On particle trajectories in linear deep-water waves

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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

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

Received December 2010 Revised June 2011 Published January 2012

Abstract
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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, (1990).
[2]	A. Constantin, On the deep water wave motion,, J. Phys. A, 34 (2001), 1405. doi: 10.1088/0305-4470/34/7/313.
[3]	A. Constantin, Edge waves along a sloping beach,, J. Phys. A, 34 (2001), 9723. doi: 10.1088/0305-4470/34/45/311.
[4]	A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5.
[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. doi: 10.1016/j.euromechflu.2010.09.008.
[6]	A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity,, J. Fluid Mech., 498 (2004), 171. doi: 10.1017/S0022112003006773.
[7]	A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity,, European J. Appl. Math., 15 (2004), 755. doi: 10.1017/S0956792504005777.
[8]	A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12.
[9]	A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Comm. Pure Appl. Math., 63 (2010), 533. doi: 10.1002/cpa.20165.
[10]	A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481. doi: 10.1002/cpa.3046.
[11]	A. Constantin, D. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity,, J. Fluid Mech., 548 (2006), 151. doi: 10.1017/S0022112005007469.
[12]	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.
[13]	A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves,, Nonlinear Anal. Real World Appl., 9 (2008), 1336. doi: 10.1016/j.nonrwa.2007.03.003.
[14]	A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity,, Duke Math. J., 140 (2007), 591. doi: 10.1215/S0012-7094-07-14034-1.
[15]	G. D. Crapper, "Introduction to Water Waves,", Ellis Horwood Ltd., (1984).
[16]	L. Debnath, "Nonlinear Water Waves,", Academic Press, (1994).
[17]	F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile,, Ann. Phys., 2 (1809), 412.
[18]	D. Henry, The trajectories of particles in deep-water Stokes waves,, Int. Math. Res. Not., (2006), 1. doi: 10.1155/IMRN/2006/21630.
[19]	D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves,, J. Nonlinear Math. Phys., 14 (2007), 1. doi: 10.2991/jnmp.2007.14.1.1.
[20]	D. Henry, On Gerstner's water wave,, J. Nonlinear Math. Phys., 15 (2008), 87. doi: 10.2991/jnmp.2008.15.s2.7.
[21]	V. Hur, Global bifurcation theory of deep-water waves with vorticity,, SIAM J. Math. Anal., 37 (2006), 1482. doi: 10.1137/040621168.
[22]	D. Ionescu-Kruse, Particle trajectories in linearized irrotational shallow water flows,, J. Nonlinear Math. Phys., 15 (2008), 13. doi: 10.2991/jnmp.2008.15.s2.2.
[23]	R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,", Cambridge Univ. Press, (1997). doi: 10.1017/CBO9780511624056.
[24]	J. Lamb, "Hydrodynamics,", Cambridge Univ. Press, (1895).
[25]	J. Lighthill, "Waves in Fluids,", Cambridge Univ. Press, (1978).
[26]	A. V. Matioc, On particle trajectories in linear water waves,, Nonlinear Anal. Real World Appl., 11 (2010), 4275. doi: 10.1016/j.nonrwa.2010.05.014.
[27]	B. V. Matioc, On the regularity of deep-water waves with general vorticity distributions,, to appear in { Quart. Appl. Math.}., ().
[28]	L. M. Milne-Thomson, "Theoretical Hydrodynamics,", The Macmillan Co., (1938).
[29]	A. Sommerfeld, "Mechanics of Deformable Bodies,", Academic Press, (1950).
[30]	J. J. Stoker, "Water Waves. The Mathematical Theory with Applications,", Interscience Publ. Inc., (1957).
[31]	G. G. Stokes, On the theory of oscillatory waves,, Trans. Cambridge Phil. Soc., 8 (1847), 441.
[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. doi: 10.1017/S0022112088002423.
[33]	J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.
[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. doi: 10.1017/S0022112096003539.

show all references

References:

[1]	D. J. Acheson, "Elementary Fluid Dynamics,", The Clarendon Press, (1990).
[2]	A. Constantin, On the deep water wave motion,, J. Phys. A, 34 (2001), 1405. doi: 10.1088/0305-4470/34/7/313.
[3]	A. Constantin, Edge waves along a sloping beach,, J. Phys. A, 34 (2001), 9723. doi: 10.1088/0305-4470/34/45/311.
[4]	A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5.
[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. doi: 10.1016/j.euromechflu.2010.09.008.
[6]	A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity,, J. Fluid Mech., 498 (2004), 171. doi: 10.1017/S0022112003006773.
[7]	A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity,, European J. Appl. Math., 15 (2004), 755. doi: 10.1017/S0956792504005777.
[8]	A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12.
[9]	A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Comm. Pure Appl. Math., 63 (2010), 533. doi: 10.1002/cpa.20165.
[10]	A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., 57 (2004), 481. doi: 10.1002/cpa.3046.
[11]	A. Constantin, D. Sattinger and W. Strauss, Variational formulations for steady water waves with vorticity,, J. Fluid Mech., 548 (2006), 151. doi: 10.1017/S0022112005007469.
[12]	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.
[13]	A. Constantin, M. Ehrnström and G. Villari, Particle trajectories in linear deep-water waves,, Nonlinear Anal. Real World Appl., 9 (2008), 1336. doi: 10.1016/j.nonrwa.2007.03.003.
[14]	A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity,, Duke Math. J., 140 (2007), 591. doi: 10.1215/S0012-7094-07-14034-1.
[15]	G. D. Crapper, "Introduction to Water Waves,", Ellis Horwood Ltd., (1984).
[16]	L. Debnath, "Nonlinear Water Waves,", Academic Press, (1994).
[17]	F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile,, Ann. Phys., 2 (1809), 412.
[18]	D. Henry, The trajectories of particles in deep-water Stokes waves,, Int. Math. Res. Not., (2006), 1. doi: 10.1155/IMRN/2006/21630.
[19]	D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves,, J. Nonlinear Math. Phys., 14 (2007), 1. doi: 10.2991/jnmp.2007.14.1.1.
[20]	D. Henry, On Gerstner's water wave,, J. Nonlinear Math. Phys., 15 (2008), 87. doi: 10.2991/jnmp.2008.15.s2.7.
[21]	V. Hur, Global bifurcation theory of deep-water waves with vorticity,, SIAM J. Math. Anal., 37 (2006), 1482. doi: 10.1137/040621168.
[22]	D. Ionescu-Kruse, Particle trajectories in linearized irrotational shallow water flows,, J. Nonlinear Math. Phys., 15 (2008), 13. doi: 10.2991/jnmp.2008.15.s2.2.
[23]	R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,", Cambridge Univ. Press, (1997). doi: 10.1017/CBO9780511624056.
[24]	J. Lamb, "Hydrodynamics,", Cambridge Univ. Press, (1895).
[25]	J. Lighthill, "Waves in Fluids,", Cambridge Univ. Press, (1978).
[26]	A. V. Matioc, On particle trajectories in linear water waves,, Nonlinear Anal. Real World Appl., 11 (2010), 4275. doi: 10.1016/j.nonrwa.2010.05.014.
[27]	B. V. Matioc, On the regularity of deep-water waves with general vorticity distributions,, to appear in { Quart. Appl. Math.}., ().
[28]	L. M. Milne-Thomson, "Theoretical Hydrodynamics,", The Macmillan Co., (1938).
[29]	A. Sommerfeld, "Mechanics of Deformable Bodies,", Academic Press, (1950).
[30]	J. J. Stoker, "Water Waves. The Mathematical Theory with Applications,", Interscience Publ. Inc., (1957).
[31]	G. G. Stokes, On the theory of oscillatory waves,, Trans. Cambridge Phil. Soc., 8 (1847), 441.
[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. doi: 10.1017/S0022112088002423.
[33]	J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.
[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. doi: 10.1017/S0022112096003539.

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