A Lagrangian approximation to the water-wave problem

AbstractWe describe the derivation by a variational approach of the Camassa-Holm model for periodic shallow water waves

A Hamiltonian structure of the {I}sobe-{K}akinuma model for water waves

We consider the Isobe-Kakinuma model for water waves, which is obtained as the system of Euler-Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. We show that the Isobe-Kakinuma model also enjoys a Hamiltonian structure analogous to the one exhibited by V. E. Zakharov on the full water wave problem and, moreover, that the Hamiltonian of the Isobe-Kakinuma model is a higher order shallow water approximation to the one of the full water wave problem

Wave propagation over a beach within a nonlinear theory

Wave propagation over a beach is considered within a nonlinear theory in shallow water. Lagrangian coordinates are used to describe the problem. The solution is expanded in double series involving a small parameter and local oscillations. Two cases are treated: The beach with appreciable inclination on the horizontal (cliff) and the beach of small inclination. We show that finite solutions are obtained, in contrast to the linear theory which involves a logarithmic singularity at the shoreline. For the cliff, it is shown that local oscillations do not appear in the first two orders of approximation, and the incident wave is totally reflected without loss of energy at this order of approximation. The case of an incident wave on the beach is considered. The deformation of this wave is investigated and explicit formulae are obtained for the reflected wave and for the local oscillations, to shed light on the energy transfer due to interaction with the beach

Variational derivation of two-component Camassa-Holm shallow water system

By a variational approach in the Lagrangian formalism, we derive the nonlinear integrable two-component Camassa-Holm system (1). We show that the two-component Camassa-Holm system (1) with the plus sign arises as an approximation to the Euler equations of hydrodynamics for propagation of irrotational shallow water waves over a flat bed. The Lagrangian used in the variational derivation is not a metric.Comment: to appear in Appl. Ana

Practical use of variational principles for modeling water waves

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.Comment: 30 pages, 1 figure, 62 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

Tsunami generation by paddle motion and its interaction with a beach: Lagrangian modelling and experiment

A 2D Lagrangian numerical wave model is presented and validated against a set of physical wave-flume experiments on interaction of tsunami waves with a sloping beach. An iterative methodology is proposed and applied for experimental generation of tsunami-like waves using a piston-type wavemaker with spectral control. Three distinct types of wave interaction with the beach are observed with forming of plunging or collapsing breaking waves. The Lagrangian model demonstrates good agreement with experiments. It proves to be efficient in modelling both wave propagation along the flume and initial stages of strongly non-linear wave interaction with a beach involving plunging breaking. Predictions of wave runup are in agreement with both experimental results and the theoretical runup law

Modeling water waves beyond perturbations

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.Comment: 15 pages, 1 figure, 39 references. This document is a contributed chapter to an upcoming volume to be published by Springer in Lecture Notes in Physics Series. Other author's papers can be downloaded at http://www.denys-dutykh.com