23,497 research outputs found
Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom
A single incompressible, inviscid, irrotational fluid medium bounded by a
free surface and varying bottom is considered. The Hamiltonian of the system is
expressed in terms of the so-called Dirichlet-Neumann operators. The equations
for the surface waves are presented in Hamiltonian form. Specific scaling of
the variables is selected which leads to approximations of Boussinesq and KdV
types taking into account the effect of the slowly varying bottom. The arising
KdV equation with variable coefficients is studied numerically when the initial
condition is in the form of the one soliton solution for the initial depth.Comment: 18 pages, 6 figures, 1 tabl
Instability of two interacting, quasi-monochromatic waves in shallow water
We study the nonlinear interactions of waves with a doubled-peaked power
spectrum in shallow water. The starting point is the prototypical equation for
nonlinear uni-directional waves in shallow water, i.e. the Korteweg de Vries
equation. Using a multiple-scale technique two defocusing coupled Nonlinear
Schr\"odinger equations are derived. We show analytically that plane wave
solutions of such a system can be unstable to small perturbations. This
surprising result suggests the existence of a new energy exchange mechanism
which could influence the behaviour of ocean waves in shallow water.Comment: 4 pages, 2 figure
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
Topographical scattering of waves: a spectral approach
The topographical scattering of gravity waves is investigated using a
spectral energy balance equation that accounts for first order wave-bottom
Bragg scattering. This model represents the bottom topography and surface waves
with spectra, and evaluates a Bragg scattering source term that is
theoretically valid for small bottom and surface slopes and slowly varying
spectral properties. The robustness of the model is tested for a variety of
topographies uniform along one horizontal dimension including nearly
sinusoidal, linear ramp and step profiles. Results are compared with
reflections computed using an accurate method that applies integral matching
along vertical boundaries of a series of steps. For small bottom amplitudes,
the source term representation yields accurate reflection estimates even for a
localized scatterer. This result is proved for small bottom amplitudes
relative to the mean water depth . Wave reflection by small amplitude bottom
topography thus depends primarily on the bottom elevation variance at the Bragg
resonance scales, and is insensitive to the detailed shape of the bottom
profile. Relative errors in the energy reflection coefficient are found to be
typically .Comment: Second revision for Journal of Waterways Ports and Coastal
Engineerin
Evolution of long water waves in variable channels
This paper applies two theoretical wave models, namely the generalized channel Boussinesq (gcB) and the channel Korteweg–de Vries (cKdV) models (Teng & Wu 1992) to investigate the evolution, transmission and reflection of long water waves propagating in a convergent–divergent channel of arbitrary cross-section. A new simplified version of the gcB model is introduced based on neglecting the higher-order derivatives of channel variations. This simplification preserves the mass conservation property of the original gcB model, yet greatly facilitates applications and clarifies the effect of channel cross-section. A critical comparative study between the gcB and cKdV models is then pursued for predicting the evolution of long waves in variable channels. Regarding the integral properties, the gcB model is shown to conserve mass exactly whereas the cKdV model, being limited to unidirectional waves only, violates the mass conservation law by a significant margin and bears no waves which are reflected due to changes in channel cross-sectional area. Although theoretically both models imply adiabatic invariance for the wave energy, the gcB model exhibits numerically a greater accuracy than the cKdV model in conserving wave energy. In general, the gcB model is found to have excellent conservation properties and can be applied to predict both transmitted and reflected waves simultaneously. It also broadly agrees well with the experiments. A result of basic interest is that in spite of the weakness in conserving total mass and energy, the cKdV model is found to predict the transmitted waves in good agreement with the gcB model and with the experimental data availabl
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