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 The dynamic pressure in deep-water extreme Stokes waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170095 ◽  
Author(s):  
Tony Lyons
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Dynamic Pressure ◽  
Stokes Wave ◽  
Wave Crest ◽  
Maximum Principles ◽  
Nonlinear Water Waves ◽  
Stagnation Points ◽  
Wave Trough ◽  
Crest Line

In this paper, we consider the dynamic pressure in a deep-water extreme Stokes wave. While the presence of stagnation points introduces a number of mathematical complications, maximum principles are applied to analyse the dynamic pressure in the fluid body by means of an excision process. It is shown that the dynamic pressure attains its maximum value beneath the wave crest and its minimum beneath the wave trough, while it decreases in moving away from the crest line along any streamline. This article is part of the theme issue ‘Nonlinear water waves’.

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.

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

2002 ◽  
Author(s):  
J. W. Kim ◽  
K. J. Bai ◽  
R. C. Ertekin ◽  
W. C. Webster
Keyword(s):  
Solitary Wave ◽  
Water Waves ◽  
Wave Train ◽  
Vertical Wall ◽  
Numerical Test ◽  
Boussinesq Equations ◽  
Approximate Model ◽  
Stokes Wave ◽  
Nonlinear Water Waves ◽  
True Solution

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, 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 the 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.

Modulation Instability and Extreme Events Beyond Initial Three Wave Systems

2016 ◽  
Author(s):  
Amin Chabchoub ◽  
Takuji Waseda
Keyword(s):  
Numerical Simulations ◽  
Extreme Events ◽  
Water Waves ◽  
Modulation Instability ◽  
Stokes Wave ◽  
Side Band ◽  
Wave Basin ◽  
Stokes Waves ◽  
A Minor ◽  
Band Pair

One possible mechanism that models the dynamics of extreme events in the ocean is the modulation instability (MI). The latter has been discovered in the 60s and significant progress in understanding the physics of modulationally unstable deep-water waves has been achieved since then. The MI instability starts its dynamics from a minor periodic perturbation of a regular Stokes wave, which enhances in amplitude, generating therefore periodic large waves, within the specific range of modulation period. In the spectral domain the same process starts from in amplitude very small symmetric side-band pair, lying in the unstable range from the main carrier frequency peak, which then starts to grow while generating by their own a side-band cascade. We report a new type of periodically modulated and unstable Stokes waves which initial dynamics starts from more that one unique unstable side-band pair. Laboratory experiments have been conducted in a large water wave basin, while numerical simulations have been performed using the modified nonlinear Schrödinger equation and the boundary element method. Both, experiments and numerical simulations are in reasonable agreement. Furthermore, the validity, limitations and applicability of such models will be discussed in detail.

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

2003 ◽  
Vol 125 (1) ◽  
pp. 25-32 ◽  
Author(s):  
J. W. Kim ◽  
K. J. Bai ◽  
R. C. Ertekin ◽  
W. C. Webster
Keyword(s):  
Solitary Wave ◽  
Water Waves ◽  
Wave Train ◽  
Vertical Wall ◽  
Numerical Test ◽  
Boussinesq Equations ◽  
Approximate Model ◽  
Stokes Wave ◽  
Nonlinear Water Waves ◽  
True Solution

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.

scholarly journals Study on the Interaction of Nonlinear Water Waves considering Random Seas

PAMM ◽  
2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Marten Hollm ◽  
Leo Dostal ◽  
Hendrik Fischer ◽  
Robert Seifried
Keyword(s):  
Water Waves ◽  
Nonlinear Water Waves ◽  
Random Seas

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

DISPERSION IN SEISMIC SURFACE WAVES

Geophysics ◽  
1951 ◽  
Vol 16 (1) ◽  
pp. 63-80 ◽  
Author(s):  
Milton B. Dobrin
Keyword(s):  
Surface Waves ◽  
Water Waves ◽  
Seismic Refraction ◽  
Wave Theory ◽  
Wave Dispersion ◽  
Surface Zone ◽  
Surface Wave Dispersion ◽  
Near Surface ◽  
Arrival Times ◽  
Ground Roll

A non‐mathematical summary is presented of the published theories and observations on dispersion, i.e., variation of velocity with frequency, in surface waves from earthquakes and in waterborne waves from shallow‐water explosions. Two further instances are cited in which dispersion theory has been used in analyzing seismic data. In the seismic refraction survey of Bikini Atoll, information on the first 400 feet of sediments below the lagoon bottom could not be obtained from ground wave first arrival times because shot‐detector distances were too great. Dispersion in the water waves, however, gave data on speed variations in the bottom sediments which made possible inferences on the recent geological history of the atoll. Recent systematic observations on ground roll from explosions in shot holes have shown dispersion in the surface waves which is similar in many ways to that observed in Rayleigh waves from distant earthquakes. Classical wave theory attributes Rayleigh wave dispersion to the modification of the waves by a surface layer. In the case of earthquakes, this layer is the earth’s crust. In the case of waves from shot‐holes, it is the low‐speed weathered zone. A comparison of observed ground roll dispersion with theory shows qualitative agreement, but it brings out discrepancies attributable to the fact that neither the theory for liquids nor for conventional solids applies exactly to unconsolidated near‐surface rocks. Additional experimental and theoretical study of this type of surface wave dispersion may provide useful information on the properties of the surface zone and add to our knowledge of the mechanism by which ground roll is generated in seismic shooting.

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.

Sign in / Sign up

Export Citation Format

Share Document