On the particle paths in solitary water waves

Constantin, Adrian

doi:10.1090/S0033-569X-09-01166-1

Author: Adrian Constantin
Journal: Quart. Appl. Math. 68 (2010), 81-90
MSC (2000): Primary 35Q35, 76B07; Secondary 35J65, 76B25
DOI: https://doi.org/10.1090/S0033-569X-09-01166-1
Published electronically: October 15, 2009
MathSciNet review: 2598882
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We provide the qualitative flow pattern beneath a solitary water wave by describing the individual particle trajectories.

References

G. B. Airy, Tides and waves, Encyc. Metropolitana 5 (1845), 241–396.

C. J. Amick, Bounds for water waves, Arch. Rational Mech. Anal. 99 (1987), no. 2, 91–114. MR 886932, DOI https://doi.org/10.1007/BF00275873

C. J. Amick and J. F. Toland, On periodic water-waves and their convergence to solitary waves in the long-wave limit, Philos. Trans. Roy. Soc. London Ser. A 303 (1981), no. 1481, 633–669. MR 647410, DOI https://doi.org/10.1098/rsta.1981.0231

C. J. Amick and J. F. Toland, On solitary water-waves of finite amplitude, Arch. Rational Mech. Anal. 76 (1981), no. 1, 9–95. MR 629699, DOI https://doi.org/10.1007/BF00250799

J. Thomas Beale, The existence of solitary water waves, Comm. Pure Appl. Math. 30 (1977), no. 4, 373–389. MR 445136, DOI https://doi.org/10.1002/cpa.3160300402

M. J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d’un canal réctangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl. 17 (1872), 55–108.

Adrian Constantin, On the deep water wave motion, J. Phys. A 34 (2001), no. 7, 1405–1417. MR 1819940, DOI https://doi.org/10.1088/0305-4470/34/7/313

Adrian Constantin, Edge waves along a sloping beach, J. Phys. A 34 (2001), no. 45, 9723–9731. MR 1876166, DOI https://doi.org/10.1088/0305-4470/34/45/311

Adrian Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), no. 3, 523–535. MR 2257390, DOI https://doi.org/10.1007/s00222-006-0002-5

Adrian Constantin, Mats Ehrnström, and Erik Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J. 140 (2007), no. 3, 591–603. MR 2362244, DOI https://doi.org/10.1215/S0012-7094-07-14034-1

Adrian Constantin and Joachim Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 3, 423–431. MR 2318158, DOI https://doi.org/10.1090/S0273-0979-07-01159-7

Adrian Constantin and Robin Stanley Johnson, On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves, J. Nonlinear Math. Phys. 15 (2008), no. suppl. 2, 58–73. MR 2434725, DOI https://doi.org/10.2991/jnmp.2008.15.s2.5

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res. 40 (2008), no. 3, 175–211. MR 2369543, DOI https://doi.org/10.1016/j.fluiddyn.2007.06.004

Adrian Constantin and David Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009), no. 1, 165–186. MR 2481064, DOI https://doi.org/10.1007/s00205-008-0128-2

Adrian Constantin, David Sattinger, and Walter Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech. 548 (2006), 151–163. MR 2264220, DOI https://doi.org/10.1017/S0022112005007469

Adrian Constantin and Walter Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004), no. 4, 481–527. MR 2027299, DOI https://doi.org/10.1002/cpa.3046

Adrian Constantin and Walter A. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math. 60 (2007), no. 6, 911–950. MR 2306225, DOI https://doi.org/10.1002/cpa.20165

Adrian Constantin and Walter Strauss, Rotational steady water waves near stagnation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365 (2007), no. 1858, 2227–2239. MR 2329144, DOI https://doi.org/10.1098/rsta.2007.2004

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math. DOI: 10.1002/cpa.20299.

Walter Craig, Non-existence of solitary water waves in three dimensions, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 360 (2002), no. 1799, 2127–2135. Recent developments in the mathematical theory of water waves (Oberwolfach, 2001). MR 1949966, DOI https://doi.org/10.1098/rsta.2002.1065

Walter Craig and Peter Sternberg, Symmetry of solitary waves, Comm. Partial Differential Equations 13 (1988), no. 5, 603–633. MR 919444, DOI https://doi.org/10.1080/03605308808820554

Alex D. D. Craik, The origins of water wave theory, Annual review of fluid mechanics. Vol. 36, Annu. Rev. Fluid Mech., vol. 36, Annual Reviews, Palo Alto, CA, 2004, pp. 1–28. MR 2062306, DOI https://doi.org/10.1146/annurev.fluid.36.050802.122118

P. G. Drazin and R. S. Johnson, Solitons: an introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989. MR 985322

Mats Ehrnström, On the streamlines and particle paths of gravitational water waves, Nonlinearity 21 (2008), no. 5, 1141–1154. MR 2412330, DOI https://doi.org/10.1088/0951-7715/21/5/012

L. E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems, Cambridge Tracts in Mathematics, vol. 128, Cambridge University Press, Cambridge, 2000. MR 1751289

K. O. Friedrichs and D. H. Hyers, The existence of solitary waves, Comm. Pure Appl. Math. 7 (1954), 517–550. MR 65317, DOI https://doi.org/10.1002/cpa.3160070305

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

M. D. Groves and E. Wahlén, Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity, Phys. D 237 (2008), no. 10-12, 1530–1538. MR 2454604, DOI https://doi.org/10.1016/j.physd.2008.03.015

David Henry, On Gerstner’s water wave, J. Nonlinear Math. Phys. 15 (2008), no. suppl. 2, 87–95. MR 2434727, DOI https://doi.org/10.2991/jnmp.2008.15.s2.7

Vera Mikyoung Hur, Exact solitary water waves with vorticity, Arch. Ration. Mech. Anal. 188 (2008), no. 2, 213–244. MR 2385741, DOI https://doi.org/10.1007/s00205-007-0064-6

R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. MR 1629555

Robin S. Johnson, The classical problem of water waves: a reservoir of integrable and nearly-integrable equations, J. Nonlinear Math. Phys. 10 (2003), no. suppl. 1, 72–92. MR 2063546, DOI https://doi.org/10.2991/jnmp.2003.10.s1.6

D. J. Korteweg and G. deVries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag. 39 (1895), 422–443.

James Lighthill, Waves in fluids, Cambridge University Press, Cambridge-New York, 1978. MR 642980

John W. Miles, Solitary waves, Annual review of fluid mechanics, Vol. 12, Annual Reviews, Palo Alto, Calif., 1980, pp. 11–43. MR 565388

W. J. M. Rankine, On the exact form of waves near the surface of deep water, Phil. Trans. Roy. Soc. London A 153 (1863), 127–138.

Lord Rayleigh, On waves, Phil. Mag. 1 (1876), 257–279.

J. S. Russell, Report on waves, Rep. Meet. Brit. Assoc. Adv. Sci. 14 (1844), 311–390.

J. J. Stoker, Water waves: The mathematical theory with applications, Pure and Applied Mathematics, Vol. IV, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR 0103672

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 1–48. MR 1422004, DOI https://doi.org/10.12775/TMNA.1996.001

Eugen Varvaruca, On some properties of traveling water waves with vorticity, SIAM J. Math. Anal. 39 (2008), no. 5, 1686–1692. MR 2377294, DOI https://doi.org/10.1137/070697513