Practical use of variational principles for modeling water waves

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

Modified shallow water equations for significantly varying seabeds

“…However, our study contains a certain number of new elements with respect to the existing state of the art. Namely, our derivation procedure relies on a generalised Lagrangian principle of the water wave problem [15] which allows easily the derivations of approximations that cannot be obtained with more conventional asymptotic expansions. Indeed, we do not introduce explicitly any small parameter and our approximation is made through the choice of a suitable ansatz.…”

“…Recently, we proposed a relaxed Lagrangian variational principle which allows much more freedom in constructing approximations compared to the classical formulations. Namely, the water wave equations can be obtained as EulerLagrange equations of the functional ∭ L d 2 x dt involving the Lagrangian density [15]:…”

“…In this ansatz, we take for simplicity the pseudo-velocities to be equal to the velocity field u = µ, v = ν. However, in other situations they can differ (see [15] for more examples). Physically, the ansatz (2.6) means that we are considering a so-called columnar flow [43], which is a sensible model for long waves in shallow water, as long as their amplitudes are not too large.…”

“…Namely, the shallow water ansatz from [15] is additionally constrained to respect the bathymetry variations in space and time. Then, applying the variational principle, we arrive naturally to some modified Saint-Venant (mSV) equations.…”

Serre-type Equations in Deep Water

“…Recently, two authors of this manuscript proposed a generalization to Luke's Lagrangian [15]. The so-called 'relaxed variational principle' will be extensively used in this study and we proceed to a brief description of the main ideas behind this generalization.…”