December  2016, 36(12): 7057-7061. doi: 10.3934/dcds.2016107

On the symmetry of spatially periodic two-dimensional water waves

1. 

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

Received  December 2015 Revised  January 2016 Published  October 2016

We show that a spatially periodic solution to the irrotational two-dimensional gravity water wave problem, with the property that the horizontal velocity component at the flat bed is symmetric, while the acceleration at the flat bed is anti-symmetric with respect to a common axis of symmetry, necessarily constitutes a traveling wave. The proof makes use complex variables and structural properties of the governing equations for nonlinear water waves.
Citation: Florian Kogelbauer. On the symmetry of spatially periodic two-dimensional water waves. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7057-7061. doi: 10.3934/dcds.2016107
References:
[1]

D. Clamond, New exact relations for easy recovery of steady wave profiles from bottom Pressure measurements,, J. Fluid Mech., 726 (2013), 547.  doi: 10.1017/jfm.2013.253.  Google Scholar

[2]

D. Clamond and A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed,, J. Fluid Mech., 714 (2013), 463.  doi: 10.1017/jfm.2012.490.  Google Scholar

[3]

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

[4]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, CBMS-NSF Regional Conference Series in Applied Mathematics, (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[5]

A. Constantin, Estimating wave heights from pressure data at the bed,, J. Fluid Mech., 743 (2014).  doi: 10.1017/jfm.2014.81.  Google Scholar

[6]

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.  Google Scholar

[7]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity,, J. Fluid Mech., 498 (2004), 171.  doi: 10.1017/S0022112003006773.  Google Scholar

[8]

M. Ehrnström, H. Holden and X. Raynaud, Symmetric Waves Are Traveling Waves,, International Mathematics Research Notices, 2009 (2009), 4578.  doi: 10.1093/imrn/rnp100.  Google Scholar

[9]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997).  doi: 10.1017/CBO9780511624056.  Google Scholar

[10]

F. Kogelbauer, Recovery of the wave profile for irrotational periodic water waves from pressure measurements,, Nonl. Anal.: Real World Appl., 22 (2015), 219.  doi: 10.1016/j.nonrwa.2014.09.003.  Google Scholar

[11]

F. Kogelbauer, Symmetric irrotational water waves are traveling waves,, J. Diff. Eq., 259 (2015), 5271.  doi: 10.1016/j.jde.2015.06.025.  Google Scholar

[12]

S. Lang, Complex Analysis,, Graduate Texts in Mathematics, (2003).   Google Scholar

[13]

B.-V. Matioc, A characterization of the symmetric steady water waves in terms of the underlying flow,, Discrete Contin. Dyn. Syst., A 34 (2014), 3125.  doi: 10.3934/dcds.2014.34.3125.  Google Scholar

[14]

H. Okamoto and M. Shoji, The Mathematical Theory of Permanent Progressive Water-waves,, World Scientific, (2001).  doi: 10.1142/4547.  Google Scholar

[15]

G. Tulzer, On the symmetry of steady periodic water waves with stagnation points,, Comm. Pure Appl. Anal., 11 (2012), 1577.  doi: 10.3934/cpaa.2012.11.1577.  Google Scholar

show all references

References:
[1]

D. Clamond, New exact relations for easy recovery of steady wave profiles from bottom Pressure measurements,, J. Fluid Mech., 726 (2013), 547.  doi: 10.1017/jfm.2013.253.  Google Scholar

[2]

D. Clamond and A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed,, J. Fluid Mech., 714 (2013), 463.  doi: 10.1017/jfm.2012.490.  Google Scholar

[3]

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

[4]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, CBMS-NSF Regional Conference Series in Applied Mathematics, (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[5]

A. Constantin, Estimating wave heights from pressure data at the bed,, J. Fluid Mech., 743 (2014).  doi: 10.1017/jfm.2014.81.  Google Scholar

[6]

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.  Google Scholar

[7]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity,, J. Fluid Mech., 498 (2004), 171.  doi: 10.1017/S0022112003006773.  Google Scholar

[8]

M. Ehrnström, H. Holden and X. Raynaud, Symmetric Waves Are Traveling Waves,, International Mathematics Research Notices, 2009 (2009), 4578.  doi: 10.1093/imrn/rnp100.  Google Scholar

[9]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Texts in Applied Mathematics, (1997).  doi: 10.1017/CBO9780511624056.  Google Scholar

[10]

F. Kogelbauer, Recovery of the wave profile for irrotational periodic water waves from pressure measurements,, Nonl. Anal.: Real World Appl., 22 (2015), 219.  doi: 10.1016/j.nonrwa.2014.09.003.  Google Scholar

[11]

F. Kogelbauer, Symmetric irrotational water waves are traveling waves,, J. Diff. Eq., 259 (2015), 5271.  doi: 10.1016/j.jde.2015.06.025.  Google Scholar

[12]

S. Lang, Complex Analysis,, Graduate Texts in Mathematics, (2003).   Google Scholar

[13]

B.-V. Matioc, A characterization of the symmetric steady water waves in terms of the underlying flow,, Discrete Contin. Dyn. Syst., A 34 (2014), 3125.  doi: 10.3934/dcds.2014.34.3125.  Google Scholar

[14]

H. Okamoto and M. Shoji, The Mathematical Theory of Permanent Progressive Water-waves,, World Scientific, (2001).  doi: 10.1142/4547.  Google Scholar

[15]

G. Tulzer, On the symmetry of steady periodic water waves with stagnation points,, Comm. Pure Appl. Anal., 11 (2012), 1577.  doi: 10.3934/cpaa.2012.11.1577.  Google Scholar

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