On the dispersion relation of random gravity waves. Part 1. Theoretical framework

1979 ◽  
Vol 92 (4) ◽  
pp. 717-730 ◽  
Author(s):  
Akira Masuda ◽  
Yi-Yu Kuo ◽  
Hisashi Mitsuyasu
Keyword(s):  
Dispersion Relation ◽  
Gravity Waves ◽  
Wind Waves ◽  
Surface Displacement ◽  
Theoretical Framework ◽  
Homogeneous Field ◽  
Third Order ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Forced Waves

A theoretical framework is given, upon which to examine the dispersion relation of random gravity waves. First a weakly nonlinear theory is developed to the third-order for a statistically stationary and homogeneous field of random gravity waves. Both the spectrum of forced waves and the nonlinear dispersion relation are expressed in terms of the spectrum of free waves under the assumption of the Gaussian process for the first-order surface displacement. Next a method is proposed by which to separate each of the spectra of free and forced waves from the measured spectrum. This gives practical and powerful means of investigating the statistical structure of wind waves.

scholarly journals On the strength of the weakly nonlinear theory for surface gravity waves

2016 ◽  
Vol 810 ◽  
pp. 1-4 ◽  
Author(s):  
Michael Stiassnie
Keyword(s):  
Gravity Waves ◽  
Nonlinear Theory ◽  
Mathematical Formulation ◽  
Wave Theory ◽  
Quantitative Agreement ◽  
Surface Gravity ◽  
Weakly Nonlinear ◽  
Surface Gravity Waves ◽  
Weakly Nonlinear Theory ◽  
Wave Basin

Recently, Bonnefoy et al. (J. Fluid Mech., vol. 805, 2016, R3) studied the resonant interaction of oblique surface gravity waves in a large $50~\text{m}\times 30~\text{m}\times 5~\text{m}$ wave basin. Their experimental results are in excellent quantitative agreement with predictions of the weakly nonlinear wave theory, and provide additional evidence to the strength of this widely used mathematical formulation. In this article, the reader is introduced to the many facets of the weakly nonlinear theory for surface gravity waves, and to its current and possible future applications, deterministic as well as stochastic.

On the rippling of small waves: a harmonic nonlinear nearly resonant interaction

1972 ◽  
Vol 52 (4) ◽  
pp. 725-751 ◽  
Author(s):  
L. F. Mcgoldrick
Keyword(s):  
Dispersion Relation ◽  
Small Parameter ◽  
Gravity Waves ◽  
Resonance Condition ◽  
Resonant Interaction ◽  
Wave System ◽  
Dispersive Wave ◽  
Wave Steepness ◽  
Weakly Nonlinear ◽  
Nonlinear Dispersive Wave

We show that the rippling often observed on small progressive gravity waves can be explained in terms of a nearly resonant harmonic nonlinear interaction. The resonance condition is that the phase speeds of the two waves must be nearly identical. The in viscid analysis is generalized to any order in a small parameter proportional to the wave steepness. Wave tank measurements provide experimental evidence for most of the predicted results. The phenomenon of resonant rippling is further shown to be not just peculiar to capillary-gravity waves, but in fact possible for any weakly nonlinear dispersive wave system whose dispersion relation has discrete pairs of solutions nearly satisfying the resonance conditions.

scholarly journals NONLINEAR THEORY ON PARTICLE VELOCITY AND PRESSURE OF RANDOM WAVES

1986 ◽  
Vol 1 (20) ◽  
pp. 34
Author(s):  
Yi-Yu Kuo ◽  
Hwar-Ming Wang
Keyword(s):  
Particle Velocity ◽  
Nonlinear Theory ◽  
Power Spectra ◽  
Surface Displacement ◽  
Nonlinear Effects ◽  
Random Waves ◽  
Weakly Nonlinear ◽  
Directional Spectrum ◽  
Weakly Nonlinear Theory ◽  
Different Characteristics

In this paper, to the third approximation, we used the Fourierstieltjes integral rather than Fourier coefficient to develop a weakly nonlinear theory. From the theory, the nonlinear spectral components for water particle velocity and wave pressure can be calculated directly from the directional spectrum of water surface displacement. Computed results based on the nonlinear theory were compared with that of experiment made by Anastasiou (1982). Furthermore, in accordance with the different characteristics of wave properties, such as wave steepness, water depth and so on, the nonlinear effects on wave kinematic and pressure properties were extensively investigated by using some standard power spectra.

Nonlinear dynamic pressure beneath waves in water of large and intermediate depth

2021 ◽  
Author(s):  
Anna Kokorina ◽  
Alexey Slunyaev ◽  
Marco Klein
Keyword(s):  
Gravity Waves ◽  
Pressure Field ◽  
Dynamic Pressure ◽  
Surface Displacement ◽  
Weakly Nonlinear ◽  
Pressure Fields ◽  
Component Theory ◽  
Theoretical Predictions ◽  
Weakly Nonlinear Waves ◽  
The Waves

<p>The data of simultaneous measurements of the surface displacement produced by propagating planar waves in experimental flume and of the dynamic pressure fields beneath the waves are compared with the theoretical predictions based on different approximations for modulated potential gravity waves. The performance of different theories to reconstruct the pressure field from the known surface displacement time series (the direct problem) is investigated. A new two-component theory for weakly modulated weakly nonlinear waves is proposed, which exhibits the best capability among the considered. Peculiarities of the vertical modes of the nonlinear pressure harmonics are discussed.</p><p> </p><p>The work was supported by the RFBR projects 19-55-15005 and 20-05-00162 (AK).</p>

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

scholarly journals The role of constant vorticity on weakly nonlinear surface gravity waves

Wave Motion ◽  
2020 ◽  
pp. 102702
Author(s):  
M.A. Manna ◽  
S. Noubissie ◽  
J. Touboul ◽  
B. Simon ◽  
R.A. Kraenkel
Keyword(s):  
Gravity Waves ◽  
Surface Gravity ◽  
Weakly Nonlinear ◽  
Surface Gravity Waves ◽  
Constant Vorticity ◽  
Nonlinear Surface

scholarly journals Weakly nonlinear theory and sea state bias estimations

1999 ◽  
Vol 104 (C4) ◽  
pp. 7641-7647 ◽  
Author(s):  
Tanos Elfouhaily ◽  
Donald Thompson ◽  
Douglas Vandemark ◽  
Bertrand Chapron
Keyword(s):  
Nonlinear Theory ◽  
Sea State ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Sea State Bias
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