The Effect of Third-Order Nonlinearities on the Statistical Distributions of Wave Heights, Crests and Troughs in Bimodal Crossing Seas

2011 ◽  
Author(s):  
Petya G. Petrova ◽  
M. Aziz Tayfun ◽  
C. Guedes Soares
Keyword(s):  
Rayleigh Distribution ◽  
Second Order ◽  
Statistical Distributions ◽  
Third Order ◽  
Weakly Nonlinear ◽  
Distribution Parameters ◽  
Wave Heights ◽  
Component Wave ◽  
The Mean ◽  
Basic Characteristics

This paper investigates the effect of third-order nonlinearities on the statistical distributions of wave heights, crests and troughs of waves mechanically generated in a deep-water basin, simulating two crossing systems characterized by bimodal spectra. Observed statistics exhibit various effects of third-order nonlinearities in a manner dependent on both the distance from the wave-maker and the angle between the mean directions of the component wave systems. In order to isolate and demonstrate the effects of third-order nonlinearities by themselves, vertically asymmetric distortions induced by second-order bound waves are removed from the observed time series. It appears then that the distributions of wave crests, troughs and heights extracted from the non-skewed series clearly deviate from the Rayleigh distribution, suggesting that waves are characterized by nonlinear corrections of higher order than the typical of second-order waves. Nonetheless, some models developed for weakly nonlinear second-order waves can still be used in describing wave heights, crests and troughs in mixed seas, provided that relevant distribution parameters are modified so as to reflect the effects of third-order corrections and some basic characteristics of mixed seas.

The Effect of Third-Order Nonlinearities on the Statistical Distributions of Wave Heights, Crests and Troughs in Bimodal Crossing Seas

2013 ◽  
Vol 135 (2) ◽  
Author(s):  
Petya G. Petrova ◽  
M. Aziz Tayfun ◽  
C. Guedes Soares
Keyword(s):  
Rayleigh Distribution ◽  
Second Order ◽  
Statistical Distributions ◽  
Third Order ◽  
Weakly Nonlinear ◽  
Distribution Parameters ◽  
Wave Heights ◽  
Component Wave ◽  
Basic Characteristics ◽  
The Waves

This paper investigates the effect of third-order nonlinearities on the statistical distributions of wave heights, crests, and troughs of waves mechanically generated in a deep-water basin and simulating two crossing systems characterized by bimodal spectra. The observed statistics exhibits various effects of third-order nonlinearities, in a manner dependent on both the distance from the wave-maker and the angle between the mean directions of the component wave systems. In order to isolate and demonstrate the effects of third-order nonlinearities by themselves, the vertically asymmetric distortions induced by second-order bound waves are removed from the observed time series. It appears then that the distributions of wave crests, troughs and heights extracted from the nonskewed records clearly deviate from the Rayleigh distribution, suggesting that the waves are characterized by nonlinear corrections of higher-order than the typical of second-order waves. Nonetheless, some models developed for weakly nonlinear second-order waves can still be used in describing wave heights, crests and troughs in mixed seas, provided that the relevant distribution parameters are modified, so as to reflect the effects of third-order corrections and some basic characteristics of the mixed seas.

Wave Statistics in Nonlinear Sea States

2011 ◽  
Author(s):  
Mohamed Latheef ◽  
Chris Swan
Keyword(s):  
Empirical Distribution ◽  
Nonlinear Effects ◽  
Rayleigh Distribution ◽  
Second Order ◽  
Wave Crest ◽  
Statistical Distributions ◽  
Sea State ◽  
Wave Statistics ◽  
Wave Basin ◽  
Wave Heights

This paper concerns the statistical distribution of both wave crest elevations and wave heights in deep water. A new set of laboratory observations undertaken in a directional wave basin located in the Hydrodynamics laboratory in the Department of Civil and Environmental Engineering at Imperial College London is presented. The resulting data were analysed and compared to a number of commonly applied statistical distributions. In respect of the wave crest elevations the measured data is compared to both linear and second-order order distributions, whilst the wave heights were compared to the Rayleigh distribution, the Forristall (1978) [1] empirical distribution and the modified Glukhovskiy distribution ([2] and [3]). Taken as a whole, the data confirms that the directionality of the sea state is critically important in determining the statistical distributions. For example, in terms of the wave crest statistics effects beyond second-order are most pronounced in uni-directional seas. However, if the sea state is sufficiently steep, nonlinear effects arising at third order and above can also be significant in directionally spread seas. Important departures from Forristall’s empirical distribution for the wave heights are also identified. In particular, the data highlights the limiting effect of wave breaking in the most severe seas suggesting that many of the commonly applied design solutions may be conservative in terms of crest height and wave height predictions corresponding to a small (10−4) probability of exceedance.

scholarly journals Gain control of synaptic transfer from second- to third-order neurons of cockroach ocelli.

1996 ◽  
Vol 107 (1) ◽  
pp. 121-131 ◽  
Author(s):  
M Mizunami
Keyword(s):  
Synaptic Transmission ◽  
Characteristic Curve ◽  
Gain Control ◽  
Second Order ◽  
The Body ◽  
Intensity Change ◽  
Third Order ◽  
The Mean ◽  
Mean Luminance ◽  
Nonlinear Nature

Synaptic transmission from second- to third-order neurons of cockroach ocelli occurs in an exponentially rising part of the overall sigmoidal characteristic curve relating pre- and postsynaptic voltage. Because of the nonlinear nature of the synapse, linear responses of second-order neurons to changes in ligh intensity are half-wave rectified, i.e., the response to a decrement in light is amplified whereas that to an increment in light is compressed. Here I report that the gain of synaptic transmission from second- to third-order neurons changes by ambient light levels and by wind stimulation applied to the cerci. Transfer characteristics of the synapse were studied by simultaneous intracellular recordings of second- and third-order neurons. Potential changes were evoked in second-order neurons by a sinusoidally modulated light with various mean luminances. With a decrease in the mean luminance (a) the mean membrane potential of second-order neurons was depolarized, (b) the synapse between the second- and third-order neurons operated in a steeper range of the exponential characteristic curve, where the gain to transmit modulatory signals was higher, and (c) the gain of third-order neurons to detect a decrement in light increased. Second-order neurons were depolarized when a wind or tactile stimulus was applied to various parts of the body including the cerci. During a wind-evoked depolarization, the synapse operated in a steeper range of the characteristic curve, which resulted in an increased gain of third-order neurons to detect light decrements. I conclude that the nonlinear nature of the synapse between the second- and third-order neurons provides an opportunity for an adjustment of gain to transmit signals of intensity change. The possibility that a similar gain control occurs in other visual systems and underlies a more advanced visual function, i.e., detection of motion, is discussed.

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.

Drift of an articulated cylinder in regular waves

1984 ◽  
Vol 394 (1807) ◽  
pp. 363-385 ◽  
Keyword(s):  
Near Field ◽  
Second Order ◽  
Regular Waves ◽  
Floating Bodies ◽  
Wave Heights ◽  
Tilt Mode ◽  
The Mean ◽  
Wave Induced ◽  
Theoretical Results ◽  
Drift Forces

Potential flow theory is used to investigate the wave induced harmonic response and the mean drift of an articulated column in regular waves. The mean drift horizontal force is evaluated by means of the Stokes expansion to second order in wave steepness. Analyses based on both near field and far field formulations are shown to give identical expressions, provided that the second-order forces at the intersection between column and seabed are included in the near field approach. The latter have not been considered in previous studies concerned with drift forces on floating bodies. It is shown that the drift forces on a column, although of second order, can excite piech responses of first order: this is because articulated columns are designed to have a low natural frequency in the tilt mode, relative to wave frequencies. Comparison of the theoretical results with experimental data, from a model tested in regular waves, suggests reasonable agreement for the drift forces over a range of frequencies and two wave heights.

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.

Distribution of Wave Height Maxima in Storm Sea States

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
Z. Cherneva ◽  
C. Guedes Soares ◽  
P. Petrova
Keyword(s):  
Wave Height ◽  
Exceedance Probability ◽  
Third Order ◽  
Wave Statistics ◽  
The North ◽  
Empirical Probability ◽  
Probability Densities ◽  
Wave Heights ◽  
The North Sea ◽  
The Mean

The effect of the coefficient of kurtosis, as a measure of third order nonlinearity, on the distribution of wave height maxima has been investigated. Measurements of the surface elevation during a storm at the North Alwyn platform in the North Sea have been used. The mean number of waves in the series is around 100. The maximum wave statistics have been compared with nonlinear theoretical distributions. It was found that the empirical probability densities of the maximum wave heights describe qualitatively the shift of the distribution modes toward higher values. The tendency for the peak of distribution to diminish with an increase in the coefficient of kurtosis up to 0.6 is also clearly seen. However, the empirical peak remains higher than the theoretically predicted one. The exceedance probability of the maximum wave heights was also estimated from the data and was compared with the theory. For the highest coefficients of kurtosis, estimated at nearly 0.6, the theoretical distribution approximates very well the empirical data. For lower coefficients of kurtosis, the theory tends to overestimate the exceedance probability of the maximum wave heights.

Overall properties of a cracked solid

1980 ◽  
Vol 88 (2) ◽  
pp. 371-384 ◽  
Author(s):  
J. A. Hudson
Keyword(s):  
Integral Equation ◽  
Solid Material ◽  
Equation Of Motion ◽  
Second Order ◽  
Number Density ◽  
Scattering Operator ◽  
Third Order ◽  
The Third ◽  
Differential Integral Equation ◽  
The Mean

AbstractThe differential-integral equation of motion for the mean wave in a solid material containing embedded cavities or inclusions is derived. It consists of a series of terms of ascending powers of the scattering operator, and is here truncated after the third term. This implies the second-order interactions between scatterers are included but those of the third order are not.The formulae are specialized to the case of thin cracks, either aligned in a single direction or randomly oriented. Expressions for the overall elastic constants are derived for the case of long wavelengths. These expressions are accurate to the second order in the number density of scatterers.

Distribution of Wave Height Maxima in Storm Sea States

2008 ◽  
Author(s):  
Zhivelina Cherneva ◽  
C. Guedes Soares ◽  
Petya Petrova
Keyword(s):  
Wave Height ◽  
Exceedance Probability ◽  
Third Order ◽  
Wave Statistics ◽  
The North ◽  
Probability Densities ◽  
Distribution Mode ◽  
Wave Heights ◽  
The North Sea ◽  
The Mean

The effect of the coefficient of kurtosis as a measure of the nonlinearity of third order on the distribution of the wave height maxima has been investigated. Measurements of the surface elevation during a storm at the North Alwyn platform in the North Sea have been used. The mean number of waves in the series is around 100. The maximum wave statistics have been compared with nonlinear theoretical distributions. It was found that the empirical probability densities of the maximum wave heights describe qualitatively the shift of the distribution mode towards higher values. The tendency for the peak of distribution to diminish with increase of the coefficient of kurtosis up to 0.6 is also clearly seen. However, the empirical peak remains higher than the theoretically predicted one. Exceedance probability of the maximum wave heights was also estimated from the data and was compared with the theory. For the highest coefficients of kurtosis nearly 0.6 the theoretical distribution approximates very well the empirical data. For lower coefficients of kurtosis the theory tends to overestimate the exceedance probability of the maximum wave heights.

Bulk Contributions Modulate the Sum-Frequency Generation Spectra of Interfacial Water on Model Sea-Spray Aerosols

2018 ◽  
Author(s):  
Sandeep K. Reddy ◽  
Raphael Thiraux ◽  
Bethany A. Wellen Rudd ◽  
Lu Lin ◽  
Tehseen Adel ◽  
...  
Keyword(s):  
Single Layer ◽  
Sum Frequency Generation ◽  
Second Order ◽  
Interfacial Structure ◽  
Sea Spray ◽  
Third Order ◽  
Systematic Analysis ◽  
Sum Frequency ◽  
Structure Of Water ◽  
Frequency Generation

Vibrational sum-frequency generation (vSFG) spectroscopy is used to determine the molecular structure of water at the interface of palmitic acid monolayers. Both measured and calculated spectra display speci c features due to third-order contributions to the vSFG response which are associated with nite interfacial electric potentials. We demonstrate that theoretical modeling enables to separate the third-order contributions, thus allowing for a systematic analysis of the strictly surface-sensitive, second-order component of the vSFG response. This study provides fundamental, molecular-level insights into the interfacial structure of water in a neutral surfactant system with relevance to single layer bio-membranes and environmentally relevant sea-spray aerosols. These results emphasize the key role that computer simulations can play in interpreting vSFG spectra and revealing microscopic details of water at complex interfaces, which can be difficult to extract from experiments due to the mixing of second-order, surface-sensitive and third-order, bulk-dependent contributions to the vSFG response.

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