Wave Statistics for Intermediate Depth Water—NewWaves and Symmetry

2004 ◽  
Vol 126 (1) ◽  
pp. 54-59 ◽  
Author(s):  
P. H. Taylor ◽  
B. A. Williams
Keyword(s):  
Linear Part ◽  
Nonlinear Behavior ◽  
Wave Theory ◽  
Order Theory ◽  
Small Degree ◽  
Second Order ◽  
Linear Wave ◽  
Intermediate Water ◽  
Wave Statistics ◽  
The Hilbert Transform

A study has been made into the average shape of large crests and troughs during several storms using wave elevation data from the WACSIS measurement program. The analysis techniques adopted were data-driven at all times, in order to test whether second-order wave theory could reproduce important features in the field data. The sea surface displayed obvious nonlinear behavior, reflected in the fact that the shapes of crests were always sharper and larger than their trough equivalents. Assuming that the dominant nonlinear correction is second order in the wave steepness (but without a knowledge of the detailed form of second-order theory), the average shapes of maxima in the underlying linear wave components were shown to match NewWave. This NewWave is the scaled auto-correlation function for a linear random process with the same power spectrum as the measured waves. Thus, NewWave was shown to be an acceptable model for the linear part of large waves on intermediate water depth (here ∼17 m). Assuming that NewWave is a good model for the linear part of large crests and troughs, a value for the second-order coefficient required to estimate crest elevation statistics was derived from the measured data for several storms. This coefficient was in good agreement with the results of the second-order random simulations of Forristall and Prevosto [1]. As well as studying vertical asymmetry, required for crest and trough statistics, horizontal asymmetry was examined using the Hilbert transform. Compared to a large amount of vertical asymmetry, the analysis showed that there was virtually no horizontal asymmetry for the bulk of the waves in the records. However, there is a very small degree of horizontal asymmetry exhibited in the largest waves in the records. Thus, given a surface elevation record, it is difficult to distinguish the direction of the time axis, again consistent with most of the nonlinearity being due to simple second-order bound waves.

Wave Statistics for Intermediate Depth Water: New Waves and Symmetry

2002 ◽  
Author(s):  
P. H. Taylor ◽  
B. A. Williams
Keyword(s):  
Linear Part ◽  
Wave Theory ◽  
Order Theory ◽  
Small Degree ◽  
Second Order ◽  
Linear Wave ◽  
Intermediate Water ◽  
Wave Statistics ◽  
Non Linear ◽  
The Hilbert Transform

A study has been made into the average shape of large crests and troughs during several storms using wave elevation data from the WACSIS measurement programme. The analysis techniques adopted were data-driven at all times, in order to test whether 2nd order wave theory could reproduce important features in the field data. The sea surface displayed obvious non-linear behaviour, reflected in the fact that the shapes of crests were always sharper and larger than their trough equivalents. Assuming that the dominant non-linear correction is second order in the wave steepness (but without a knowledge of the detailed form of 2nd order theory), the average shapes of maxima in the underlying linear wave components were shown to match NewWave. This NewWave is the scaled auto-correlation function for a linear random process with the same power spectrum as the measured waves. Thus, NewWave was shown to be an acceptable model for the linear part of large waves on intermediate water depth (here ∼17m). Assuming that NewWave is a good model for the linear part of large crests and troughs, a value for the second order coefficient required to estimate crest elevation statistics was derived from the measured data for several storms. This coefficient was in good agreement with the results of the 2nd order random simulations of Forristall and Prevosto. As well as studying vertical asymmetry, required for crest and trough statistics, horizontal asymmetry was examined using the Hilbert transform. Compared to a large amount of vertical asymmetry, the analysis showed that there was virtually no horizontal asymmetry for the bulk of the waves in the records. However, there is a very small degree of horizontal asymmetry exhibited in the largest waves in the records. Thus, given a surface elevation record, it is difficult to distinguish the direction of the time axis, again consistent with most of the non-linearity being due to simple 2nd order bound waves.

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.

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.

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.

Physics-Based Approach to Wave Statistics and Probability

2013 ◽  
Author(s):  
Alexander V. Babanin
Keyword(s):  
Extreme Events ◽  
Modulational Instability ◽  
Probability Distributions ◽  
Rayleigh Distribution ◽  
Order Theory ◽  
Second Order ◽  
Ocean Engineering ◽  
Actual Distribution ◽  
Wave Statistics ◽  
Wave Heights

Design criteria in ocean engineering, whether this is one in 50 years or one in 5000 years event, are hardly ever based on measurements, and rather on statistical distributions of relevant metocean properties. Of utmost interest is the tail of these distributions, that is rare events such as the highest waves with low probability. Engineers have long since realised that the superposition of linear waves with narrow-banded spectrum as depicted by the Rayleigh distribution underestimates the probability of extreme wave crests, and is not adequate for wave heights either, which is a critical shortcoming as far as the engineering design is concerned. Ongoing theoretical and experimental efforts have been under way for decades to address this issue. Here, we will concentrate on short-term statistics, i.e. probability of crests/heights of individual waves. Typical approach is to treat all possible waves in the ocean or at a particular location as a single ensemble for which some comprehensive solution can be found. The oceanographic knowledge, however, now indicates that no single and united comprehensive solution is possible. Probability distributions in different physical circumstances should be different, and by combining them together the inevitable scatter is introduced. The scatter and the accuracy will not improve by increasing the bulk data quality and quantity, and it hides the actual distribution of extreme events. The groups have to be separated and their probability distributions treated individually. The paper offers a review of physical conditions, from simple one-dimensional trains of free waves to realistic two-dimensional wind-forced wave fields, in order to understand where different probability distributions can be expected. If the wave trains/fields in the wave records are stable, distributions for the second-order waves should serve well. If modulational instability is active, rare extreme events not predicted by the second-order theory should become possible. This depends on wave steepness, bandwidth and directionality. Mean steepness also defines the wave breaking and therefore the upper limit for wave heights in this group of conditions. Under hurricane-like circumstances, the instability gives way to direct wind forcing, and yet another statistics is to be expected.

The Consistent Second-Order Wave Theory and Its Application to a Submerged Spheroid

1970 ◽  
Vol 14 (01) ◽  
pp. 23-50
Author(s):  
Young H. Chey
Keyword(s):  
Free Surface ◽  
Solid Wall ◽  
Three Dimensional ◽  
Wave Theory ◽  
Order Theory ◽  
Wave Resistance ◽  
Second Order ◽  
Surface Phenomena ◽  
First Order ◽  
Theoretical Results

Because of the recognized inadequacy of first-order linearized surface-wave theory, the author has developed, for a three-dimensional body, a new second-order theory which provides a better description of free-surface phenomena. The new theory more accurately satisfies the kinematic boundary condition on the solid wall, and takes into account the nonlinearity of the condition at the free surface. The author applies the new theory to a submerged spheroid, to calculate wave resistance. Experiments were conducted to verify the theory, and their results are compared with the theoretical results. The comparison indicates that the use of the new theory leads to more accurate prediction of wave resistance.

On higher-order wave theory for submerged two-dimensional bodies

1969 ◽  
Vol 38 (2) ◽  
pp. 415-432 ◽  
Author(s):  
Nils Salvesen
Keyword(s):  
Free Surface ◽  
Wave Theory ◽  
Order Theory ◽  
Wave Drag ◽  
Higher Order ◽  
Second Order ◽  
Two Dimensional ◽  
Third Order ◽  
Free Surface Waves ◽  
Body Shapes

The importance of non-linear free-surface effects on potential flow past two-dimensional submerged bodies is investigated by the use of higher-order perturbation theory. A consistent second-order solution for general body shapes is derived. A comparison between experimental data and theory is presented for the free-surface waves and for the wave resistance of a foil-shaped body. The agreement is good in general for the second-order theory, while the linear theory is shown to be inadequate for predicting the wave drag at the relatively small submergence treated here. It is also shown, by including the third-order freesurface effects, how the solution to the general wave theory breaks down at low speeds.

Extension of second-order Stokes theory to variable bathymetry

2002 ◽  
Vol 464 ◽  
pp. 35-80 ◽  
Author(s):  
K. A. BELIBASSAKIS ◽  
G. A. ATHANASSOULIS
Keyword(s):  
Numerical Solutions ◽  
Second Harmonic ◽  
Bottom Topography ◽  
Wave Theory ◽  
Present Theory ◽  
Second Order ◽  
Intermediate Water ◽  
Interesting Phenomenon ◽  
Weakly Nonlinear ◽  
Harmonic Interaction

In the present work second-order Stokes theory has been extended to the case of a generally shaped bottom profile connecting two half-strips of constant (but possibly different) depths, initiating a method for generalizing the Stokes hierarchy of second- and higher-order wave theory, without the assumption of spatial periodicity. In modelling the wave–bottom interaction three partial problems arise: the first order, the unsteady second order and the steady second order. The three problems are solved by using appropriate extensions of the consistent coupled-mode theory developed by the present authors for the linearized problem. Apart from the Stokes small-amplitude expansibility assumption, no additional asymptotic assumptions have been introduced. Thus, bottom slope and curvature may be arbitrary, provided that the resulting wave dynamics is Stokes-compatible. Accordingly, the present theory can be used for the study of various wave phenomena (propagation, reflection, diffraction) arising from the interaction of weakly nonlinear waves with a general bottom topography, in intermediate water depth. An interesting phenomenon, that is also very naturally resolved, is the net mass flux induced by the depth variation, which is consistently calculated by means of the steady second-order potential. The present method has been validated against experimental results and fully nonlinear numerical solutions. It has been found that it correctly predicts the second-order harmonic generation, the amplitude nonlinearity, and the amplitude variation due to non-resonant first-and-second harmonic interaction, up to the point where the energy transfer to the third and higher harmonics can no longer be neglected. Under the restriction of weak nonlinearity, the present model can be extended to treat obliquely incident waves and the resulting second-order refraction patterns, and to study bichromatic and/or bidirectional wave–wave interactions, with application to the transformation of second-order random seas in variable bathymetry regions.

scholarly journals SECOND ORDER THEORY OF MANOMETER WAVE MEASUREMENT

1982 ◽  
Vol 1 (18) ◽  
pp. 8
Author(s):  
F. Biesel
Keyword(s):  
Gravity Wave ◽  
Wave Theory ◽  
Order Theory ◽  
Second Order ◽  
Order Effects ◽  
Mean Values ◽  
Surface Level ◽  
First Order ◽  
Wave Measurements ◽  
Wave Measurement

The paper refers to pressure gage wave measurements . First order transformation of the pressure spectrum into a surface level spectrum leads to hitherto unexplained discrepancies with prototype simultaneous pressure and level measurements . Use of second order gravity wave theory allows to draw the following conclusions » Second order effects appear to give a reasonable explanation of the observed discrepancies . A complete check would require specially made wave measurements and analyses . Second order corrections do not significantly affect mean values, such as significant height, if the manometer depth is not unduly large.

An Experimental Study of the Systematic Underestimation of Wave Crests Measured by Lagrangian Buoys, and a Retrospective Correction Method

2020 ◽  
Author(s):  
Mark McAllister ◽  
Ton van den Bremer
Keyword(s):  
Approximate Expression ◽  
Ocean Waves ◽  
Order Theory ◽  
Correction Method ◽  
Second Order ◽  
Spectral Parameters ◽  
Peak Period ◽  
Wave Statistics ◽  
Buoy Data ◽  
Crest Height

<p>Wave-following buoys are used to provide measurements of free surface elevation across the oceans. The measurements they produce are widely used to derive wave-averaged parameters such as significant wave height and peak period, alongside wave-by-wave statistics such as crest height distributions. Particularly concerning the measurement of extreme wave crests, these measurements are often perceived to be less accurate. We directly assess this through a side-by-side laboratory comparison of measurements made using Eulerian wave gauges and model wave-following buoys for directionally spread waves representative of extreme conditions on deep water. Our experimental measurements are compared to exact (Herbers and Janssen 2016, J. Phys. Oceanogr, 46, 1009-1021) and new approximate expression for Lagrangian second-order theory derived herein. We derive simple closed-form expressions for the second-order contribution to crest height representative of extreme ocean waves. Our experiments confirm that the motion a wave-following buoy should not significantly affect the measurements of wave crests or spectral parameters, and that discrepancies observed for in-situ buoy data are most likely a result of filtering. This filtering occurs when accelerations that are measured by the sensors within a buoy are converted to displacements. We present an approximate means of correcting the resulting measured crest height distributions, which is shown to be effective using our experimental data.</p>

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