# Modeling Water Waves Beyond Perturbations

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Part of the Lecture Notes in Physics book series (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.

### Keywords

• Velocity Field
• Variational Principle
• Water Wave
• Lagrangian Density
• Gordon Equation

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.

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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.
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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Correspondence to Didier Clamond .

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