August  2014, 34(8): 3171-3182. doi: 10.3934/dcds.2014.34.3171

Progressive waves on a blunt interface

1. 

Faculty of Civil and Environmental Engineering, Technion - Israel Institute of Technology, 32000 Haifa, Israel

2. 

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

Received  August 2013 Revised  September 2013 Published  January 2014

We present a new exact solution describing progressive waves on a blunt interface based on Gerstner's trochoidal wave. The second-order irrotational theory is developed for a sharp interface, and subsequently for three fluid layers, the upper and lower of which may approach one another to form the so-called blunt interface. This situation is captured analogously by our exact rotational solution. We establish remarkable agreement between the exact and second-order theories, and present applications to surface water waves.
Citation: Michael Stiassnie, Raphael Stuhlmeier. Progressive waves on a blunt interface. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3171-3182. doi: 10.3934/dcds.2014.34.3171
References:
[1]

A. Aleman and A. Constantin, Harmonic maps and ideal fluid flows,, Arch. Rat. Mech. Anal., 204 (2012), 479. doi: 10.1007/s00205-011-0483-2. Google Scholar

[2]

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

[3]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, SIAM, (2012). doi: 10.1137/1.9781611971873. Google Scholar

[4]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Commun. Pure Appl. Math., 63 (2010), 533. doi: 10.1002/cpa.20299. Google Scholar

[5]

P. G. Drazin, Introduction to Hydrodynamic Stability,, Cambridge University Press, (2002). Google Scholar

[6]

F. Gerstner, Theorie der Wellen Samt Einer Daraus Abgeleiteten Theorie der Deichprofile,, Abhandlungen der kön. böhmischen Gesellschaft der Wissenschaften, (1804). Google Scholar

[7]

D. Henry., On the deep-water stokes wave flow., Int. Math. Res. Not., 2008 (2008). doi: 10.1093/imrn/rnn071. Google Scholar

[8]

B. Kinsman, Wind Waves,, Dover, (1984). doi: 10.1029/JZ066i008p02411. Google Scholar

[9]

H. Lamb, Hydrodynamics,, Cambridge University Press, (1895). Google Scholar

[10]

S. Leblanc, Local stability of Gerstner's waves,, J. Fluid Mech., 506 (2004), 245. doi: 10.1017/S0022112004008444. Google Scholar

[11]

C. C. Mei, M. Stiassnie and D. K.-P. Yue, Theory and Applications of Ocean Surface Waves,, World Scientific Publishing Co., (2005). Google Scholar

[12]

E. Mollo-Christensen, Gravitational and geostrophic billows: Some exact solutions,, J. Atmos. Sci., 35 (1978), 1395. doi: 10.1175/1520-0469(1978)035<1395:GAGBSE>2.0.CO;2. Google Scholar

[13]

R. Stuhlmeier, Internal Gerstner waves: Applications to dead water,, Appl. Anal., (). doi: 10.1080/00036811.2013.833609. Google Scholar

show all references

References:
[1]

A. Aleman and A. Constantin, Harmonic maps and ideal fluid flows,, Arch. Rat. Mech. Anal., 204 (2012), 479. doi: 10.1007/s00205-011-0483-2. Google Scholar

[2]

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

[3]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, SIAM, (2012). doi: 10.1137/1.9781611971873. Google Scholar

[4]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Commun. Pure Appl. Math., 63 (2010), 533. doi: 10.1002/cpa.20299. Google Scholar

[5]

P. G. Drazin, Introduction to Hydrodynamic Stability,, Cambridge University Press, (2002). Google Scholar

[6]

F. Gerstner, Theorie der Wellen Samt Einer Daraus Abgeleiteten Theorie der Deichprofile,, Abhandlungen der kön. böhmischen Gesellschaft der Wissenschaften, (1804). Google Scholar

[7]

D. Henry., On the deep-water stokes wave flow., Int. Math. Res. Not., 2008 (2008). doi: 10.1093/imrn/rnn071. Google Scholar

[8]

B. Kinsman, Wind Waves,, Dover, (1984). doi: 10.1029/JZ066i008p02411. Google Scholar

[9]

H. Lamb, Hydrodynamics,, Cambridge University Press, (1895). Google Scholar

[10]

S. Leblanc, Local stability of Gerstner's waves,, J. Fluid Mech., 506 (2004), 245. doi: 10.1017/S0022112004008444. Google Scholar

[11]

C. C. Mei, M. Stiassnie and D. K.-P. Yue, Theory and Applications of Ocean Surface Waves,, World Scientific Publishing Co., (2005). Google Scholar

[12]

E. Mollo-Christensen, Gravitational and geostrophic billows: Some exact solutions,, J. Atmos. Sci., 35 (1978), 1395. doi: 10.1175/1520-0469(1978)035<1395:GAGBSE>2.0.CO;2. Google Scholar

[13]

R. Stuhlmeier, Internal Gerstner waves: Applications to dead water,, Appl. Anal., (). doi: 10.1080/00036811.2013.833609. Google Scholar

[1]

Raphael Stuhlmeier. Internal Gerstner waves on a sloping bed. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3183-3192. doi: 10.3934/dcds.2014.34.3183

[2]

Anatoly Abrashkin. Wind generated equatorial Gerstner-type waves. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4443-4453. doi: 10.3934/dcds.2019181

[3]

Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Ⅱ. Bore propagation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5543-5569. doi: 10.3934/dcds.2019244

[4]

José Raúl Quintero, Juan Carlos Muñoz Grajales. Solitary waves for an internal wave model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5721-5741. doi: 10.3934/dcds.2016051

[5]

Jerry L. Bona, Henrik Kalisch. Models for internal waves in deep water. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 1-20. doi: 10.3934/dcds.2000.6.1

[6]

Mark Jones. The bifurcation of interfacial capillary-gravity waves under O(2) symmetry. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1183-1204. doi: 10.3934/cpaa.2011.10.1183

[7]

Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239

[8]

Hung-Chu Hsu. Exact azimuthal internal waves with an underlying current. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4391-4398. doi: 10.3934/dcds.2017188

[9]

Tony Lyons. Geophysical internal equatorial waves of extreme form. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4471-4486. doi: 10.3934/dcds.2019183

[10]

Mateusz Kluczek. Nonhydrostatic Pollard-like internal geophysical waves. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5171-5183. doi: 10.3934/dcds.2019210

[11]

Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i

[12]

Ralph Lteif, Samer Israwi, Raafat Talhouk. An improved result for the full justification of asymptotic models for the propagation of internal waves. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2203-2230. doi: 10.3934/cpaa.2015.14.2203

[13]

Dmitry Treschev. Travelling waves in FPU lattices. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 867-880. doi: 10.3934/dcds.2004.11.867

[14]

Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607

[15]

Grégoire Allaire, Carlos Conca, Luis Friz, Jaime H. Ortega. On Bloch waves for the Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 1-28. doi: 10.3934/dcdsb.2007.7.1

[16]

Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103

[17]

Mikhail Kovalyov. On the nature of large and rogue waves. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3061-3093. doi: 10.3934/dcds.2014.34.3061

[18]

Lutz Recke, Anatoly Samoilenko, Alexey Teplinsky, Viktor Tkachenko, Serhiy Yanchuk. Frequency locking of modulated waves. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 847-875. doi: 10.3934/dcds.2011.31.847

[19]

Stephen Coombes, Helmut Schmidt, Carlo R. Laing, Nils Svanstedt, John A. Wyller. Waves in random neural media. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951

[20]

Paolo Paoletti. Acceleration waves in complex materials. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 637-659. doi: 10.3934/dcdsb.2012.17.637

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]