# American Institute of Mathematical Sciences

August  2014, 34(8): 3109-3123. doi: 10.3934/dcds.2014.34.3109

## Dispersion relations for periodic water waves with surface tension and discontinuous vorticity

 1 Institut für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria

Received  August 2013 Revised  September 2013 Published  January 2014

We derive the dispersion relation for water waves with surface tension and having a piecewise constant vorticity distribution. More precisely, we consider here two scenarios; the first one is that of a flow with constant non-zero vorticity adjacent to the flat bed while above this layer of vorticity we assume the flow to be irrotational. The second type of flow has a layer of non-vanishing vorticity adjacent to the free surface and is irrotational below.
Citation: Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109
##### References:
 [1] 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. [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, 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (2011). doi: 10.1137/1.9781611971873. [4] A. Constantin, Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity, Commun. Pure Appl. Anal., 11 (2012), 1397-1406. doi: 10.3934/cpaa.2012.11.1397. [5] A. Constantin, Mean velocities in a Stokes wave, Arch. Ration. Mech. Anal., 207 (2013), 907-917. doi: 10.1007/s00205-012-0584-6. [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] 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. [9] A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046. [10] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299. [11] A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4. [12] A. Constantin and W. A. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950. doi: 10.1002/cpa.20165. [13] A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation, Arch. Ration. Mech. Anal., 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x. [14] M. Ehrnström, J. Escher and E. Wahlén, Steady water waves with multiple critical layers, SIAM J. Math. Anal., 43 (2011), 1436-1456. doi: 10.1137/100792330. [15] J. Escher, Regularity of rotational travelling water waves, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370 (2012), 1602-1615. doi: 10.1098/rsta.2011.0458. [16] J. Escher, A.-V. Matioc and B.-V. Matioc, On stratified steady periodic water waves with linear density distribution and stagnation points, J. Differential Equations, 251 (2011), 2932-2949. doi: 10.1016/j.jde.2011.03.023. [17] D. Henry, Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity, SIAM J. Math. Anal, 42 (2010), 3103-3111. doi: 10.1137/100801408. [18] D. Henry, Analyticity of the free surface for periodic travelling capillary-gravity water waves with vorticity, J. Math. Fluid Mech., 14 (2012), 249-254. doi: 10.1007/s00021-011-0056-z. [19] D. Henry, Dispersion relations for steady periodic water waves with an isolated layer of vorticity at the surface, Nonlinear Anal. Real World Appl., 14 (2013), 1034-1043. doi: 10.1016/j.nonrwa.2012.08.015. [20] D. Henry and B.-V. Matioc, On the regularity of steady periodic stratified water waves, Commun. Pure Appl. Anal., 11 (2012), 1453-1464. doi: 10.3934/cpaa.2012.11.1453. [21] R. S. Johnson, A modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056. [22] I. G. Jonsson, Wave-current interactions. In: The Sea , Wiley, New York, 1990. [23] J. Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2001, Reprint of the 1978 original. doi: 10.1007/s002050100160. [24] C. I. Martin, Local bifurcation and regularity for steady periodic capillary-gravity water waves with constant vorticity, Nonlinear Anal. Real World Appl., 14 (2013), 131-149. doi: 10.1016/j.nonrwa.2012.05.007. [25] C. I. Martin, Local bifurcation for steady periodic capillary water waves with constant vorticity, J. Math. Fluid Mech., 15 (2013), 155-170. doi: 10.1007/s00021-012-0096-z. [26] C. I. Martin and B.-V. Matioc, Existence of capillary-gravity water waves with piecewise constant vorticity, arXiv:1302.5523. [27] A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity, Differential Integral Equations, 26 (2013), 129-140. [28] B.-V. Matioc, Analyticity of the streamlines for periodic traveling water waves with bounded vorticity, Int. Math. Res. Not., 17 (2011), 3858-3871. doi: 10.1093/imrn/rnq235. [29] G. Thomas and G. Klopman, Wave-Current Interactions in the Nearshore Region, WIT, Southampton, United Kingdom, 1997. [30] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. [31] E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal., 38 (2006), 921-943 (electronic). doi: 10.1137/050630465. [32] E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005. [33] S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583. [34] L.-J. Wang, Regularity of traveling periodic stratified water waves with vorticity, Nonlinear Anal., 81 (2013), 247-263. doi: 10.1016/j.na.2012.11.009.

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##### References:
 [1] 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. [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, 81 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (2011). doi: 10.1137/1.9781611971873. [4] A. Constantin, Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity, Commun. Pure Appl. Anal., 11 (2012), 1397-1406. doi: 10.3934/cpaa.2012.11.1397. [5] A. Constantin, Mean velocities in a Stokes wave, Arch. Ration. Mech. Anal., 207 (2013), 907-917. doi: 10.1007/s00205-012-0584-6. [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] 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. [9] A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046. [10] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299. [11] A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4. [12] A. Constantin and W. A. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., 60 (2007), 911-950. doi: 10.1002/cpa.20165. [13] A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation, Arch. Ration. Mech. Anal., 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x. [14] M. Ehrnström, J. Escher and E. Wahlén, Steady water waves with multiple critical layers, SIAM J. Math. Anal., 43 (2011), 1436-1456. doi: 10.1137/100792330. [15] J. Escher, Regularity of rotational travelling water waves, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 370 (2012), 1602-1615. doi: 10.1098/rsta.2011.0458. [16] J. Escher, A.-V. Matioc and B.-V. Matioc, On stratified steady periodic water waves with linear density distribution and stagnation points, J. Differential Equations, 251 (2011), 2932-2949. doi: 10.1016/j.jde.2011.03.023. [17] D. Henry, Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity, SIAM J. Math. Anal, 42 (2010), 3103-3111. doi: 10.1137/100801408. [18] D. Henry, Analyticity of the free surface for periodic travelling capillary-gravity water waves with vorticity, J. Math. Fluid Mech., 14 (2012), 249-254. doi: 10.1007/s00021-011-0056-z. [19] D. Henry, Dispersion relations for steady periodic water waves with an isolated layer of vorticity at the surface, Nonlinear Anal. Real World Appl., 14 (2013), 1034-1043. doi: 10.1016/j.nonrwa.2012.08.015. [20] D. Henry and B.-V. Matioc, On the regularity of steady periodic stratified water waves, Commun. Pure Appl. Anal., 11 (2012), 1453-1464. doi: 10.3934/cpaa.2012.11.1453. [21] R. S. Johnson, A modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511624056. [22] I. G. Jonsson, Wave-current interactions. In: The Sea , Wiley, New York, 1990. [23] J. Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2001, Reprint of the 1978 original. doi: 10.1007/s002050100160. [24] C. I. Martin, Local bifurcation and regularity for steady periodic capillary-gravity water waves with constant vorticity, Nonlinear Anal. Real World Appl., 14 (2013), 131-149. doi: 10.1016/j.nonrwa.2012.05.007. [25] C. I. Martin, Local bifurcation for steady periodic capillary water waves with constant vorticity, J. Math. Fluid Mech., 15 (2013), 155-170. doi: 10.1007/s00021-012-0096-z. [26] C. I. Martin and B.-V. Matioc, Existence of capillary-gravity water waves with piecewise constant vorticity, arXiv:1302.5523. [27] A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity, Differential Integral Equations, 26 (2013), 129-140. [28] B.-V. Matioc, Analyticity of the streamlines for periodic traveling water waves with bounded vorticity, Int. Math. Res. Not., 17 (2011), 3858-3871. doi: 10.1093/imrn/rnq235. [29] G. Thomas and G. Klopman, Wave-Current Interactions in the Nearshore Region, WIT, Southampton, United Kingdom, 1997. [30] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. [31] E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal., 38 (2006), 921-943 (electronic). doi: 10.1137/050630465. [32] E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005. [33] S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583. [34] L.-J. Wang, Regularity of traveling periodic stratified water waves with vorticity, Nonlinear Anal., 81 (2013), 247-263. doi: 10.1016/j.na.2012.11.009.
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