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Mathematische Annalen

, Volume 357, Issue 1, pp 23–30 | Cite as

On the pressure transfer function for solitary water waves with vorticity

  • David HenryEmail author
Article

Abstract

In this paper we analyse the role which the pressure function on the sea-bed plays in determining solitary waves with vorticity. We prove that the pressure function on the flat bed determines a unique, real analytic solitary wave solution to the governing equations, given a real analytic vorticity distribution. In particular, the pressure function on the flat bed prescribes a unique surface profile for the resulting solitary water wave.

Mathematics Subject Classification (1991)

35Q35 76B25 76B03 

Notes

Acknowledgments

The author would like to thank the anonymous referee for useful suggestions.

References

  1. 1.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12, 623–727 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Amick, C.J., Fraenkel, L.E., Toland, J.F.: On the Stokes conjecture for the wave of extreme form. Acta Math. 148, 193–214 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Baquerizo, A., Losada, M.A.: Transfer function between wave height and wave pressure for progressive waves. Coastal Eng. 24, 351–353 (1995)CrossRefGoogle Scholar
  4. 4.
    Bishop, C.T., Donelan, M.A.: Measuring waves with pressure transducers. Coastal Eng. 11, 309–328 (1987)CrossRefGoogle Scholar
  5. 5.
    Constantin, A.: On the recovery of solitary wave profiles from pressure measurements. J. Fluid Mech. 699, 376–384 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Constantin, A.: Nonlinear water waves with applications to wave–current interactions and tsunamis. CBMS-NSF Conference Series in Applied Mathematics, vol. 81. SIAM, Philadelphia (2011)Google Scholar
  7. 7.
    Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Constantin, A., Escher, J.: Analyticity of periodic travelling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Constantin, A., Escher, J., Hsu, H.-C.: Pressure beneath a solitary water wave: mathematical theory and experiments. Arch. Ration. Mech. Anal. 201, 251–269 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Constantin, A., Strauss, W.: Pressure and trajectories beneath a Stokes wave. Comm. Pure Appl. Math. 63, 533–557 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Constantin, A., Strauss, W.: Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57, 481–527 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Deconinck, B., Henderson, D., Oliveras, K.L., Vasan, V.: Recovering the water-wave surface from pressure measurements. In: Proceedings of 10th International Conference on WAVES, Vancouver, July 25–29, (2011)Google Scholar
  14. 14.
    Escher, J., Schlurmann, T.: On the recovery of the free surface from the pressure within periodic traveling water waves. J. Nonlinear Math. Phys. 15, 50–57 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Groves, M.D., Wahlén, E.: Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity. Physica D 237, 1530–1538 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Henry, D.: Pressure in a deep-water Stokes wave. J. Math. Fluid Mech. 13, 251–257 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Henry, D.: On the deep-water Stokes wave flow. Int. Math. Res. Not. Art. ID rnn071 (2008)Google Scholar
  18. 18.
    Hur, V.M.: Exact solitary water waves with vorticity. Arch. Ration. Mech. Anal. 188, 213–244 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    John, F.: Partial Differential Equations. Springer, Berlin (1991)Google Scholar
  20. 20.
    Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions. Birkhauser, Basel (1992)zbMATHCrossRefGoogle Scholar
  21. 21.
    Kuo, Y.-Y., Chiu, J.-F.: Transfer function between the wave height and wave pressure for progressive waves. Coastal Eng. 23, 81–93 (1994)CrossRefGoogle Scholar
  22. 22.
    Lewy, H.: A note on harmonic functions and a hydrodynamical application. Proc. Am. Math. Soc. 3, 111–113 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966)zbMATHGoogle Scholar
  24. 24.
    Oliveras, K.L., Vasan, V., Deconinck, B., Henderson, D.: Recovering surface elevation from pressure data. SIAM J. Appl. Math (in press)Google Scholar
  25. 25.
    Thomas, G., Klopman, G.: Wave–current interactions in the nearshore region. In: Gravity Waves in Water of Finite Depth, Advances in Fluid Mechanics, vol. 10, pp. 215–319. WIT, Southampton (1997)Google Scholar
  26. 26.
    Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Tsai, C.-H., Huang, M.-C., Young, F.-J., Lin, Y.-C., Li, H.-W.: On the recovery of surface wave by pressure transfer function. Ocean Eng. 32, 1247–1259 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity College CorkCorkIreland

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