On the limiting Stokes wave of extreme height in arbitrary water depth

Zhong, Xiaoxu; Liao, Shijun

doi:10.1017/jfm.2018.171

Cited by 33 publications

(44 citation statements)

References 60 publications

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“…There are also other interesting aspects of Stokes waves that have been investigated by various researchers. For example, Longuet-Higgins [16] studied the integral properties of Stokes waves, including mean surface, mean velocity, momentum and mean energy flux; Crew & Trinh [17] presented the findings on the singularities of Stokes waves; and Zhong & Liao [18] made use of the homotopy analysis method and gained convergent results of limiting Stokes waves in arbitrary depth. These topics are, however, beyond the scope of the present paper.…”

Section: Introductionmentioning

confidence: 99%

On Stokes wave solutions

Zhao

Liu

2022

Proc. R. Soc. A.

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Using a perturbation method with the aid of MATLAB ® Symbolic Toolbox, a new set of Stokes wave solutions, up to the fifth order, are derived. The new solutions are expressed in terms of free surface profile, velocity potential and wave celerity (or frequency dispersion relation). The solutions in the deep water limit are also deduced and discussed. The new solutions are compared with the existing solutions up to the fifth order. Differences appear among the existing and the present solutions, starting at the third order. The causes for the differences are identified. The characteristic differences were highlighted in the figures for high order solutions of Stokes waves. Numerical examples are provided to quantify the differences between the new fifth-order solutions and Fenton’s, and Skjelbreia and Hendrickson’s solutions.

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Section: Introductionmentioning

confidence: 99%

On Stokes wave solutions

Zhao

Liu

2022

Proc. R. Soc. A.

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“…For purely travelling waves, breaking has known implications on maximum wave height. Stokes (1880) first proposed that a crest enclosing an angle of 120 • was the limiting form prior to breaking for a progressive surface gravity wave on deep water (see Zhong & Liao (2018) for a study of all water depths). This limit provides an upper bound of wave height h for a given wavenumber k and corresponds to a steepness of kh/2 = 0.44.…”

Section: Introductionmentioning

confidence: 99%

Wave breaking and jet formation on axisymmetric surface gravity waves

et al. 2022

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Axisymmetric standing waves occur across a wide range of free surface flows. When these waves reach a critical height (steepness), wave breaking and jet formation occur. For travelling surface gravity waves, wave breaking is generally considered to limit wave height and reversible wave motion. In the ocean, the behaviour of directionally spread waves lies between the limits of purely travelling (two dimensions) and axisymmetric (three dimensions). Hence, understanding wave breaking and jet formation on axisymmetric surface gravity waves is an important step in understanding extreme and breaking waves in the ocean. We examine an example of axisymmetric wave breaking and jet formation colloquially known as the ‘spike wave’, created in the FloWave circular wave tank at the University of Edinburgh, UK. We generate this spike wave with maximum crest amplitudes of 0.15–6.0 m (0.024–0.98 when made non-dimensional by characteristic radius), with wave breaking occurring for crest amplitudes greater than 1.0 m (0.16 non-dimensionalised). Unlike two-dimensional travelling waves, wave breaking does not limit maximum crest amplitude, and our measurements approximately follow the jet height scaling proposed by Ghabache et al. (J. Fluid Mech., vol. 761, 2014, pp. 206–219) for cavity collapse. The spike wave is predominantly created by linear dispersive focusing. A trough forms, then collapses producing a jet, which is sensitive to the trough's shape. The evolution of the jets that form in our experiments is predicted well by the hyperbolic jet model proposed by Longuet–Higgins (J. Fluid Mech., vol. 127, 1983, pp. 103–121), previously applied to jets forming on bubbles.

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“…For example, the convergent series solution of Von Kàrmàn plate under arbitrary uniform pressure (i.e., with arbitrary deformation) are obtained by means of the HAM [30], and besides it has been proved that all previous perturbative approaches for Von Kàrmàn plate are special cases of the HAM. In addition, using the HAM, Zhong and Liao [31] successfully gained the convergent series solution of the limiting Stokes wave of extreme height in arbitrary water depth (including the extremely shallow water), which could not been found by perturbation methods and even by numerical techniques. All of these illustrate that the HAM is indeed valid for highly nonlinear problems.…”

Section: Motivationmentioning

confidence: 99%

A new non-perturbative approach in quantum mechanics for time-independent Schrödinger equations

Liao

2020

Sci. China Phys. Mech. Astron.

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A new non-perturbative approach is proposed to solve time-independent Schrödinger equations in quantum mechanics and chromodynamics (QCD). It is based on the homotopy analysis method (HAM), which was developed by the author for highly nonlinear equations since 1992 and has been widely applied in many fields. Unlike perturbative methods, this HAM-based approach has nothing to do with small/large physical parameters. Besides, convergent series solution can be obtained even if the disturbance is far from the known status. A nonlinear harmonic oscillator is used as an example to illustrate the validity of this approach for disturbances that might be more than hundreds larger than the possible superior limit of the perturbative approach. This HAM-based approach could provide us rigorous theoretical results in quantum mechanics and chromodynamics (QCD), which can be directly compared with experimental data. Obviously, this is of great benefit not only for improving the accuracy of experimental measurements but also for validating physical theories.

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On the limiting Stokes wave of extreme height in arbitrary water depth

Cited by 33 publications

References 60 publications

On Stokes wave solutions

On Stokes wave solutions

Wave breaking and jet formation on axisymmetric surface gravity waves

A new non-perturbative approach in quantum mechanics for time-independent Schrödinger equations

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