Hamiltonian long‐wave expansions for free surfaces and interfaces
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Bibliographic record
Abstract
Abstract The theory of internal waves between two bodies of immiscible fluid is important both for its interest to ocean engineering and as a source of numerous interesting mathematical model equations that exhibit nonlinearity and dispersion. In this paper we derive a Hamiltonian formulation of the problem of a dynamic free interface (with rigid lid upper boundary conditions), and of a free surface and a free interface, this latter situation occurring more commonly in experiment and in nature. From the formulation, we develop a Hamiltonian perturbation theory for the long‐wave limits, and we carry out a systematic analysis of the principal long‐wave scaling regimes. This analysis provides a uniform treatment of the classical works of Peters and Stoker (28), Benjamin (3, 4), Ono (26), and many others. Our considerations include the Boussinesq and Korteweg–de Vries (KdV) regimes over finite‐depth fluids, the Benjamin‐Ono regimes in the situation in which one fluid layer is infinitely deep, and the intermediate long‐wave regimes. In addition, we describe a novel class of scaling regimes of the problem, in which the amplitude of the interface disturbance is of the same order as the mean fluid depth, and the characteristic small parameter corresponds to the slope of the interface. Our principal results are that we highlight the discrepancies between the case of rigid lid and of free surface upper boundary conditions, which in some circumstances can be significant. Motivated by the recent results of Choi and Camassa (6, 7), we also derive novel systems of nonlinear dispersive long‐wave equations in the large‐amplitude, small‐slope regime. Our formulation of the dynamical free‐surface, free‐interface problem is shown to be very effective for perturbation calculations; in addition, it holds promise as a basis for numerical simulations. © 2005 Wiley Periodicals, Inc.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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