The Exact Mathematical Models of Nonlinear Surface Waves

Conference paper
Part of the Springer Geology book series (SPRINGERGEOL)

Abstract

The problem of exact mathematical models of potential surface waves propagation is considered. The existing models converting the original system of hydrodynamic equations and boundary conditions into some new form are analyzed. A new approach to the above mentioned problem is presented resulting in the single equation for the waveform. The relations permitting to reconstruct all physical fields on the base of this waveform are formulated. It is shown how on the base of the equation received to construct all well-known results of approximate theories of stationary surface waves.

Keywords

Potential waves Waveform Stream function 

References

  1. 1.
    Nekrasov, A.I.: The exact theory of steady waves on the surface of heavy liquid, collection of works by A.I. Nekrasov. Academy of Science, Moscow, USSR, pp. 358–439 (1961). (In Russian)Google Scholar
  2. 2.
    Levi-Civita, T.: Determinazione rigorosa delle onde irrotazionale periodiche in aqua profonda. Atti. Accad. Licei. 33(5), 141–150 (1924)MATHGoogle Scholar
  3. 3.
    Struik, D.J.: Dtermination rigoureuse des ondes irrotationelles permanentes dans un canal profondeur finie. Math. Ann. 95, 595–634 (1926)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Byatt-Smith, J.G.B.: An integral equation for unsteady surface waves and a comment on the Boussinesq equation. J. Fluid Mech. 49, 625–633 (1971)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ablowitz, M.J., Fokas, A.S., Musslimani, Z.H.: On an new non-local theory of water waves. J. Fluid Mech. 562, 313–343 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kistovich, A.V., Chashechkin, Y.D.: An integral description of the propagation of steady-stable perturbations of the heavy liquid surface. Izv. Atmos. Ocean. Phys. 45, 654–659 (2009).  https://doi.org/10.1134/S0001433809050120 CrossRefMATHGoogle Scholar
  7. 7.
    Kistovich, A.V., Chashechkin, Y.D.: Analytical models of stationary non-linear gravitational waves. Water Resour. 43(1), 86–94 (2016).  https://doi.org/10.1134/S0097807816120083 CrossRefGoogle Scholar
  8. 8.
    Kistovich, A.V.: The exact complex-valued solution for steady surface waves. Procedia IUTAM 8, 161–165 (2013).  https://doi.org/10.1016/j.piutam.2013.04.020 CrossRefGoogle Scholar
  9. 9.
    Kistovich, A.V.: A new approach to description of potential surface waves. Process. in Geomedia 3(3), 34–40 (2015). IPMech. RAS, Moscow (In Russian)Google Scholar
  10. 10.
    Stokes, G.G.: On the theory of oscillatory waves. Trans. Cam. Phil. Soc. 8, 441–455 (1847)Google Scholar
  11. 11.
    Lord Rayleigh: On waves. Phil. Mag. and J. Sci. 5, 1(4), 257–279 (1876)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.All-Russian Scientific Research Institute of Physico-Technical and Radio-Technical MeasurementsMoscowRussia

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