A variational principle for nonlinear water waves

Debnath, Lokenath

doi:10.1007/bf01176549

Cited by 7 publications

(13 citation statements)

References 4 publications

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“…Although not easily observed in Figures 4 and 6, the leading wave propagates at higher speed and exhibits a larger curvature for a ¼ 1 than for a ¼ 0.1, in accord with the characteristic lines of the first-order partial differential operator of equation ( 1), for m ¼ d ¼ t ¼ 0 (Whitham, 1974;Johnson, 1997;Lannes, 2013;Debnath, 1994;Dingemans, 1997).…”

Section: Results For Gaussian Initial Conditionssupporting

confidence: 70%

“…When ju(t, x)j is large, the nonlinear advection term associated with e = 0 and p = 1 in equation (1) results in wave steepening for u x < 0 and e > 0 that would lead to the formation of a (discontinuous) shock wave for t ¼ m ¼ d ¼ 0, and the formation of a Taylor's (smooth) shock wave for t ¼ d ¼ 0 and m > 0 (Whitham, 1974;Johnson, 1997;Lannes, 2013;Debnath, 1994;Dingemans, 1997)). In the absence of linear drift and for p ¼ 2, e ¼ 1 2 and d ¼ 0, i.e.…”

Section: Hff 343mentioning

confidence: 99%

“…Most models developed to date for the study of wave propagation in shallow water, such as, for example, the Korteweg-de Vries (Korteweg and de Vries, 1895) and Benjamin-Bona-Mahony (BBM) or regularized long-wave (RLW) equations and modifications and generalizations thereof (Peregrine, 1966(Peregrine, , 1967Benjamin et al, 1972;Ramos, 2016;Ramos and García L opez, 2017;Johnson, 1997;Lannes, 2013;Debnath, 1994;Dingemans, 1997), have been obtained by means of asymptotic methods, Taylor's series expansions, depthaveraging techniques (Whitham, 1974;Ramos, 2016;Ramos and García L opez, 2017), etc., and are one-dimensional and unidirectional, i.e. waves propagate only in one direction.…”

Section: Introductionmentioning

confidence: 99%

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Effect of initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water. II: Blowup for nonsmooth conditions

Ramos,

García López

2023

HFF

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Purpose The purpose of this paper is to analyze numerically the blowup in finite time of the solutions to a one-dimensional, bidirectional, nonlinear wave model equation for the propagation of small-amplitude waves in shallow water, as a function of the relaxation time, linear and nonlinear drift, power of the nonlinear advection flux, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of three types of initial conditions. Design/methodology/approach An implicit, first-order accurate in time, finite difference method valid for semipositive relaxation times has been used to solve the equation in a truncated domain for three different initial conditions, a first-order time derivative initially equal to zero and several constant wave speeds. Findings The numerical experiments show a very rapid transient from the initial conditions to the formation of a leading propagating wave, whose duration depends strongly on the shape, amplitude and width of the initial data as well as on the coefficients of the bidirectional equation. The blowup times for the triangular conditions have been found to be larger than those for the Gaussian ones, and the latter are larger than those for rectangular conditions, thus indicating that the blowup time decreases as the smoothness of the initial conditions decreases. The blowup time has also been found to decrease as the relaxation time, degree of nonlinearity, linear drift coefficient and amplitude of the initial conditions are increased, and as the width of the initial condition is decreased, but it increases as the viscosity coefficient is increased. No blowup has been observed for relaxation times smaller than one-hundredth, viscosity coefficients larger than ten-thousandths, quadratic and cubic nonlinearities, and initial Gaussian, triangular and rectangular conditions of unity amplitude. Originality/value The blowup of a one-dimensional, bidirectional equation that is a model for the propagation of waves in shallow water, longitudinal displacement in homogeneous viscoelastic bars, nerve conduction, nonlinear acoustics and heat transfer in very small devices and/or at very high transfer rates has been determined numerically as a function of the linear and nonlinear drift coefficients, power of the nonlinear drift, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of the initial conditions for nonzero relaxation times.

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Section: Results For Gaussian Initial Conditionssupporting

confidence: 70%

Section: Hff 343mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Effect of initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water. II: Blowup for nonsmooth conditions

Ramos,

García López

2023

HFF

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“…Modulation instability (MI) of capillary waves has been identified for the first time in [11]. It develops when the Lighthill criterion, (∂ω/∂|a| 2 )(∂ 2 ω k /∂k 2 ) < 0, (where ω = ω k [1 − (ka) 2 /16] includes a nonlinear frequency correction [18]) is satisfied. The development of the MI is observed at a given point in space as the modulation of the envelope which, at higher wave amplitudes (forcing) leads to breaking of the initially continuous waves into sequence of the envelope solitons [11].…”

mentioning

confidence: 99%

Modulation instability and capillary wave turbulence

Xia

Shats

Punzmann

2010

EPL

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Formation of turbulence of capillary waves is studied in laboratory experiments. The spectra show multiple exponentially decreasing harmonics of the parametrically excited wave which nonlinearly broaden with the increase in forcing. Spectral broadening leads to the development of the spectral continuum which scales as ∝ f −2.8 , in agreement with the weak turbulence theory (WTT) prediction. Modulation instability of capillary waves is shown to be responsible for the transition from discrete to broadband spectrum. The instability leads to spectral broadening of the harmonics, randomization of their phases, it isolates the wave field from the wall, eventually allows the transition from 4-to 3-wave interactions as the dominant nonlinear process, thus creating the prerequisites assumed in WTT.

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“…Evidently the Laplace equation, two free surface conditions, and the bottom boundary condition constitute the two-dimensional water wave equation. This system of equation has been used by Stoker [17], Debnath [18] for the investigation of the linearized initial value problem for the generation and propagation of water waves.…”

Section: Two Dimensional Water Wave Equationsmentioning

confidence: 99%

Hamiltonian Formulation for Water Wave Equation

Sultana¹,

Rahman²

2013

OJFD

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This paper concerns the development and application of the Hamiltonian function which is the sum of kinetic energy and potential energy of the system. Two dimensional water wave equations for irrotational, incompressible, inviscid fluid have been constructed in cartesian coordinates and also in cylindrical coordinates. Then Lagrangian function within a certain flow region is expanded under the assumption that the dispersion μ and the nonlinearity ε satisfied   2 O   . Using Hamilton's principle for water wave evolution Hamiltonian formulation is derived. It is obvious that the motion of the system is conservative. Then Hamilton's canonical equation of motion is also derived.

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A variational principle for nonlinear water waves

Cited by 7 publications

References 4 publications

Effect of initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water. II: Blowup for nonsmooth conditions

Effect of initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water. II: Blowup for nonsmooth conditions

Modulation instability and capillary wave turbulence

Hamiltonian Formulation for Water Wave Equation

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