# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 547-558. doi: 10.1017/jfm.2013.253. [2] D. Clamond and A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed, J. Fluid Mech., 714 (2013), 463-475. doi: 10.1017/jfm.2012.490. [3] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [4] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2011. doi: 10.1137/1.9781611971873. [5] A. Constantin, Estimating wave heights from pressure data at the bed, J. Fluid Mech., 743 (2014), 10pp. doi: 10.1017/jfm.2014.81. [6] A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1. [7] A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773. [8] M. Ehrnström, H. Holden and X. Raynaud, Symmetric Waves Are Traveling Waves, International Mathematics Research Notices, 2009 (2009), 4578-4596. doi: 10.1093/imrn/rnp100. [9] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge, 1997. doi: 10.1017/CBO9780511624056. [10] F. Kogelbauer, Recovery of the wave profile for irrotational periodic water waves from pressure measurements, Nonl. Anal.: Real World Appl., 22 (2015), 219-224. doi: 10.1016/j.nonrwa.2014.09.003. [11] F. Kogelbauer, Symmetric irrotational water waves are traveling waves, J. Diff. Eq., 259 (2015), 5271-5275. doi: 10.1016/j.jde.2015.06.025. [12] S. Lang, Complex Analysis, Graduate Texts in Mathematics, Springer, 2003. [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), 3125-3133. doi: 10.3934/dcds.2014.34.3125. [14] H. Okamoto and M. Shoji, The Mathematical Theory of Permanent Progressive Water-waves, World Scientific, 2001. doi: 10.1142/4547. [15] G. Tulzer, On the symmetry of steady periodic water waves with stagnation points, Comm. Pure Appl. Anal., 11 (2012), 1577-1586. doi: 10.3934/cpaa.2012.11.1577.

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##### References:
 [1] D. Clamond, New exact relations for easy recovery of steady wave profiles from bottom Pressure measurements, J. Fluid Mech., 726 (2013), 547-558. doi: 10.1017/jfm.2013.253. [2] D. Clamond and A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed, J. Fluid Mech., 714 (2013), 463-475. doi: 10.1017/jfm.2012.490. [3] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [4] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2011. doi: 10.1137/1.9781611971873. [5] A. Constantin, Estimating wave heights from pressure data at the bed, J. Fluid Mech., 743 (2014), 10pp. doi: 10.1017/jfm.2014.81. [6] A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1. [7] A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181. doi: 10.1017/S0022112003006773. [8] M. Ehrnström, H. Holden and X. Raynaud, Symmetric Waves Are Traveling Waves, International Mathematics Research Notices, 2009 (2009), 4578-4596. doi: 10.1093/imrn/rnp100. [9] R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge, 1997. doi: 10.1017/CBO9780511624056. [10] F. Kogelbauer, Recovery of the wave profile for irrotational periodic water waves from pressure measurements, Nonl. Anal.: Real World Appl., 22 (2015), 219-224. doi: 10.1016/j.nonrwa.2014.09.003. [11] F. Kogelbauer, Symmetric irrotational water waves are traveling waves, J. Diff. Eq., 259 (2015), 5271-5275. doi: 10.1016/j.jde.2015.06.025. [12] S. Lang, Complex Analysis, Graduate Texts in Mathematics, Springer, 2003. [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), 3125-3133. doi: 10.3934/dcds.2014.34.3125. [14] H. Okamoto and M. Shoji, The Mathematical Theory of Permanent Progressive Water-waves, World Scientific, 2001. doi: 10.1142/4547. [15] G. Tulzer, On the symmetry of steady periodic water waves with stagnation points, Comm. Pure Appl. Anal., 11 (2012), 1577-1586. doi: 10.3934/cpaa.2012.11.1577.
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