Practical use of variational principles for modeling water waves

TL;DR

This paper introduces a flexible variational approach for deriving approximate equations of irrotational water waves, enabling the development of both known and novel models across different water depths with analytical solutions.

Contribution

It presents a relaxed variational principle framework that enhances model derivation flexibility while maintaining the variational structure, applicable to shallow and deep water scenarios.

Findings

Derived several water wave models, including new ones.

Constructed exact travelling wave solutions.

Demonstrated advantages of the relaxed variational formulation.

Abstract

This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a 'relaxed' variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is particularly suitable for the construction of approximate water wave models, since it allows more freedom while preserving the variational structure. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters, as well as arbitrary depths. Using subordinate constraints (e.g., irrotationality or free surface impermeability) in various combinations, several model equations are derived, some being well-known, other being new. The models obtained are studied analytically and exact travelling wave solutions are constructed when possible.