.2991/jnmp.2007.14.1.1. [24] D. Henry, On the deep-water Stokes wave flow, Int. Math. Res. Not., 2008 (2008), 1-7. doi: 10.1093/imrn/rnn071. [25] D. Henry, Pressure in a deep-water Stokes wave, J. Math. Fluid. Mech., 13 (2011), 251-257. doi: 10.1007/s00021-009-0015-0. [26] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, 1997. doi: 10.1017/CBO9780511624056. [27] P. I. Plotnikov and J. F. Toland, Convexity of Stokes waves of extreme form, Arch. Ration. Mech. Anal., 171 (2004), 349-416. doi: 10.1007/s00205-003-0292-3. [28] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Berlin: Srpringer, 1992. [29] Ch. Pommerenke., Conformal Maps at the Boundary, in Handbook of Complex Analysis: Geometric Function Theory, 1, Amsterdam: North-Holland, 2002. doi: 10.1016/S1874-5709(02)80004-X. [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, I, Cambridge, 1880, 225-228. [31] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (2006), 1-48. [32] E. Varvaruca, Some geometric and analytic properties of solutions of Bernoulli free-boundary problems, Interfaces Free Bound., 9 (2007), 367-381. doi: 10.4171/IFB/169. [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.
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