Higher–order Boussinesq–type equations for surface gravity waves: derivation and analysis

Madsen, Peter Hauge; Schäffer, Hemming A.

doi:10.1098/rsta.1998.0309

Cited by 285 publications

(243 citation statements)

References 25 publications

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“…Such a series expansion is very common in hydrodynamics and was first introduced by Lagrange [28]. Indeed, using Laplace's equation together with the boundary condition (A), one can write the velocity potential at any point as [29] ψðt; x ∥ ; zÞ…”

Section: Background Flowmentioning

confidence: 99%

Rotating black holes in a draining bathtub: Superradiant scattering of gravity waves

et al. 2015

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In a draining rotating fluid flow background, surface perturbations behave as a scalar field on a rotating effective black hole spacetime. We propose a new model for the background flow which takes into account the varying depth of the water. Numerical integration of the associated Klein-Gordon equation using accessible experimental parameters shows that gravity waves in an appropriate frequency range are amplified through the mechanism of superradiance. Our numerical results suggest that the observation of this phenomenon in a common fluid mechanical system is within experimental reach. Unlike the case of wave scattering around Kerr black holes, which depends only on one dimensionless background parameter (the ratio a=M between the specific angular momentum and the mass of the black hole), our system depends on two dimensionless background parameters, namely the normalized angular velocity and surface gravity at the effective black hole horizon.

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Section: Background Flowmentioning

confidence: 99%

Rotating black holes in a draining bathtub: Superradiant scattering of gravity waves

et al. 2015

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“…The general two-dimensional equations valid on a sloping bottom can be found in Madsen & Schaffer (1998a), and they have previously been solved in one-dimension by Madsen et al (1996) …”

Section: Boussinesq Formulationsmentioning

confidence: 99%

Nearshore Wave Dynamics Simulated by Boussinesq Type Models

Sørensen¹,

Madsen²,

Schäffer³

1999

Coastal Engineering 1998

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This paper presents a numerical study of nearshore wave dynamics including the surf zone. Two different time-domain Boussinesq type formulations are applied for this purpose. The first model incoporates Pade [2,2] dispersion characteristics and lowest order nonlinearity. The second and more sophisticated model incorporates Pade [4,4] characteristics and higher-order nonlinearity in the dispersive terms. The models are validated on the shoaling and breaking of regular and irregular waves with and without an ambient current.

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“…The core of the general procedure [17] to obtain BTEs to be used for this class of BTMs can be summarized as follows (dimensionless variables used):…”

Section: (I) Equations For Irrotational Flowsmentioning

confidence: 99%

A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics

2013

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This paper, which is largely the fruit of an invited talk on the topic at the latest International Conference on Coastal Engineering, describes the state of the art of modelling by means of Boussinesq-type models (BTMs). Motivations for using BTMs as well as their fundamentals are illustrated, with special attention to the interplay between the physics to be described, the chosen model equations and the numerics in use. The perspective of the analysis is that of a physicist/engineer rather than of an applied mathematician. The chronological progress of the currently available BTMs from the pioneering models of the late 1960s is given. The main applications of BTMs are illustrated, with reference to specific models and methods. The evolution in time of the numerical methods used to solve BTMs (e.g. finite differences, finite elements, finite volumes) is described, with specific focus on finite volumes. Finally, an overview of the most important BTMs currently available is presented, as well as some indications on improvements required and fields of applications that call for attention.

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Higher–order Boussinesq–type equations for surface gravity waves: derivation and analysis

Cited by 285 publications

References 25 publications

Rotating black holes in a draining bathtub: Superradiant scattering of gravity waves

Rotating black holes in a draining bathtub: Superradiant scattering of gravity waves

Nearshore Wave Dynamics Simulated by Boussinesq Type Models

A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics

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