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 and Continuous Dynamical Systems, 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), 479-513. doi: 10.1007/s00205-011-0483-2.

[2]

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

[3]

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

[4]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Commun. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299.

[5]

P. G. Drazin, Introduction to Hydrodynamic Stability, Cambridge University Press, 2002.

[6]

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

[7]

D. Henry., On the deep-water stokes wave flow. Int. Math. Res. Not., 2008 (2008), 7 pp. doi: 10.1093/imrn/rnn071.

[8]

B. Kinsman, Wind Waves, Dover, New York, 1984. doi: 10.1029/JZ066i008p02411.

[9]

H. Lamb, Hydrodynamics, Cambridge University Press, Cambridge, 1895.

[10]

S. Leblanc, Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245-254. doi: 10.1017/S0022112004008444.

[11]

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

[12]

E. Mollo-Christensen, Gravitational and geostrophic billows: Some exact solutions, J. Atmos. Sci., 35 (1978), 1395-1398. doi: 10.1175/1520-0469(1978)035<1395:GAGBSE>2.0.CO;2.

[13]

R. Stuhlmeier, Internal Gerstner waves: Applications to dead water, Appl. Anal., to appear. doi: 10.1080/00036811.2013.833609.

show all references

References:
[1]

A. Aleman and A. Constantin, Harmonic maps and ideal fluid flows, Arch. Rat. Mech. Anal., 204 (2012), 479-513. doi: 10.1007/s00205-011-0483-2.

[2]

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

[3]

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

[4]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Commun. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299.

[5]

P. G. Drazin, Introduction to Hydrodynamic Stability, Cambridge University Press, 2002.

[6]

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

[7]

D. Henry., On the deep-water stokes wave flow. Int. Math. Res. Not., 2008 (2008), 7 pp. doi: 10.1093/imrn/rnn071.

[8]

B. Kinsman, Wind Waves, Dover, New York, 1984. doi: 10.1029/JZ066i008p02411.

[9]

H. Lamb, Hydrodynamics, Cambridge University Press, Cambridge, 1895.

[10]

S. Leblanc, Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245-254. doi: 10.1017/S0022112004008444.

[11]

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

[12]

E. Mollo-Christensen, Gravitational and geostrophic billows: Some exact solutions, J. Atmos. Sci., 35 (1978), 1395-1398. doi: 10.1175/1520-0469(1978)035<1395:GAGBSE>2.0.CO;2.

[13]

R. Stuhlmeier, Internal Gerstner waves: Applications to dead water, Appl. Anal., to appear. doi: 10.1080/00036811.2013.833609.

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