Modeling water waves beyond perturbations

TL;DR

This paper discusses a variational principle-based method for modeling water waves that does not depend on small parameters, enabling more flexible and accurate approximations across different water depths.

Contribution

It introduces a relaxed variational principle approach for water wave modeling, allowing approximations beyond traditional perturbation methods and unifying shallow and deep water equations.

Findings

Applicable to shallow and deep water scenarios

Unifies different water wave equations

Provides a flexible modeling framework

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.