scholarly journals COUPLING STOKES AND CNOIDAL WAVE THEORIES IN A NONLINEAR REFRACTION MODEL

1988 ◽  
Vol 1 (21) ◽  
pp. 42
Author(s):  
Thomas A. Hardy ◽  
Nicholas C. Kraus
Keyword(s):  
Nonlinear Refraction ◽  
Wave Model ◽  
Wave Theory ◽  
Quantitative Agreement ◽  
Wave Mode ◽  
Finite Amplitude ◽  
Second Order ◽  
Stokes Wave ◽  
Linear Wave ◽  
Cnoidal Wave

An efficient numerical model is presented for calculating the refraction and shoaling of finite-amplitude waves over an irregular sea bottom. The model uses third-order Stokes wave theory in relatively deep water and second-order cnoidal wave theory in relatively shallow water. It can also be run using combinations of lower-order wave theories, including a pure linear wave mode. The problem of the connection of Stokes and cnoidal theories is investigated, and it is found that the use of second-order rather than first-order cnoidal theory greatly reduces the connection discontinuity. Calculations are compared with physical model measurements of the height and direction of waves passing over an elliptical shoal. The finite-amplitude wave model gives better qualitative and quantitative agreement with the measurements than the linear model.

A Second Order Lagrangian Model for Irregular Ocean Waves

2005 ◽  
Vol 128 (3) ◽  
pp. 177-183 ◽  
Author(s):  
Sébastien Fouques ◽  
Harald E. Krogstad ◽  
Dag Myrhaug
Keyword(s):  
Specular Reflection ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Wave Theory ◽  
Ocean Waves ◽  
Second Order ◽  
Lagrangian Model ◽  
Lagrangian Description ◽  
Linear Wave ◽  
Irregular Solution

Synthetic aperture radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, slope, and mean curvature are studied.

scholarly journals HORIZONTAL WATER PARTICLE VELOCITY OF FINITE AMPLITUDE WAVES

1970 ◽  
Vol 1 (12) ◽  
pp. 19 ◽  
Author(s):  
Yuichi Iwagaki ◽  
Tetsuo Sakai
Keyword(s):  
Particle Velocity ◽  
Wave Length ◽  
Wave Theory ◽  
Approximate Expression ◽  
Finite Amplitude ◽  
Stokes Wave ◽  
Wave Crest ◽  
Amplitude Wave ◽  
Water Particle ◽  
Hot Film

This paper firstly describes two methods to measure vertical distribution and time variation of horizontal water particle velocity induced "by surface waves in a wave tank These two methods consist of tracing hydrogen bubbles and using hot film anemometers, respectively Secondly, the experimental results by the two methods are presented with the theoretical curves derived from the small amplitude wave theory, Stokes wave theory of 3rd order, and the hyperbolic wave theory as an approximate expression of the cnoidal wave theory Finally, based on the comparison of the experimental data with the theoretical curves, the applicability of the finite amplitude wave theories, which has been studied for the wave profile, wave velocity, wave length and wave crest height, is discussed from view point of the water particle velocity.

A Second Order Lagrangian Model for Irregular Ocean Waves

2004 ◽  
Author(s):  
Se´bastien Fouques ◽  
Harald E. Krogstad ◽  
Dag Myrhaug
Keyword(s):  
Specular Reflection ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Wave Theory ◽  
Ocean Waves ◽  
Second Order ◽  
Lagrangian Model ◽  
Lagrangian Description ◽  
Linear Wave ◽  
Irregular Solution

Synthetic Aperture Radar (SAR) imaging of ocean waves involves both the geometry and the kinematics of the sea surface. However, the traditional linear wave theory fails to describe steep waves, which are likely to bring about specular reflection of the radar beam, and it may overestimate the surface fluid velocity that causes the so-called velocity bunching effect. Recently, the interest for a Lagrangian description of ocean gravity waves has increased. Such an approach considers the motion of individual labeled fluid particles and the free surface elevation is derived from the surface particles positions. The first order regular solution to the Lagrangian equations of motion for an inviscid and incompressible fluid is the so-called Gerstner wave. It shows realistic features such as sharper crests and broader troughs as the wave steepness increases. This paper proposes a second order irregular solution to these equations. The general features of the first and second order waves are described, and some statistical properties of various surface parameters such as the orbital velocity, the slope and the mean curvature are studied.

Crest-Height Distribution in Nonlinear Random Wave

2009 ◽  
Author(s):  
Cuilin Li ◽  
Dingyong Yu ◽  
Yangyang Gao ◽  
Junxian Yang
Keyword(s):  
Nonlinear Wave ◽  
Wave Theory ◽  
Distribution Functions ◽  
Theoretical Distribution ◽  
Distribution Model ◽  
Stokes Wave ◽  
Wave Crest ◽  
Linear Wave ◽  
Height Distribution ◽  
Crest Height

Many empirical and theoretical distribution functions for wave crest heights have been proposed, but there is a lack of agreement. With the development of ocean exploitation, waves crest heights represent a key point in the design of coastal structures, both fixed and floating, for shoreline protection and flood prevention. Waves crest height is the dominant parameter in assessing the likelihood of wave-in-deck impact and its resulting severe damage. Unlike wave heights, wave crests generally appear to be affected by nonlinearities; therefore, linear wave theory could not be satisfied to practical application. It is great significant to estimate a new nonlinear wave crest height distribution model correctly. This paper derives an approximation distribution formula based on Stokes wave theory. The resulting theoretical forms for nonlinear wave crest are compared with observed data and discussed in detail. The results are shown to be in good agreement. Furthermore, the results indicate that the new theoretical distribution has more accurate than other methods presented in this paper (e.g. Rayleigh distribution and Weibull distribution) and appears to have a greater range of applicability.

A Note on the Second-Order Contribution to Extreme Waves Generated During Hurricanes

2019 ◽  
Vol 141 (4) ◽  
Author(s):  
Mark L. McAllister ◽  
Thomas A. A. Adcock ◽  
Paul H. Taylor ◽  
Ton S. van den Bremer
Keyword(s):  
Wind Waves ◽  
Low Frequency ◽  
Nonlinear Behavior ◽  
Wave Theory ◽  
Linear Range ◽  
Second Order ◽  
Linear Wave ◽  
Extreme Waves ◽  
Order Difference ◽  
Hurricane Wind

High wind speeds generated during hurricanes result in the formation of extreme waves. Extreme waves by nature are steep meaning that linear wave theory alone is insufficient in understanding and predicting their occurrence. The complex, highly transient nature of the direction of wind and hence of waves generated during hurricanes affects this nonlinear behavior. Herein, we examine how this directionality can affect the second-order nonlinearity of extreme waves generated during hurricanes. This is achieved through both deterministic calculations and experiments based on the observations of Young (2006, “Directional Spectra of Hurricane Wind Waves,” J. Geophys. Res. Oceans, 111(C8), epub). Our calculations show that interactions between the tail and peak of the spectrum can become significant when they travel in different directions, resulting in second-order difference components that exist in the linear range of frequencies. These calculations are generally supported by experimental observations, but we note the difficulty of generating and focusing the high-frequency tail of the spectrum experimentally. Bound second-order difference components or subharmonics typically exist as low frequency infra-gravity waves. Components that exist in the linear range of frequencies may be missed by conventional methods of processing field data where low-pass filtering is used and hence overlooked. In this note, we show that in idealized directional spreading conditions representative of a hurricane, failing to account for second-order difference components may lead to underestimation of extreme wave height.

Reconstruction of Ocean Surfaces From Randomly Distributed Measurements Using a Grid-Based Method

2021 ◽  
Author(s):  
Nicolas Desmars ◽  
Moritz Hartmann ◽  
Jasper Behrendt ◽  
Marco Klein ◽  
Norbert Hoffmann
Keyword(s):  
Wave Model ◽  
Wave Theory ◽  
Linear Wave ◽  
Surface Dynamics ◽  
Wave Measurements ◽  
Model Parameterization ◽  
Grid Points ◽  
High Flexibility ◽  
Similar Accuracy ◽  
Grid Based

Abstract In view of deterministic ocean wave prediction, we introduce and investigate a new method to reconstruct ocean surfaces based on randomly distributed wave measurements. Instead of looking for the optimal parameters of a wave model through the minimization of a cost function, our approach directly solves the free surface dynamics — coupled with an interpolation operator — for the quantities of interest (i.e., surface elevation and velocity potential) at grid points that are used to compute the relevant operators. This method allows a high flexibility in terms of desired accuracy and ensures the physical consistency of the solution. Using the linear wave theory and unidirectional wave fields, we validate the applicability of the proposed method. In particular, we show that our grid-based method is able to reach similar accuracy than the wave-model parameterization method at a reasonable cost.

scholarly journals Wave-Generated Current: A Second-Order Formulation

2020 ◽  
Vol 8 (6) ◽  
pp. 418
Author(s):  
Anne Katrine Bratland
Keyword(s):  
Wave Theory ◽  
Second Order ◽  
Stokes Drift ◽  
Wave Number ◽  
Stokes Wave ◽  
Wave Loads ◽  
Third Order ◽  
The Third ◽  
Computational Fluid Dynamics Cfd ◽  
The Mean

In Stokes’ wave theory, wave numbers are corrected in the third order solution. A change in wave number is also associated with a change in current velocity. Here, it will be argued that the current is the reason for the wave number correction, and that wave-generated current at the mean free surface in infinite depth equals half the Stokes drift. To demonstrate the validity of this second-order formulation, comparisons to computational fluid dynamics (CFD) results are shown; to indicate its effect on wave loads on structures, model tests and analyses are compared.

A high-order cnoidal wave theory

1979 ◽  
Vol 94 (1) ◽  
pp. 129-161 ◽  
Author(s):  
J. D. Fenton
Keyword(s):  
Shallow Water ◽  
Wave Height ◽  
Water Depth ◽  
Wave Theory ◽  
High Order ◽  
Stokes Wave ◽  
Cnoidal Wave ◽  
Expansion Parameter ◽  
Wave Solutions ◽  
Fifth Order

A method is outlined by which high-order solutions are obtained for steadily progressing shallow water waves. It is shown that a suitable expansion parameter for these cnoidal wave solutions is the dimensionless wave height divided by the parameter m of the cn functions: this explicitly shows the limitation of the theory to waves in relatively shallow water. The corresponding deep water limitation for Stokes waves is analysed and a modified expansion parameter suggested.Cnoidal wave solutions to fifth order are given so that a steady wave problem with known water depth, wave height and wave period or length may be solved to give expressions for the wave profile and fluid velocities, as well as integral quantities such as wave power and radiation stress. These series solutions seem to exhibit asymptotic behaviour such that there is no gain in including terms beyond fifth order. Results from the present theory are compared with exact numerical results and with experiment. It is concluded that the fifth-order cnoidal theory should be used in preference to fifth-order Stokes wave theory for wavelengths greater than eight times the water depth, when it gives quite accurate results.

Integral properties of periodic gravity waves of finite amplitude

1975 ◽  
Vol 342 (1629) ◽  
pp. 157-174 ◽  
Keyword(s):  
Solitary Wave ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Wave Theory ◽  
Finite Amplitude ◽  
Cnoidal Wave ◽  
The Mean ◽  
Exact Relations ◽  
Kinetic And Potential Energies

A number of exact relations are proved for periodic water waves of finite amplitude in water of uniform depth. Thus in deep water the mean fluxes of mass, momentum and energy are shown to be equal to 2T(4T—3F) and (3T—2V) crespectively, where T and V denote the kinetic and potential energies and c is the phase velocity. Some parametric properties of the solitary wave are here generalized, and some particularly simple relations are proved for variations of the Lagrangian The integral properties of the wave are related to the constants Q, R and S which occur in cnoidal wave theory. The speed, momentum and energy of deep-water waves are calculated numerically by a method employing a new expansion parameter. With the aid of Padé approximants, convergence is obtained for waves having amplitudes up to and including the highest. For the highest wave, the computed speed and amplitude are in agreement with independent calculations by Yamada and Schwartz. At the same time the computations suggest that the speed and energy, for waves of a given length, are greatest when the height is less than the maximum. In this respect the present results tend to confirm previous computations on solitary waves.

On the radiation damping of finite-amplitude progressive edge waves

1990 ◽  
Vol 431 (1883) ◽  
pp. 419-431 ◽  
Keyword(s):  
Wave Theory ◽  
Wave Dispersion ◽  
Wave Mode ◽  
Finite Amplitude ◽  
Edge Wave ◽  
Water Model ◽  
Edge Waves ◽  
Transfer Energy ◽  
Oblique Waves ◽  
Weakly Nonlinear

Using small-amplitude expansions, it is demonstrated that weakly nonlinear periodic edge waves, travelling along the shoreline of a beach, can be attenuated owing to radiation of oblique waves out to sea. A few beach profiles, for which edge-wave dispersion relations are known in closed form, are discussed, and necessary conditions are determined for such radiation to occur due to nonlinear self-interactions. In particular, it is shown that quadratic nonlinear interactions cause the second edge-wave mode on a uniformly sloping beach of slope α to radiate when 1/18π < α < ⅙π; a detailed derivation to find the amplitude of the radiated wave and the attendant decay rate of the edge wave is presented, using the full water-wave theory. Also, it is pointed out that a concomitant nonlinear mechanism can transfer energy from incoming oblique waves to subharmonic edge waves – a plausible mechanism for the generation of travelling edge waves in coastal waters – and the details of this process are discussed within the framework of a shallow-water model.

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