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.

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.

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.

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.

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.

Evidence for general base catalysis by protic solvents in a kinetic study of alcoholyses and hydrolyses of 1-(phenoxycarbonyl)pyridinium ions under both solvolytic and non-solvolytic conditions

2009 ◽  
Vol 74 (1) ◽  
pp. 43-55 ◽  
Author(s):  
Dennis N. Kevill ◽  
Byoung-Chun Park ◽  
Jin Burm Kyong
Keyword(s):  
Second Order ◽  
Base Catalysis ◽  
Substitution Reactions ◽  
Third Order ◽  
Methoxy Derivative ◽  
First Order ◽  
General Base ◽  
Protic Solvents ◽  
Kinetics Of ◽  
Pyridinium Ions

The kinetics of nucleophilic substitution reactions of 1-(phenoxycarbonyl)pyridinium ions, prepared with the essentially non-nucleophilic/non-basic fluoroborate as the counterion, have been studied using up to 1.60 M methanol in acetonitrile as solvent and under solvolytic conditions in 2,2,2-trifluoroethan-1-ol (TFE) and its mixtures with water. Under the non- solvolytic conditions, the parent and three pyridine-ring-substituted derivatives were studied. Both second-order (first-order in methanol) and third-order (second-order in methanol) kinetic contributions were observed. In the solvolysis studies, since solvent ionizing power values were almost constant over the range of aqueous TFE studied, a Grunwald–Winstein equation treatment of the specific rates of solvolysis for the parent and the 4-methoxy derivative could be carried out in terms of variations in solvent nucleophilicity, and an appreciable sensitivity to changes in solvent nucleophilicity was found.

scholarly journals Wave Directions in a Narrow Bay

2010 ◽  
Vol 40 (1) ◽  
pp. 155-169 ◽  
Author(s):  
Heidi Pettersson ◽  
Kimmo K. Kahma ◽  
Laura Tuomi
Keyword(s):  
Wave Model ◽  
Wave Climate ◽  
Spectral Peak ◽  
Step Size ◽  
Weakly Nonlinear ◽  
Spectral Wave Model ◽  
Directional Coupling ◽  
Directional Wave ◽  
The Baltic ◽  
The Waves

Abstract In slanting fetch conditions the direction of actively growing waves is strongly controlled by the fetch geometry. The effect was found to be pronounced in the long and narrow Gulf of Finland in the Baltic Sea, where it significantly modifies the directional wave climate. Three models with different assumptions on the directional coupling between the wave components were used to analyze the physics responsible for the directional behavior of the waves in the gulf. The directionally decoupled model produced the direction at the spectral peak correctly when the slanting fetch geometry was narrow but gave a weaker steering than observed when the fetch geometry was broader. The method of Donelan estimated well the direction at the spectral peak in well-defined slanting fetch conditions, but overestimated the longer fetch components during wave growth from a more complex shoreline. Neither the decoupled nor the Donelan model reproduced the observed shifting of direction with the frequency. The performance of the third-generation spectral wave model (WAM) in estimating the wave directions was strongly dependent on the grid resolution of the model. The dominant wave directions were estimated satisfactorily when the grid-step size was dropped to 5 km in the gulf, which is 70 km in its narrowest part. A mechanism based on the weakly nonlinear interactions is proposed to explain the strong steering effect in slanting fetch conditions.

Towards a Technique for Nonlinear Modal Analysis

2012 ◽  
Author(s):  
Simon A. Neild ◽  
Andrea Cammarano ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Transformation Process ◽  
Second Order ◽  
Identity Transformation ◽  
Modal Testing ◽  
System Response ◽  
Weakly Nonlinear ◽  
Nonlinear Modal Analysis

In this paper we discuss a theoretical technique for decomposing multi-degree-of-freedom weakly nonlinear systems into a simpler form — an approach which has parallels with the well know method for linear modal analysis. The key outcome is that the system resonances, both linear and nonlinear are revealed by the transformation process. For each resonance, parameters can be obtained which characterise the backbone curves, and higher harmonic components of the response. The underlying mathematical technique is based on a near identity normal form transformation. This is an established technique for analysing weakly nonlinear vibrating systems, but in this approach we use a variation of the method for systems of equations written in second-order form. This is a much more natural approach for structural dynamics where the governing equations of motion are written in this form as standard practice. In fact the first step in the method is to carry out a linear modal transformation using linear modes as would typically done for a linear system. The near identity transform is then applied as a second step in the process and one which identifies the nonlinear resonances in the system being considered. For an example system with cubic nonlinearities, we show how the resulting transformed equations can be used to obtain a time independent representation of the system response. We will discuss how the analysis can be carried out with applied forcing, and how the approximations about response frequencies, made during the near-identity transformation, affect the accuracy of the technique. In fact we show that the second-order normal form approach can actually improve the predictions of sub- and super-harmonic responses. Finally we comment on how this theoretical technique could be used as part of a modal testing approach in future work.

Panicle, spikelet, and floret development in orchardgrass (Dactylis glomerata)

1993 ◽  
Vol 71 (4) ◽  
pp. 523-532 ◽  
Author(s):  
Joanna Fraser ◽  
Eric G. Kokko
Keyword(s):  
Electron Microscopy ◽  
Scanning Electron Microscopy ◽  
Unique Feature ◽  
Dactylis Glomerata ◽  
Bilateral Symmetry ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Lateral Branches ◽  
Scanning Electron

The initial stages of panicle, spikelet, and floret development in field-grown 'Kay' orchardgrass were examined using scanning electron microscopy. Spikelets arose from a complex multilevelled sequence of initiation from branch apices. Spikelets developed indirectly in a two-tiered progression: (i) an acropetal and basipetal sequence of first order, second-order, and third-order inflorescence apices, and (ii) an acropetal development within subclusters of higher-order lateral branch inflorescence apices. The panicle had the unique feature of dorsiventrality as well as bilateral symmetry. The basal apex from first-order, second-order, or third-order apices developed on the same side of the main axis as the first-order apex. The two glumes subtending each spikelet primordium developed alternately and acropetally. Development and initiation of florets within spikelets was basipetal within the panicle, basipetal within clusters and subclusters of spikelets on lateral branches, and acropetal within spikelets. Within florets, paleas developed later than lemmas. Key words: Dactylis glomerata, cocksfoot, scanning electron microscopy, development, panicle.

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