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On particle trajectories in linear deep-water waves
Effects of shear flow on KdV balance - applications to tsunami
1. | Fakultät für Mathematik, University of Vienna, Nordbergstr. 15, 1090 Vienna, Austria |
References:
[1] |
B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water waves and asymptotics, Invent. Math., 171 (2008), 485-541.
doi: 10.1007/s00222-007-0088-4. |
[2] |
T. B. Benjamin, The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech., 12 (1962), 97-116.
doi: 10.1017/S0022112062000063. |
[3] |
D. J. Benney and R. F. Bergeron, A new class of nonlinear waves in parallel flows (Nonlinear waves in parallel shear flows, discussing laminar flow breakdown due to free stream disturbances), Stud. Appl. Math., 48 (1969), 181-204. |
[4] |
N. Bowditch, "The American Practical Navigator," National Imagery and Mapping Agency, Bethesda, MD, 1995. |
[5] |
J. C. Burns, Long waves in running water, Proc. Camb. Phil. Soc., 4 (1953), 695-706.
doi: 10.1017/S0305004100028899. |
[6] |
A. Constantin, On the deep water wave motion, J. Phys. A, Math. Gen., 34 (2001), 1405-1417.
doi: 10.1088/0305-4470/34/7/313. |
[7] |
A. Constantin, On the propagation of tsunami waves, with emphasis on the tsunami of 2004, Discrete Cont. Dyn.-B, 12 (2009), 525-537.
doi: 10.3934/dcdsb.2009.12.525. |
[8] |
A. Constantin, On the relevance of soliton theory to tsunami modelling, Wave Motion, 46 (2009), 420-426.
doi: 10.1016/j.wavemoti.2009.05.002. |
[9] |
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. |
[10] |
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. |
[11] |
A. Constantin and D. Henry, Solitons and Tsunamis, Z. Naturforsch, 64a (2009), 65-68. |
[12] |
A. Constantin and R. S. Johnson, Modelling tsunamis, J. Phys. A, 39 (2006), L215-L217.
doi: 10.1088/0305-4470/39/14/L01. |
[13] |
A. Constantin and R. S. Johnson, On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves, J. Nonlinear Math. Phys., 15 (2008), 58-73.
doi: 10.2991/jnmp.2008.15.s2.5. |
[14] |
A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dyn. Res., 40 (2008), 175-211.
doi: 10.1088/0169-5983/42/3/038901. |
[15] |
A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Commun. Pure Appl. Math., 57 (2004), 481-527.
doi: 10.1002/cpa.3046. |
[16] |
A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Rational Mech. Anal., 199 (2011), 33-67.
doi: 10.1007/s00205-010-0314-x. |
[17] |
N. C. Freeman and R. S. Johnson, Shallow water waves on shear flows, J. Fluid Mech., 42 (1970), 401-409.
doi: 10.1017/S0022112070001349. |
[18] |
F. Gerstner, Theorie der Wellen samt einer abgeleiteten Theorie der Deichprofile, Abh. böhm. Ges. Wiss., 1 (1804). |
[19] |
J. L. Hammack and H. Segur, The Korteweg-de Vries equation and water waves. Part 2: Comparison with experiments, J. Fluid Mech., 65 (1974), 289-314.
doi: 10.1017/S002211207400139X. |
[20] |
J. L. Hammack and H. Segur, The Korteweg-de Vries equation and water waves. Part 3: Oscillatory waves, J. Fluid Mech., 84 (1978), 337-358.
doi: 10.1017/S0022112078000208. |
[21] |
D. Henry, On Gerstner's Water Wave, J. Nonlinear Math. Phys., 15 (2008), 87-95.
doi: 10.2991/jnmp.2008.15.s2.7. |
[22] |
R. S. Johnson, On the nonlinear critical layer below a nonlinear unsteady surface wave, J. Fluid Mech., 167 (1986), 327-351.
doi: 10.1017/S0022112086002847. |
[23] |
R. S. Johnson, On solutions of the Burns condition (which determines the speed of propagation of linear long waves on a shear flow with or without a critical layer), Geophys. Astrophys. Fluid. Dyn., 57 (1991), 115-133.
doi: 10.1080/03091929108225231. |
[24] |
R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,'' Cambridge University Press, Cambridge, 1997.
doi: 10.1017/CBO9780511624056. |
[25] |
M. Lakshmanan, Integrable Nonlinear wave equations and possible connections to tsunami dynamics, in "Tsunami and Nonlinear Waves'' (A. Kundu, ed.), Springer, (2007), pp. 31-49.
doi: 10.1007/978-3-540-71256-5_2. |
[26] |
J. Lighthill, "Waves in fluids,'' Cambridge University Press, Cambridge, 1978. |
[27] |
F. Omori, On tsunamis around Japan, Rep. Imp. Earthquake Comm., 34 (1902), 5-79. |
[28] |
W. J. M. Rankine, On the exact form of waves near the surface of deep water, Philos. Trans. Roy. Soc. London Ser. A, 153 (1863), 127-138. |
[29] |
H. Segur, The Korteweg-de Vries equation and water waves. Solutions of the equation. Part 1, J. Fluid Mech., 59 (1973), 721-736.
doi: 10.1017/S0022112073001813. |
[30] |
H. Segur, Waves in shallow water, with emphasis on the tsunami of 2004, in "Tsunami and Nonlinear Waves'' (A. Kundu, ed.), Springer, (2007), pp. 3-29.
doi: 10.1007/978-3-540-71256-5_1. |
[31] |
H. Segur, Integrable models of waves in shallow water, in "Probability, Geometry, and Integrable Systems for Henry McKean's Seventy-Fifth Birthday'' (eds. M. Pinsky and B. Birnir), MSRI Publ. (2008), 345-371. |
[32] |
R. Stuhlmeier, KdV theory and the Chilean tsunami of 1960, Discrete Cont. Dyn.-B, 12 (2009), 623-632.
doi: 10.3934/dcdsb.2009.12.623. |
[33] |
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. |
[34] |
F. Ursell, The long-wave paradox in the theory of gravity waves, Math. Proc. Cambridge Philos. Soc., 49 (1953), 685-694.
doi: 10.1017/S0305004100028887. |
[35] |
E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483.
doi: 10.1016/j.jde.2008.10.005. |
[36] |
C. S. Yih, Surface waves in flowing water, J. Fluid Mech., 51 (1972), 209-220. |
show all references
References:
[1] |
B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water waves and asymptotics, Invent. Math., 171 (2008), 485-541.
doi: 10.1007/s00222-007-0088-4. |
[2] |
T. B. Benjamin, The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech., 12 (1962), 97-116.
doi: 10.1017/S0022112062000063. |
[3] |
D. J. Benney and R. F. Bergeron, A new class of nonlinear waves in parallel flows (Nonlinear waves in parallel shear flows, discussing laminar flow breakdown due to free stream disturbances), Stud. Appl. Math., 48 (1969), 181-204. |
[4] |
N. Bowditch, "The American Practical Navigator," National Imagery and Mapping Agency, Bethesda, MD, 1995. |
[5] |
J. C. Burns, Long waves in running water, Proc. Camb. Phil. Soc., 4 (1953), 695-706.
doi: 10.1017/S0305004100028899. |
[6] |
A. Constantin, On the deep water wave motion, J. Phys. A, Math. Gen., 34 (2001), 1405-1417.
doi: 10.1088/0305-4470/34/7/313. |
[7] |
A. Constantin, On the propagation of tsunami waves, with emphasis on the tsunami of 2004, Discrete Cont. Dyn.-B, 12 (2009), 525-537.
doi: 10.3934/dcdsb.2009.12.525. |
[8] |
A. Constantin, On the relevance of soliton theory to tsunami modelling, Wave Motion, 46 (2009), 420-426.
doi: 10.1016/j.wavemoti.2009.05.002. |
[9] |
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. |
[10] |
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. |
[11] |
A. Constantin and D. Henry, Solitons and Tsunamis, Z. Naturforsch, 64a (2009), 65-68. |
[12] |
A. Constantin and R. S. Johnson, Modelling tsunamis, J. Phys. A, 39 (2006), L215-L217.
doi: 10.1088/0305-4470/39/14/L01. |
[13] |
A. Constantin and R. S. Johnson, On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves, J. Nonlinear Math. Phys., 15 (2008), 58-73.
doi: 10.2991/jnmp.2008.15.s2.5. |
[14] |
A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dyn. Res., 40 (2008), 175-211.
doi: 10.1088/0169-5983/42/3/038901. |
[15] |
A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Commun. Pure Appl. Math., 57 (2004), 481-527.
doi: 10.1002/cpa.3046. |
[16] |
A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Rational Mech. Anal., 199 (2011), 33-67.
doi: 10.1007/s00205-010-0314-x. |
[17] |
N. C. Freeman and R. S. Johnson, Shallow water waves on shear flows, J. Fluid Mech., 42 (1970), 401-409.
doi: 10.1017/S0022112070001349. |
[18] |
F. Gerstner, Theorie der Wellen samt einer abgeleiteten Theorie der Deichprofile, Abh. böhm. Ges. Wiss., 1 (1804). |
[19] |
J. L. Hammack and H. Segur, The Korteweg-de Vries equation and water waves. Part 2: Comparison with experiments, J. Fluid Mech., 65 (1974), 289-314.
doi: 10.1017/S002211207400139X. |
[20] |
J. L. Hammack and H. Segur, The Korteweg-de Vries equation and water waves. Part 3: Oscillatory waves, J. Fluid Mech., 84 (1978), 337-358.
doi: 10.1017/S0022112078000208. |
[21] |
D. Henry, On Gerstner's Water Wave, J. Nonlinear Math. Phys., 15 (2008), 87-95.
doi: 10.2991/jnmp.2008.15.s2.7. |
[22] |
R. S. Johnson, On the nonlinear critical layer below a nonlinear unsteady surface wave, J. Fluid Mech., 167 (1986), 327-351.
doi: 10.1017/S0022112086002847. |
[23] |
R. S. Johnson, On solutions of the Burns condition (which determines the speed of propagation of linear long waves on a shear flow with or without a critical layer), Geophys. Astrophys. Fluid. Dyn., 57 (1991), 115-133.
doi: 10.1080/03091929108225231. |
[24] |
R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,'' Cambridge University Press, Cambridge, 1997.
doi: 10.1017/CBO9780511624056. |
[25] |
M. Lakshmanan, Integrable Nonlinear wave equations and possible connections to tsunami dynamics, in "Tsunami and Nonlinear Waves'' (A. Kundu, ed.), Springer, (2007), pp. 31-49.
doi: 10.1007/978-3-540-71256-5_2. |
[26] |
J. Lighthill, "Waves in fluids,'' Cambridge University Press, Cambridge, 1978. |
[27] |
F. Omori, On tsunamis around Japan, Rep. Imp. Earthquake Comm., 34 (1902), 5-79. |
[28] |
W. J. M. Rankine, On the exact form of waves near the surface of deep water, Philos. Trans. Roy. Soc. London Ser. A, 153 (1863), 127-138. |
[29] |
H. Segur, The Korteweg-de Vries equation and water waves. Solutions of the equation. Part 1, J. Fluid Mech., 59 (1973), 721-736.
doi: 10.1017/S0022112073001813. |
[30] |
H. Segur, Waves in shallow water, with emphasis on the tsunami of 2004, in "Tsunami and Nonlinear Waves'' (A. Kundu, ed.), Springer, (2007), pp. 3-29.
doi: 10.1007/978-3-540-71256-5_1. |
[31] |
H. Segur, Integrable models of waves in shallow water, in "Probability, Geometry, and Integrable Systems for Henry McKean's Seventy-Fifth Birthday'' (eds. M. Pinsky and B. Birnir), MSRI Publ. (2008), 345-371. |
[32] |
R. Stuhlmeier, KdV theory and the Chilean tsunami of 1960, Discrete Cont. Dyn.-B, 12 (2009), 623-632.
doi: 10.3934/dcdsb.2009.12.623. |
[33] |
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. |
[34] |
F. Ursell, The long-wave paradox in the theory of gravity waves, Math. Proc. Cambridge Philos. Soc., 49 (1953), 685-694.
doi: 10.1017/S0305004100028887. |
[35] |
E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483.
doi: 10.1016/j.jde.2008.10.005. |
[36] |
C. S. Yih, Surface waves in flowing water, J. Fluid Mech., 51 (1972), 209-220. |
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