Shoaling of Finite-Amplitude Waves on Plane Beaches

Chan, R.K.C.; Street, Robert L.

doi:10.1061/9780872620285.021

Cited by 3 publications

(1 citation statement)

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“…Madsen & Mei [9] examined the same problem by using a system of partial differential equations (which is essentially equivalent to Peregrine's model) derived by Mei & Le Méhauté [10]. On the other hand, Chan & Street [11] directly approximated the Navier-Stokes equations (instead of using Peregrine's model) with finite difference equations to observe the solitary wave transformation.…”

Section: Simulating the Solitary Wave Transformationmentioning

confidence: 99%

A Regular Integral Equation Formalism for Solving the Standard Boussinesq’s Equations for Variable Water Depth

Jang

2017

J Sci Comput

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This paper begins with a question of existence of a regular integral equation formalism, but different from the existing usual ones, for solving the standard Boussinesq's equations for variable water depth (or Peregrine's model). For the question, a pseudo-water depth parameter, suggested by Jang (Commun Nonlinear Sci Numer Simul 43:118-138, 2017), is introduced to alter the standard Boussinesq's equations into an integral formalism. This enables us to construct a regular (nonlinear) integral equations of second kind (as required), being equivalent to the standard Boussinesq's equations (of Peregrine's model). The (constructed) integral equations are, of course, inherently different from the usual integral equation formalisms. For solving them, the successive approximation (or the fixed point iteration) is applied (Jang 2017), whereby a new iterative formula is immediately derived, in this paper, for numerical solutions of the standard Boussinesq's equations for variable water depth. The formula, semi-analytic and derivative-free, is shown to be useful to observe especially the nonlinear wave phenomena of shallow water waves on a beach. In fact, a numerical experiment is performed on a solitary wave approaching a sloping beach. It shows clearly the main feature of nonlinear wave characteristics, which has reached good agreement with the known (numerical) solutions. Hence, while being theoretical but fundamental in nonlinear computational partial differential equations, the question raised in the study may be solved.Keywords A regular integral equation formalism · The standard Boussinesq's equations for variable water depth · A pseudo-water depth parameter · The successive approximation

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Section: Simulating the Solitary Wave Transformationmentioning

confidence: 99%

A Regular Integral Equation Formalism for Solving the Standard Boussinesq’s Equations for Variable Water Depth

Jang

2017

J Sci Comput

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Run-up of non-breaking waves — A finite-element approach

Gopalakrishnan

Tung

1980

Coastal Engineering

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A Strongly-Nonlinear Model for Water Waves in Water of Variable Depth—The Irrotational Green-Naghdi Model

Kim

Bai

Ertekin

et al. 2003

Journal of Offshore Mechanics and Arctic Engineering

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Recently, the authors have derived a new approximate model for the nonlinear water waves, the Irrotational Green-Naghdi (IGN) model. In this paper, we first derive the IGN equations applicable to variable water depth, and then perform numerical tests to show whether and how fast the solution of the IGN model converges to the true solution as its level increases. The first example given is the steady solution of progressive waves of permanent form, which includes the small-amplitude sinusoidal wave, the solitary wave and the nonlinear Stokes wave. The second example is the run-up of a solitary wave on a vertical wall. The last example is the shoaling of a wave train over a sloping beach. In each numerical test, the self-convergence of the IGN model is shown first. Then the converged solution is compared to the known analytic solutions and/or solutions of other approximate models such as the KdV and the Boussinesq equations.

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Shoaling of Finite-Amplitude Waves on Plane Beaches

Cited by 3 publications

References 0 publications

A Regular Integral Equation Formalism for Solving the Standard Boussinesq’s Equations for Variable Water Depth

A Regular Integral Equation Formalism for Solving the Standard Boussinesq’s Equations for Variable Water Depth

Run-up of non-breaking waves — A finite-element approach

A Strongly-Nonlinear Model for Water Waves in Water of Variable Depth—The Irrotational Green-Naghdi Model

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