Long waves on a beach

Peregrine, D. H.

doi:10.1017/s0022112067002605

Cited by 1,235 publications

(888 citation statements)

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“…Additional work with these equations can be found in Madsen and Mei (1969), and Madsen, Mei and Savage (1970) for solitary waves and solitons, respectively. Peregrine (1967) The Boussinesq term gives frequency dispersion and is related to the parameter o2 , where a is the relative depth o = water depth/wave lenth…”

Section: Asymptotic Expansion Methodsmentioning

confidence: 99%

“…The region where U is comparable to unity is cited as applicable to cnoidal wave theory (e.g., Peregrine, 1967) which has been shown to match experimental water surface and velocity profiles (Wiegel, 1964) for finite amplitude, progressive waves in shallow water. Therefore, the Boussinesq equations can be considered the more general set with the Saint Venant equations a special case for long waves such as tides and sluggish river hydrographs.…”

Section: Waves Of Permanent and Nonpermanent Formmentioning

confidence: 99%

“…This restriction limits the applicability of the Boussinesq equations derived by expansion methods to slowly-varying bathymetry. Using the mean velocity as reference, Peregrine (1967Peregrine ( , 1972 obtained the following equations for variable depths in two planar dimensions (x,y)…”

Section: Variable Water Depthmentioning

confidence: 99%

“…The key aspect of the expansion derivation by Peregrine (1967Peregrine ( , 1972 or Whitham (1974) is the consistency in the order-of -magnitude of terms retained. All terms of higher order ( o2 , 2 , a4 , etc.)…”

Section: Variable Water Depthmentioning

confidence: 99%

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Computation of rapidly varied unsteady, free-surface flow

Basco¹

1987

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Section: Asymptotic Expansion Methodsmentioning

confidence: 99%

Section: Waves Of Permanent and Nonpermanent Formmentioning

confidence: 99%

Section: Variable Water Depthmentioning

confidence: 99%

Section: Variable Water Depthmentioning

confidence: 99%

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Computation of rapidly varied unsteady, free-surface flow

Basco¹

1987

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“…As a step up in accuracy, various Boussinesq formulations are available, e.g. Peregrine (1967), Su & Gardner (1969), Madsen & Sørensen (1992), Nwogu (1993), and Madsen, Bingham & Liu (2002) and Madsen, Bingham & Schäffer (2003), and despite their very different levels of sophistication and accuracy with respect to dispersion and deep water capacities, they all simplify to the NSW equations in the dispersion-free shallow water limit.…”

Section: A New Characteristic Formulation Of the Unsteady Nsw Equationsmentioning

confidence: 99%

Transient waves generated by a moving bottom obstacle: a new near-field solution

Madsen

Hansen

2012

J. Fluid Mech.

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We consider the classical problem of a single-layer homogeneous fluid at rest and a low, slowly varying, long and positive bottom obstacle, which is abruptly started from rest to move with a constant speed V. As a result a system of transient waves will develop, and we assume that locally in the region over the obstacle dispersion can be ignored while nonlinearity cannot. The relevant governing equations for the nearfield solution are therefore the nonlinear shallow water (NSW) equations. These are bidirectional and can be formulated in terms of a two-family system of characteristics. We analytically integrate and eliminate the backward-going family and achieve a versatile unidirectional single-family formulation, which covers subcritical, transcritical and supercritical conditions with relatively high accuracy. The formulation accounts for the temporal and spatial evolution of the bound waves in the vicinity of the obstacle as well as the development of the transient free waves generated at the onset of the motion. At some distance from the obstacle, dispersion starts to play a role and undular bores develop, but up to this point the new formulation agrees very well with numerical simulations based on a high-order Boussinesq formulation. Finally, we derive analytical asymptotic solutions to the new equations, providing estimates of the asymptotic surface levels in the vicinity of the obstacle as well as the crest levels of the leading non-dispersive free waves. These estimates can be used to predict the height and speed of the leading waves in the undular bores. The numerical and analytical solutions to the new single-family formulation of the NSW equations are compared to results based on the forced Korteweg-de Vries/Hopf equation and to numerical Boussinesq simulations.

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A New Two-Dimensional Finite Element Model for the Shallow Water Equations Using a Lagrangian Framework Constructed Along Fluid Particle Trajectories

Petera

Nassehi

1996

Int. J. Numer. Meth. Engng.

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SUMMARYIn this paper we describe a new finite element model for the tidal hydrodynamics in estuaries. The mathematical model is based on the solution of the two-dimensional shallow water equations in a Lagrangian framework which is defined along the trajectories of fluid particles. This method gives a flexible and robust numerical scheme for moving boundary flows encountered in tidal water systems. In order to validate the developed model we have. at first instance, compared our numerical results with analytical solutions obtained for domains with simple geometries. Further tests are then conducted to demonstrate the model's ability to cope with conditions such as hydraulic shock, abrupt changes in the flow domain geometry and gradual changes of water surface breadth. The change in the water surface breadth corresponds to the drying and wetting of the plains along the banks of a typical tidal riverlestuary reach. The drying and wetting of flood plains result in the existence of very shallow depth of water at some sections of the flow domain during a tidal cycle. The flow equations under these conditions are strongly convection dominated. Previously published tidal models rely on either, some form of upwinding or the use o f extremely fine meshes to give stable results for the convection dominated very shallow depth computations in estuaries. We show that our model can yield stable and accurate results for very shallow depths in the tidal flow domains without using any kind of artifical damping or excessive mesh refinement. Computational costs of simulating hydrodynamical conditions in a natural water course, even using a depth averaged two-dimensional approach, can be very high. The ability of our scheme to cope with convection dominated conditions has enabled us to economize the computational efforts by using coarse meshes in our finite element calculations. After the validation stage. the developed model is applied to simulate the tidal conditions in a real estuary. The comparison of the model results with the field observations shows a close agreement between these sets of data KEY WORDS Tay cstunry shallow water equations. finite element method; Lagrangian framewoik; fluid parlicle trajectories:

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Long waves on a beach

Cited by 1,235 publications

References 6 publications

Computation of rapidly varied unsteady, free-surface flow

Computation of rapidly varied unsteady, free-surface flow

Transient waves generated by a moving bottom obstacle: a new near-field solution

A New Two-Dimensional Finite Element Model for the Shallow Water Equations Using a Lagrangian Framework Constructed Along Fluid Particle Trajectories

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