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Modeling Water Waves Beyond Perturbations

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New Approaches to Nonlinear Waves

Part of the book series: Lecture Notes in Physics ((LNP,volume 908))

Abstract

In this chapter, we illustrate the advantage of variational principles for modeling water waves from an elementary practical viewpoint. The method is based on a ‘relaxed’ variational principle, i.e., on a Lagrangian involving as many variables as possible, and imposing some suitable subordinate constraints. This approach allows the construction of approximations without necessarily relying on a small parameter. This is illustrated via simple examples, namely the Serre equations in shallow water, a generalization of the Klein–Gordon equation in deep water and how to unify these equations in arbitrary depth. The chapter ends with a discussion and caution on how this approach should be used in practice.

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Notes

  1. 1.

    For example \(\tilde{\boldsymbol{u}} = \boldsymbol{u}(y =\eta )\), \(\breve{v} = v(y = -d)\).

  2. 2.

    For two-dimensional vectors \(\boldsymbol{a} = (a_{1},a_{2})\) and \(\boldsymbol{b} = (b_{1},b_{2})\), \(\boldsymbol{a \times b} = a_{1}b_{2} - a_{2}b_{1}\) is a scalar.

  3. 3.

    http://en.wikipedia.org/wiki/Luke’s_variational_principle.

  4. 4.

    \(\bar{u} = \frac{1} {h}\int _{-d}^{\eta }u\,\mathrm{d}y\).

  5. 5.

    R. B. Laughlin et al. earned the 1998 Physics Nobel price.

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Authors and Affiliations

  1. Laboratoire J. A. Dieudonné, Université de Nice – Sophia Antipolis, Parc Valrose, 06108, Nice Cedex 2, France

    Didier Clamond

  2. Université Savoie Mont Blanc, LAMA, UMR 5127 CNRS, Campus Scientifique, 73376, Le Bourget-du-Lac Cedex, France

    Denys Dutykh

Authors
  1. Didier Clamond
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  2. Denys Dutykh
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Correspondence to Didier Clamond .

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Editors and Affiliations

  1. Institute for Analysis, Johannes Kepler University, Linz, Austria

    Elena Tobisch

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Clamond, D., Dutykh, D. (2016). Modeling Water Waves Beyond Perturbations. In: Tobisch, E. (eds) New Approaches to Nonlinear Waves. Lecture Notes in Physics, vol 908. Springer, Cham. https://doi.org/10.1007/978-3-319-20690-5_7

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