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On the decrease of velocity with depth in irrotational periodic water waves

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Abstract

We give an alternative proof for a classical result (due to Longuet-Higgins) that provides an estimate for the decay rate with depth of the velocity beneath two-dimensional, spatially periodic, irrotational water waves over a flat bed. Furthermore, an improvement to the same estimate is presented.

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References

  1. Aleman, A., Constantin, A.: On the decrease of kinetic energy with depth in wave-current interactions. Math. Ann. (2019). https://doi.org/10.1007/s00208-019-01910-8

    Article  Google Scholar 

  2. Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Constantin, A.: Nonlinear water waves with applications to wave-current interactions and Tsunamis. In: CBMS-NSF Regional Conference Series in Applied Mathematics, 81, SIAM, Philadelphia, PA (2011)

  4. Constantin, A.: Extrema of the dynamic pressure in an irrotational regular wave train. Phys. Fluids 28, 113604 (2016). https://doi.org/10.1063/1.4967362

    Article  Google Scholar 

  5. Constantin, A., Strauss, W.A.: Pressure beneath a Stokes wave. Comm. Pure Appl. Math. 53, 533–557 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Da Silva, A.F.T., Peregrine, D.H.: Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 192, 281–302 (1988)

    MathSciNet  Google Scholar 

  7. Dean, R.G., Dalrymple, R.A.: Water Wave Mechanics for Engineers and Scientists. World Scientific, Singapore (2000)

    Google Scholar 

  8. Garrison, T.S.: Essentials of Oceanography. Cengage Learning, Brooks/Cole (2009)

    Google Scholar 

  9. Lannes, D.: Well-posedness of the water wave equations. J. Am. Math. Soc. 18, 605–654 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Longuet-Higgins, M.S.: On the decrease of velocity with depth in an irrotational water wave. Math. Proc. Camb. Philos. Soc. 49, 552–560 (1953)

    MathSciNet  MATH  Google Scholar 

  11. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  12. Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39–72 (1997)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is grateful for insightful comments and suggestions from the referees.

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  1. Faculty of Mathematics, University of Vienna, Oskar–Morgenstern–Platz 1, 1090, Vienna, Austria

    Luigi Roberti

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  1. Luigi Roberti
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Correspondence to Luigi Roberti.

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Communicated by Adrian Constantin.

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Roberti, L. On the decrease of velocity with depth in irrotational periodic water waves. Monatsh Math 193, 671–682 (2020). https://doi.org/10.1007/s00605-020-01451-2

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  • DOI: https://doi.org/10.1007/s00605-020-01451-2

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