Nonlinear Periodic Waves in Shallow Water

  • Alexander Shermenev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2630)


Two classical types of periodic wave motion in polar coordinates has been studied using a computer algebra system. In the case of polar coordinates, the usual perturbation techniques for the nonlinear shallow water equation leads to overdetermined systems of linear algebraic equations for unknown coefficients. The compatibility of the systems is the key point of the investigation. The found coefficients allow to construct solutions to the shallow water equation which are periodic in time. The accuracy of the solutions is the same as of the shallow water equation. Expanding the potential and surface elevation in Fourier series, we express explicitly the coefficients of the first two harmonics as polynomials of Bessel functions. One may speculate that the obtained expressions are the first two terms of an expanded exact three-dimensional solution to the surface wave equations, which describe the axisymmetrical and the simplest unaxisymmetrical surface waves in shallow water.


Surface Elevation Periodic Wave Shallow Water Equation Computer Algebra System Surface Gravity Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boussinesq, J. (1872): Theorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horisontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 2nd Series 17, 55–108Google Scholar
  2. 2.
    Friedrichs, K. O. (1948): On the derivation of the shallow water theory. Comm. Pure Appl. Math. 1, 81–85Google Scholar
  3. 3.
    Grange, J. L. de la (Lagrange) (1788): Mecanique Analitique. v. 2, ParisGoogle Scholar
  4. 4.
    Lamb (1932): Hydrodynamics. Sixth Ed., Cambridge Univ. PressGoogle Scholar
  5. 5.
    Madsen, P. A., Schäffer, H. A. (1998): Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis. Phil. Trans. R. Soc. Lond. A, 8, 441–455Google Scholar
  6. 6.
    Mei, C. C. (1983): The Applied Dynamics of Ocean Surface Waves. WileyGoogle Scholar
  7. 7.
    Shermenev, A., Shermeneva, M. (2000): Long periodic waves on an even beach. Physical Review E, No. 5, 6000–6002MathSciNetGoogle Scholar
  8. 8.
    Shermenev, A. (2001): Nonlinear periodic waves on a beach. Geophysical and Astrophysical Fluid Dynamics, No. 1–2, 1–14Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Alexander Shermenev
    • 1
  1. 1.Wave Research CenterRussian Academy of SciencesMoscowRussia

Personalised recommendations