Properties of short-crested waves in water of finite depth

Marchant, Timothy R.; Roberts, A. J.

doi:10.1017/s0334270000005658

Cited by 40 publications

(48 citation statements)

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“…Not only does this method enjoy the stability properties of the TFE method for DNO (as compared to the OE and FE implementations), but also, and in contrast to the situation for computing DNO, it has a greatly reduced operation count in comparison to the Roberts algorithm. In fact, we showed that the computational complexity is O (NN 1 N 2 log(N y )N y + N 2 N 1 N 2 N y ) to be compared with O(N 3 N 2 1 N 2 2 ) for Roberts [70] and Marchant & Roberts [47]. With this efficiency and high-order stability we are now able to produce extremely accurate simulations of highly nonlinear waveforms as we now demonstrate.…”

Section: For Demonstrations)mentioning

confidence: 67%

“…(6)) provided c 0 satisfies (5) and, again, the linear operator (3) appears on the left-hand side of (27). As with the approach of Roberts et al [70,47], the velocity c n−1 is used to satisfy the solvability condition required by the singular nature of the system (27), while several parameterizations of the bifurcation curve can be given which produce a unique solution at every order; please see [61] for details. On the theoretical side, Reeder & Shinbrot [68] proved the existence of solution branches to (26) and parametric (i.e.…”

Section: For Demonstrations)mentioning

confidence: 93%

“…In the case of the original Euler equations (2) the work of Roberts [70], Roberts & Peregrine [71], and Marchant & Roberts [47] is definitive in the absence of surface tension (it appears, however, that the subtleties of the existence theory for pure gravity waves in three dimensions was not properly appreciated by these authors, see § 2.2). We also note the closely related work of Roberts & Schwartz [72] which also considered the computation of threedimensional traveling waves, but did not utilize a BP approach.…”

Section: Boundary Perturbation Methods For Traveling Wavesmentioning

confidence: 99%

“…Among the former are the traveling wave computations of Roberts, Schwartz, & Marchant [72,70,71,47], and Nicholls & Reitich [60,61]. Among the latter, the initial value problem has been studied by Watson & West [81]; West, Brueckner, Janda, Milder, & Milton [82]; and Milder [49] using surface integral operators related to the Dirichlet-Neumann operator.…”

Section: Introductionmentioning

confidence: 99%

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Boundary Perturbation Methods for Water Waves

Nicholls

2007

GAMM-Mitteilungen

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Key words Water waves, free-surface fluid flows, ideal fluid flows, boundary perturbation methods, spectral methods.The most successful equations for the modeling of ocean wave phenomena are the freesurface Euler equations. Their solutions accurately approximate a wide range of physical problems from open-ocean transport of pollutants, to the forces exerted upon oil platforms by rogue waves, to shoaling and breaking of waves in nearshore regions. These equations provide numerous challenges for theoreticians and practitioners alike as they couple the difficulties of a free boundary problem with the subtle balancing of nonlinearity and dispersion in the absence of dissipation. In this paper we give an overview of, what we term, "Boundary Perturbation" methods for the analysis and numerical simulation of this "water wave problem." Due to our own research interests this review is focused upon the numerical simulation of traveling water waves, however, the extensive literature on the initial value problem and additional theoretical developments are also briefly discussed.

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Section: For Demonstrations)mentioning

confidence: 67%

Section: For Demonstrations)mentioning

confidence: 93%

Section: Boundary Perturbation Methods For Traveling Wavesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Boundary Perturbation Methods for Water Waves

Nicholls

2007

GAMM-Mitteilungen

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“…Analytical solutions have been provided by Chappelear (1961) and Hsu, Tsuchiya & Silvester (1979) to third order and by Ioualalen (1993) to fourth order. Roberts (1983), Roberts & Schwartz (1983) and Marchant & Roberts (1987) computed short-crested waves using a numerical perturbation method up to 27th and 35th order for deep water and finite depth, respectively. Roberts & Peregrine (1983) treated the important limit of grazing angles, where the short-crested deepwater waves become long-crested.…”

Section: Third-order Theory For Multi-directional Irregular Wavesmentioning

confidence: 99%

Third-order theory for multi-directional irregular waves

Madsen

Fuhrman

2012

J. Fluid Mech.

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A new third-order solution for multi-directional irregular water waves in finite water depth is presented. The solution includes explicit expressions for the surface elevation, the amplitude dispersion and the vertical variation of the velocity potential. Expressions for the velocity potential at the free surface are also provided, and the formulation incorporates the effect of an ambient current with the option of specifying zero net volume flux. Harmonic resonance may occur at third order for certain combinations of frequencies and wavenumber vectors, and in this situation the perturbation theory breaks down due to singularities in the transfer functions. We analyse harmonic resonance for the case of a monochromatic short-crested wave interacting with a plane wave having a different frequency, and make long-term simulations with a high-order Boussinesq formulation in order to study the evolution of wave trains exposed to harmonic resonance.Key words: surface gravity waves, waves/free-surface flows IntroductionNonlinear irregular multi-directional water waves are commonly described to second order using the formulation by Sharma & Dean (1981). This is based on a double summation over all possible pairs of wave components, utilizing a second-order solution for bi-directional bichromatic waves as the kernel in the summation. In this work, we present a third-order analytical solution for trichromatic tri-directional waves, which is then used as the kernel in a triple summation over all triplets, plus a double summation over the relevant doublets. The outcome of this work is thus a third-order theory for multi-directional irregular waves in finite water depth, including the effect of ambient or wave-induced currents. The formulation is an extension of the work by Zhang & Chen (1999) from deep water to arbitrary depth, and from collinear interactions to multi-directional interactions. It is also an extension of the work by from third-order bi-directional bichromatic waves to multi-directional irregular waves. demonstrated the importance of accounting for third-order effects in boundary conditions for monochromatic short-crested waves in connection with numerical or laboratory experiments. They made numerical simulations using a high-order Boussinesq-type formulation and performed an analysis

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Dynamics of Surface Waves in Coastal Waters

Huang¹

2009

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vjjj PrefaceChapter 1 makes a concise, historical review of water wave theories with the emphasis on current mild-slope equations and Boussinesq-type equations. An outline of three kinds of fundamental formulations of the surface water wave Iproblems then follows.Chapter 2, on the Liu-Dingemans evolution equations, gives a modification to that, and extensions of that, from third-to fourth-order with stability analysis, and from pure waves to wave-current interactions.Chapter 3 deals with resonant wave-current-bottom interactions, concentrating on subharmonic resonance concerning the modulated wave groups and second-order long wavesIChapter 4 rs dedIcated to SIX kmds of the mIld-slope equations dependmg on the variation of ambient currents and bottom topography. In particular, a new !preface jx my first battle call for the two great things not sought or conjectured by Imj manuel Kant in 1788: the starry heavens above me and the moral law within mEJ '{lu Huang Shanghai Apri1200~

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Properties of short-crested waves in water of finite depth

Cited by 40 publications

References 16 publications

Boundary Perturbation Methods for Water Waves

Boundary Perturbation Methods for Water Waves

Third-order theory for multi-directional irregular waves

Dynamics of Surface Waves in Coastal Waters

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