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.
Keywords
- Velocity Field
- Variational Principle
- Water Wave
- Lagrangian Density
- Gordon Equation
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- 1.
For example \(\tilde{\boldsymbol{u}} = \boldsymbol{u}(y =\eta )\), \(\breve{v} = v(y = -d)\).
- 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.
- 4.
\(\bar{u} = \frac{1} {h}\int _{-d}^{\eta }u\,\mathrm{d}y\).
- 5.
R. B. Laughlin et al. earned the 1998 Physics Nobel price.
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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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DOI: https://doi.org/10.1007/978-3-319-20690-5_7
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