scholarly journals On kinematics of very steep waves

2013 ◽  
Vol 13 (8) ◽  
pp. 2101-2107 ◽  
Author(s):  
L. Shemer
Keyword(s):  
Wave Breaking ◽  
Wide Spectrum ◽  
Wave Train ◽  
Vertical Acceleration ◽  
Wave Group ◽  
Third Order ◽  
Large Wave ◽  
The Third ◽  
Steep Waves ◽  
Group Velocities

Abstract. Experiments on extremely steep deterministic waves generated in a large wave tank by focusing of a broad-banded wave train serve as a motivation for the theoretical analysis of the conditions leading to wave breaking. Particular attention is given to the crest of the steepest wave where both the horizontal velocity and the vertical acceleration attain their maxima. Analysis is carried out up to the third order in wave steepness. The apparent, Eulerian and Lagrangian accelerations are computed for wave parameters observed in experiments. It is demonstrated that for a wave group with a wide spectrum, the crest propagation velocity differs significantly from both the phase and the group velocities of the peak wave. Conclusions are drawn regarding the applicability of various criteria for wave breaking.

scholarly journals On kinematics of very steep waves

2013 ◽  
Vol 1 (2) ◽  
pp. 1487-1506
Author(s):  
L. Shemer
Keyword(s):  
Wave Breaking ◽  
Wide Spectrum ◽  
Wave Train ◽  
Vertical Acceleration ◽  
Wave Group ◽  
Wave Steepness ◽  
Large Wave ◽  
Wave Parameters ◽  
Steep Waves ◽  
Group Velocities

Abstract. Experiments on extremely steep waves generated in a large wave tank by focusing of a broad-banded wave train serve as a motivation for the theoretical analysis of the conditions leading to wave breaking. Particular attention is given to the crest of the steepest wave where both the horizontal velocity and the vertical acceleration attain their maxima. Analysis is carried out up to the 3rd order in wave steepness. The apparent, Eulerian and Lagrangian accelerations are computed for wave parameters observed in experiments. It is demonstrated that for a wave group with a wide spectrum, the crest propagation velocity differs significantly from both the phase and the group velocities of the peak wave. Conclusions are drawn regarding the applicability of various criteria for wave breaking.

An experiment on third-order resonant wave interactions

1966 ◽  
Vol 25 (3) ◽  
pp. 417-435 ◽  
Author(s):  
M. S. Longuet-Higgins ◽  
N. D. Smith
Keyword(s):  
Gravity Waves ◽  
Response Curve ◽  
Resonant Interaction ◽  
Third Wave ◽  
Theoretical Discussion ◽  
Wave Interactions ◽  
Third Order ◽  
Large Wave ◽  
The Third ◽  
Resonant Wave

An experiment has been carried out to verify the existence of the resonant interaction between trains of gravity waves, predicted by Phillips (1960). As suggested by Longuet-Higgins (1962), two trains of waves in mutually perpendicular directions were generated in a rectangular wave tank. The ratio σ1/σ2of the wave frequencies was varied (1·4 < σ1/σ2< 2·1). When σ1/σ2[eDot ] 1·7357 it was expected that a resonant interaction would take place, generating a wave of frequency (2σ1−σ2). The amplitude of the third wave was expected to increase almost linearly in the direction of wave propagation. The shape of the response curve as a function of σ1/σ2was also predicted.In the present experiments rather large wave amplitudes had to be used, and the theoretical shape of the response curve was distorted by non-linear detuning. Nevertheless the peak amplitude of the resonant wave was found to increase with distance in very nearly the manner predicted.These experiments were carried out in 1961 but publication was deferred pending a similar but more accurate investigation by McGoldrick, Phillips, Huang & Hodgson (1966). Much of the theoretical discussion given in the present paper is relevant to their work.

scholarly journals Nonlinear Evolution of a Steep, Focusing Wave Group in Deep Water Simulated With oceanwave3d

2019 ◽  
Vol 142 (2) ◽  
Author(s):  
Dylan Barratt ◽  
Harry B. Bingham ◽  
Thomas A. A. Adcock
Keyword(s):  
Energy Transfer ◽  
Deep Water ◽  
Nonlinear Evolution ◽  
Initial Conditions ◽  
Amplitude Spectrum ◽  
Wave Group ◽  
Third Order ◽  
The Third ◽  
Nonlinear Potential ◽  
Wave Event

Abstract Steep, focusing waves can experience fast and local nonlinear evolution of the spectrum due to wave–wave interactions resulting in energy transfer to both higher and lower wavenumber components. The shape and kinematics of a steep wave may, thus, differ substantially from the predictions of linear theory. We have investigated the role of nonlinear interactions on group shape for a steep, narrow-banded, directionally spread wave group focusing in deep water using the fully nonlinear potential flow solver, oceanwave3d. Exact second-order correction of the initial conditions has been implemented together with a novel third-order approximate correction based on a Stokes-type formulation for surface elevation combined with a scaling argument for the third-order velocity potential. Four-phase separation reveals that the third-order scheme provides a good estimate for the third-order superharmonics. A quantitative assessment of numerical error has also been performed for the spatial and temporal discretization, including energy conservation, a reversibility check, and validation against previous simulations performed with a higher-order spectral (HOS) code. The initially narrow-banded amplitude spectrum exhibits the formation of “sidelobes” at angles of approximately ±35deg to the spectral peak during the simulated extreme wave event, occurring in approximately ten wave periods, with a preferential energy transfer to high-wavenumber components. The directional energy transfer is attributed to resonant third-order interactions with a discussion of the engineering implications.

Third-Order Hydrodynamic Loads on an Oscillating Vertical Cylinder in Water

1999 ◽  
Vol 121 (1) ◽  
pp. 16-21 ◽  
Author(s):  
M. Markiewicz ◽  
P. Łe¸tkowski ◽  
O. Mahrenholtz
Keyword(s):  
Harmonic Component ◽  
Vertical Cylinder ◽  
Offshore Structures ◽  
Third Order ◽  
Earthquake Excitation ◽  
Hydrodynamic Loads ◽  
Third Harmonic ◽  
The Third ◽  
Steep Waves ◽  
Order Components

The third-harmonic component of the third-order hydrodynamic loads on a vertical circular cylinder oscillating in water is calculated by a conventional perturbation method within the framework of a potential theory. Although the third-order forces are expressed in terms of the first, second, and third-order components of the velocity potential, the latter is not directly required for the calculation. It is replaced by a properly defined linearized radiation potential via Haskind-like theorem. The results of the study are applicable to the analysis of high-frequency resonances of deepwater offshore structures under earthquake excitation or under steep waves (ringing problem).

scholarly journals Nonlinear Evolution of a Steep, Focusing Wave Group in Deep Water Simulated With OceanWave3D

2019 ◽  
Author(s):  
Dylan Barratt ◽  
Harry B. Bingham ◽  
Thomas A. A. Adcock
Keyword(s):  
Energy Transfer ◽  
Deep Water ◽  
Nonlinear Evolution ◽  
Initial Conditions ◽  
Amplitude Spectrum ◽  
Wave Group ◽  
Third Order ◽  
The Third ◽  
Nonlinear Potential ◽  
Wave Event

Abstract Steep, focusing waves can experience fast and local nonlinear evolution of the spectrum due to wave-wave interactions resulting in energy transfer to both higher and lower wavenumber components. The shape and kinematics of a steep wave may, thus, differ substantially from the predictions of linear theory. We have investigated the role of nonlinear interactions on group-shape for a steep, narrow-banded, directionally-spread wave group focusing in deep water using the fully-nonlinear potential flow solver, OceanWave3D. Exact second-order correction of the initial conditions has been implemented together with a novel third-order approximate correction based on a Stokes-type formulation for surface elevation combined with a scaling-argument for the third-order velocity potential. Four-phase separation reveals that the third-order scheme provides a good estimate for the third-order superharmonics. A quantitative assessment of numerical error has also been performed for the spatial and temporal discretization, including energy conservation, a reversibility check and validation against previous simulations performed with a higher-order spectral (HOS) code. The initially narrow-banded amplitude spectrum exhibits the formation of sidelobes at angles of approximately ±35° to the spectral peak during the simulated extreme wave event, occurring in approximately 10 wave periods, with a preferential energy transfer to high-wavenumber components. The directional energy transfer is attributed to resonant third-order interactions with a discussion of the engineering implications.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

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