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July  2012, 11(4): 1537-1547. doi: 10.3934/cpaa.2012.11.1537

On particle trajectories in linear deep-water waves

Anca-Voichita Matioc 1,

1. 

Institute for Applied Mathematics, Leibniz University Hanover, Welfengarten 1, 30167 Hanover

Received  December 2010 Revised  June 2011 Published  January 2012

We determine the phase portrait of a Hamiltonian system of equations describing the motion of the particles in linear deep-water waves. The particles experience in each period a forward drift which decreases with greater depth.
Keywords: phase portrait., particle trajectory, Linear deep-water waves.
Mathematics Subject Classification: Primary: 76B15, 34C25; Secondary: 35Q3.
Citation: Anca-Voichita Matioc. On particle trajectories in linear deep-water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1537-1547. doi: 10.3934/cpaa.2012.11.1537
References:
[1]

D. J. Acheson, "Elementary Fluid Dynamics," The Clarendon Press, Oxford Univ. Press, New York, 1990.  Google Scholar

[2]

A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[3]

A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731. doi: 10.1088/0305-4470/34/45/311.  Google Scholar

[4]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[5]

A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16. doi: 10.1016/j.euromechflu.2010.09.008.  Google Scholar

[6]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773.  Google Scholar

[7]

A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math., 15 (2004), 755-768. doi: 10.1017/S0956792504005777.  Google Scholar

[8]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[9]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20165.  Google Scholar

[10]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.  Google Scholar

[11]

A. Constantin, D. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech., 548 (2006), 151-163. doi: 10.1017/S0022112005007469.  Google Scholar

[12]

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.  Google Scholar

[13]

A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves, Nonlinear Anal. Real World Appl., 9 (2008), 1336-1344. doi: 10.1016/j.nonrwa.2007.03.003.  Google Scholar

[14]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.  Google Scholar

[15]

G. D. Crapper, "Introduction to Water Waves," Ellis Horwood Ltd., Chichester, 1984. Google Scholar

[16]

L. Debnath, "Nonlinear Water Waves," Academic Press, Inc., Boston, MA, 1994.  Google Scholar

[17]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445. Google Scholar

[18]

D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not., Art. ID 23405 (2006), 1-13. doi: 10.1155/IMRN/2006/21630.  Google Scholar

[19]

D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves, J. Nonlinear Math. Phys., 14 (2007), 1-7. doi: 10.2991/jnmp.2007.14.1.1.  Google Scholar

[20]

D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95. doi: 10.2991/jnmp.2008.15.s2.7.  Google Scholar

[21]

V. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math. Anal., 37 (2006), 1482-1521. doi: 10.1137/040621168.  Google Scholar

[22]

D. Ionescu-Kruse, Particle trajectories in linearized irrotational shallow water flows, J. Nonlinear Math. Phys., 15 (2008), 13-27. doi: 10.2991/jnmp.2008.15.s2.2.  Google Scholar

[23]

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

[24]

J. Lamb, "Hydrodynamics," Cambridge Univ. Press, Cambridge, 1895. Google Scholar

[25]

J. Lighthill, "Waves in Fluids," Cambridge Univ. Press, Cambridge, 1978.  Google Scholar

[26]

A. V. Matioc, On particle trajectories in linear water waves, Nonlinear Anal. Real World Appl., 11 (2010), 4275-4284. doi: 10.1016/j.nonrwa.2010.05.014.  Google Scholar

[27]

B. V. Matioc, On the regularity of deep-water waves with general vorticity distributions,, to appear in { Quart. Appl. Math.}., ().   Google Scholar

[28]

L. M. Milne-Thomson, "Theoretical Hydrodynamics," The Macmillan Co., London, 1938. Google Scholar

[29]

A. Sommerfeld, "Mechanics of Deformable Bodies," Academic Press, Inc., Boston, MA, 1950.  Google Scholar

[30]

J. J. Stoker, "Water Waves. The Mathematical Theory with Applications," Interscience Publ. Inc., New York, 1957.  Google Scholar

[31]

G. G. Stokes, On the theory of oscillatory waves, Trans. Cambridge Phil. Soc., 8 (1847), 441-455. Google Scholar

[32]

A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302. doi: 10.1017/S0022112088002423.  Google Scholar

[33]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  Google Scholar

[34]

C. S. Yih, The role of drift mass in the kinetic energy and momentum of periodic water waves and sound waves, J. Fluid Mech, 331 (1997), 429-438. doi: 10.1017/S0022112096003539.  Google Scholar

show all references

References:
[1]

D. J. Acheson, "Elementary Fluid Dynamics," The Clarendon Press, Oxford Univ. Press, New York, 1990.  Google Scholar

[2]

A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417. doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[3]

A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), 9723-9731. doi: 10.1088/0305-4470/34/45/311.  Google Scholar

[4]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[5]

A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16. doi: 10.1016/j.euromechflu.2010.09.008.  Google Scholar

[6]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773.  Google Scholar

[7]

A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math., 15 (2004), 755-768. doi: 10.1017/S0956792504005777.  Google Scholar

[8]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[9]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20165.  Google Scholar

[10]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.  Google Scholar

[11]

A. Constantin, D. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech., 548 (2006), 151-163. doi: 10.1017/S0022112005007469.  Google Scholar

[12]

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.  Google Scholar

[13]

A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves, Nonlinear Anal. Real World Appl., 9 (2008), 1336-1344. doi: 10.1016/j.nonrwa.2007.03.003.  Google Scholar

[14]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.  Google Scholar

[15]

G. D. Crapper, "Introduction to Water Waves," Ellis Horwood Ltd., Chichester, 1984. Google Scholar

[16]

L. Debnath, "Nonlinear Water Waves," Academic Press, Inc., Boston, MA, 1994.  Google Scholar

[17]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445. Google Scholar

[18]

D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not., Art. ID 23405 (2006), 1-13. doi: 10.1155/IMRN/2006/21630.  Google Scholar

[19]

D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves, J. Nonlinear Math. Phys., 14 (2007), 1-7. doi: 10.2991/jnmp.2007.14.1.1.  Google Scholar

[20]

D. Henry, On Gerstner's water wave, J. Nonlinear Math. Phys., 15 (2008), 87-95. doi: 10.2991/jnmp.2008.15.s2.7.  Google Scholar

[21]

V. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math. Anal., 37 (2006), 1482-1521. doi: 10.1137/040621168.  Google Scholar

[22]

D. Ionescu-Kruse, Particle trajectories in linearized irrotational shallow water flows, J. Nonlinear Math. Phys., 15 (2008), 13-27. doi: 10.2991/jnmp.2008.15.s2.2.  Google Scholar

[23]

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

[24]

J. Lamb, "Hydrodynamics," Cambridge Univ. Press, Cambridge, 1895. Google Scholar

[25]

J. Lighthill, "Waves in Fluids," Cambridge Univ. Press, Cambridge, 1978.  Google Scholar

[26]

A. V. Matioc, On particle trajectories in linear water waves, Nonlinear Anal. Real World Appl., 11 (2010), 4275-4284. doi: 10.1016/j.nonrwa.2010.05.014.  Google Scholar

[27]

B. V. Matioc, On the regularity of deep-water waves with general vorticity distributions,, to appear in { Quart. Appl. Math.}., ().   Google Scholar

[28]

L. M. Milne-Thomson, "Theoretical Hydrodynamics," The Macmillan Co., London, 1938. Google Scholar

[29]

A. Sommerfeld, "Mechanics of Deformable Bodies," Academic Press, Inc., Boston, MA, 1950.  Google Scholar

[30]

J. J. Stoker, "Water Waves. The Mathematical Theory with Applications," Interscience Publ. Inc., New York, 1957.  Google Scholar

[31]

G. G. Stokes, On the theory of oscillatory waves, Trans. Cambridge Phil. Soc., 8 (1847), 441-455. Google Scholar

[32]

A. F. Teles da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), 281-302. doi: 10.1017/S0022112088002423.  Google Scholar

[33]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  Google Scholar

[34]

C. S. Yih, The role of drift mass in the kinetic energy and momentum of periodic water waves and sound waves, J. Fluid Mech, 331 (1997), 429-438. doi: 10.1017/S0022112096003539.  Google Scholar

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