Refraction-diffraction model for weakly nonlinear water waves

1984 ◽  
Vol 141 ◽  
pp. 265-274 ◽  
Author(s):  
Philip L.-F. Liu ◽  
Ting-Kuei Tsay
Keyword(s):  
Water Waves ◽  
Model Equation ◽  
Harmonic Wave ◽  
Nonlinear Effects ◽  
Variable Coefficients ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Diffraction Model ◽  
Stokes Waves ◽  
Higher Harmonic

A model equation is derived for calculating transformation and propagation of Stokes waves. With the assumption that the water depth is slowly varying, the model equation, which is a nonlinear Schrödinger equation with variable coefficients, describes the forward-scattering wavefield. The model equation is used to investigate the wave convergence over a semicircular shoal. Numerical results are compared with experimental data (Whalin 1971). Nonlinear effects, which generate higher-harmonic wave components, are definitely important in the focusing zone. Mean free-surface set-downs over the shoal are also computed.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

scholarly journals “Extraordinary” modulation instability in optics and hydrodynamics

2021 ◽  
Vol 118 (14) ◽  
pp. e2019348118
Author(s):  
Guillaume Vanderhaegen ◽  
Corentin Naveau ◽  
Pascal Szriftgiser ◽  
Alexandre Kudlinski ◽  
Matteo Conforti ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Stability Analysis ◽  
Linear Stability ◽  
Nonlinear Theory ◽  
Linear Stability Analysis ◽  
Water Waves ◽  
Modulation Instability ◽  
Nonlinear Effects ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

The classical theory of modulation instability (MI) attributed to Bespalov–Talanov in optics and Benjamin–Feir for water waves is just a linear approximation of nonlinear effects and has limitations that have been corrected using the exact weakly nonlinear theory of wave propagation. We report results of experiments in both optics and hydrodynamics, which are in excellent agreement with nonlinear theory. These observations clearly demonstrate that MI has a wider band of unstable frequencies than predicted by the linear stability analysis. The range of areas where the nonlinear theory of MI can be applied is actually much larger than considered here.

scholarly journals Dynamic-pressure distributions under Stokes waves with and without a current

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170103 ◽  
Author(s):  
Motohiko Umeyama
Keyword(s):  
Water Waves ◽  
Dynamic Pressure ◽  
Water Pressure ◽  
Travelling Waves ◽  
Stokes Wave ◽  
Front Face ◽  
Nonlinear Water Waves ◽  
Near Surface ◽  
Stokes Waves ◽  
A Current

To investigate changes in the instability of Stokes waves prior to wave breaking in shallow water, pressure data were recorded vertically over the entire water depth, except in the near-surface layer (from 0 cm to −3 cm), in a recirculating channel. In addition, we checked the pressure asymmetry under several conditions. The phase-averaged dynamic-pressure values for the wave–current motion appear to increase compared with those for the wave-alone motion; however, they scatter in the experimental range. The measured vertical distributions of the dynamic pressure were plotted over one wave cycle and compared to the corresponding predictions on the basis of third-order Stokes wave theory. The dynamic-pressure pattern was not the same during the acceleration and deceleration periods. Spatially, the dynamic pressure varies according to the faces of the wave, i.e. the pressure on the front face is lower than that on the rear face. The direction of wave propagation with respect to the current directly influences the essential features of the resulting dynamic pressure. The results demonstrate that interactions between travelling waves and a current lead more quickly to asymmetry. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Nonlinear Numerical Investigation on Higher Harmonics at Lee Side of a Submerged Bar

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
D. Ning ◽  
X. Zhuo ◽  
L. Chen ◽  
B. Teng
Keyword(s):  
Nonlinear Wave ◽  
Wave Amplitude ◽  
Harmonic Wave ◽  
Incoming Wave ◽  
Higher Harmonics ◽  
Wave Condition ◽  
Weakly Nonlinear ◽  
Higher Harmonic ◽  
Wave Nonlinearity ◽  
Method Model

The decomposition of a monochromatic wave over a submerged object is investigated numerically in a flume, based on a fully nonlinear HOBEM (higher-order boundary element method) model. Bound and free higher-harmonic waves propagating downstream the structure are discriminated by means of a two-point method. The developed numerical model is verified very well by comparison with the available data. Further numerical experiments are carried out to study the relations between free higher harmonics and wave nonlinearity. It is found that thenth-harmonic wave amplitude is growing proportional to thenth power of the incoming wave amplitude for weakly nonlinear wave condition, but higher-harmonic free wave amplitudes tend to a constant value for strong nonlinear wave condition.

Evolution of weakly nonlinear water waves in the presence of viscosity and surfactant

1991 ◽  
Vol 229 (-1) ◽  
pp. 135 ◽  
Author(s):  
S. W. Joo ◽  
A. F. Messiter ◽  
W. W. Schultz
Keyword(s):  
Water Waves ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear

scholarly journals Water waves, nonlinear Schrödinger equations and their solutions

1983 ◽  
Vol 25 (1) ◽  
pp. 16-43 ◽  
Author(s):  
D. H. Peregrine
Keyword(s):  
Water Waves ◽  
Rational Solution ◽  
Three Dimensional ◽  
Length Scale ◽  
Nls Equation ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Nonlinear Schrödinger ◽  
Wave Induced ◽  
Modulation Length

AbstractEquations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrödinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

Higher harmonics of electron plasma waves

1977 ◽  
Vol 17 (3) ◽  
pp. 357-368 ◽  
Author(s):  
E. Märk ◽  
N. Sato
Keyword(s):  
Harmonic Wave ◽  
Collisionless Plasma ◽  
Plasma Waves ◽  
Electron Plasma ◽  
Excited Electron ◽  
Harmonic Waves ◽  
Dispersive Waves ◽  
Weakly Nonlinear ◽  
Nonlinear Mixing ◽  
Higher Harmonic

A model based on nonlinear mixing of dispersive waves is used to predict higher harmonic waves generated by weakly nonlinear electron plasma waves. The total harmonic wave is given by superposition of modes which lie at different points (with the same frequency) in the dispersion diagram. The model well explains the experimental results concerning the harmonic waves produced by externally excited electron plasma waves propagating along a collisionless plasma column.

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