Measurements of third-order resonant wave interactions

This paper presents the results of experiments on the resonant interaction of gravity waves. Two mutually-orthogonal primary wave trains are generated in a tank and their interaction products studied at various positions on the surface. Under suitable conditions, the growing resonant third-order interaction product is identified; its amplitude is shown to be a linear function of the interaction distance. The band-width of the response decreases with increasing distance, as is characteristic of the phenomenon of resonance. The ratio of the frequencies of the primary waves at resonance is very close to that predicted theoretically; the growth rate of the third component is close to, though about 20% higher than, the predicted value. Conditions far from resonance are also studied; it is found that the growing tertiary wave is absent in this case.These results offer the first unambiguous experimental demonstration of resonant wave interactions.

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An experiment on third-order resonant wave interactions

Journal of Fluid Mechanics ◽

10.1017/s0022112066000168 ◽

1966 ◽

Vol 25 (3) ◽

pp. 417-435 ◽

Cited By ~ 49

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.

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An experiment on second-order capillary gravity resonant wave interactions

Journal of Fluid Mechanics ◽

10.1017/s0022112070000162 ◽

1970 ◽

Vol 40 (2) ◽

pp. 251-271 ◽

Cited By ~ 65

Author(s):

L. F. McGoldrick

Keyword(s):

Spatial Variation ◽

Viscous Dissipation ◽

Gravity Waves ◽

Initial Conditions ◽

Resonant Interaction ◽

Experimental Measurements ◽

Wave Interactions ◽

Resonant Wave ◽

At Resonance ◽

Wave Maker

This paper presents the results of a set of detailed experimental measurements on the resonant interaction of capillary-gravity waves for a case in which the entire propagation is in one direction. The influence of viscous attenuation is accounted for in the analysis. The measurements trace the entire spatial variation, or modulation envelope, of the amplitudes of the interacting modes from their inception near a wave-maker to their ultimate extinction through viscous dissipation, in excellent agreement with the theory. This is an unambiguous demonstration that at resonance and for the initial conditions specified at the wave-maker, a wave of uniform profile cannot exist.

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Theoretical and experimental studies of gravity wave interactions

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1967.0125 ◽

1967 ◽

Vol 299 (1456) ◽

pp. 104-119 ◽

Cited By ~ 21

Keyword(s):

Gravity Waves ◽

Experimental Studies ◽

Internal Gravity Waves ◽

Stokes Wave ◽

Wave Interactions ◽

Principal Characteristics ◽

Resonant Wave ◽

Resonant Interactions ◽

Interaction Distance ◽

Wave Modes

This paper is concerned with various aspects of the resonant interactions among waves. An experiment was suggested by Longuet-Higgins (1962) to detect this type of interaction among surface waves. This was subsequently performed by Longuet-Higgins & Smith (1966) and by McGoldrick, Phillips, Huang & Hodgson (1966). The results of the two sets of experiments are compared. Together they demonstrate very clearly the principal characteristics of the interaction; the maximum response at resonance and the linear growth with interaction distance, the decrease in band width with interaction distance and the shift of the resonance point that results from the amplitude dispersion. It is shown further that the instability of the Stokes wave, discovered and analysed by Benjamin & Feir, can be described in terms of these interactions and that it is not restricted to purely two dimensional motion. A Stokes wave is unstable to a disturbance containing a pair of wavenumbers defined by any point in the zone just inside the figure-of-eight loop shown in figure 12. Another example of resonant wave interactions is provided by short, internal gravity waves in a stratified fluid with constant Brunt-Väisälä frequency. The interactions among Fourier modes are considered, and it is shown that there arise both free and forced modes. In the latter, the dispersion relation for internal waves is not satisfied; there is no particular relation between wavenumber and frequency. The amplitudes of these are small compared with those of the internal wave modes provided the harmonic mean of the vorticity in the two interacting waves is small compared with the Brunt-Väisälä frequency. The motion then consists of interacting internal gravity waves, whose interaction sets are closed. On the other hand, if the forced components are comparable in magnitude with the wave modes, these interact strongly and indiscriminately; a ‘cascade’, characteristic of turbulence, develops.

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Resonant wave interactions in a stratified shear flow

Journal of Fluid Mechanics ◽

10.1017/s0022112088001351 ◽

1988 ◽

Vol 190 ◽

pp. 357-374 ◽

Cited By ~ 11

Author(s):

R. Grimshaw

Keyword(s):

Shear Flow ◽

Gravity Waves ◽

Critical Level ◽

Internal Gravity Waves ◽

Wave Interactions ◽

Resonant Wave ◽

Resonant Interactions ◽

Stratified Shear Flow ◽

Weak Resonance ◽

Background Shear

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

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Resonant wave interactions near a critical level in a stratified shear flow

Journal of Fluid Mechanics ◽

10.1017/s0022112094001461 ◽

1994 ◽

Vol 269 ◽

pp. 1-22 ◽

Cited By ~ 5

Author(s):

R. Grimshaw

Keyword(s):

Shear Flow ◽

Critical Level ◽

Harmonic Component ◽

Internal Gravity Waves ◽

Resonant Interaction ◽

Wave Interactions ◽

Incoming Wave ◽

Resonant Wave ◽

Background Density ◽

Stratified Shear Flow

Resonant interactions between internal gravity waves propagating in a stratified shear flow are considered for the case when the background density and shear flow vary slowly with respect to the waves. In Grimshaw (1988) triad resonances were considered, and interaction equations derived for the case when the resonance conditions are met only on certain space-time surfaces, being resonance sites. Here this analysis is extended to include higher-order resonances, with the aim of studying resonant wave interactions near a critical level. It is shown that a secondary resonant interaction between two incoming waves, in which two harmonic components of one incoming wave interact with a single harmonic component of another incoming wave, produces a reflected wave. This result is shown to agree with the study of Brown & Stewartson (1980, 1982a, b) who obtained this same result by a different approach.

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Modal and Nonmodal Perturbations of Monochromatic High-Frequency Gravity Waves: Primary Nonlinear Dynamics

Journal of the Atmospheric Sciences ◽

10.1175/jas3940.1 ◽

2007 ◽

Vol 64 (6) ◽

pp. 1977-1994 ◽

Cited By ~ 7

Author(s):

Ulrich Achatz

Keyword(s):

Nonlinear Dynamics ◽

Gravity Waves ◽

High Frequency ◽

Normal Modes ◽

Wave Interactions ◽

Resonant Wave ◽

Spatial Dimensions ◽

Phase Direction ◽

Parallel Mode ◽

Boussinesq Fluid

The primary nonlinear dynamics of high-frequency gravity waves (HGWs) perturbed by their most prominent normal modes (NMs) or singular vectors (SVs) in a rotating Boussinesq fluid have been studied by direct numerical simulations (DNSs), with wave scales and values of viscosity and diffusivity characteristic for the upper mesosphere. The DNS is 2.5D in that it has only two spatial dimensions, defined by the direction of propagation of the HGW and the direction of propagation of the perturbation in the plane orthogonal to the HGW phase direction, but describes a fully 3D velocity field. Many results of the more comprehensive fully 3D simulations in the literature are reproduced. So it is found that statically unstable HGWs are subject to wave breaking ending in a wave amplitude with respect to the overturning threshold near 0.3. It is shown that this is a result of a perturbation of the HGW by its leading transverse NM. For statically stable HGWs, a parallel NM has the strongest effect, quite in line with previous results on the predominantly 2D instability of such HGWs. This parallel mode is, however, not the leading NM but a larger-scale pattern, seemingly driven by resonant wave–wave interactions, leading eventually to energy transfer from the HGW into another gravity wave with steeper phase propagation. SVs turn out to be less effective in triggering HGW decay but they can produce turbulence of a strength that is (as that from the NMs) within the range of measured values, however with a more pronounced spatial confinement.

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Third-order resonant interaction of atmospheric gravity waves

Journal of Geophysical Research Atmospheres ◽

10.1002/jgrd.50252 ◽

2013 ◽

Vol 118 (5) ◽

pp. 2197-2206 ◽

Cited By ~ 10

Author(s):

Kai Ming Huang ◽

Shao Dong Zhang ◽

Fan Yi ◽

Chun Ming Huang ◽

Quan Gan ◽

...

Keyword(s):

Gravity Waves ◽

Resonant Interaction ◽

Third Order ◽

Atmospheric Gravity Waves ◽

Atmospheric Gravity

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The Transformation Between the Eulerian and Lagrangian Solutions for Irrotational Standing Gravity Waves

Volume 6: Materials Technology; C.C. Mei Symposium on Wave Mechanics and Hydrodynamics; Offshore Measurement and Data Interpretation ◽

10.1115/omae2009-79927 ◽

2009 ◽

Author(s):

Yang-Yih Chen ◽

Hung-Chu Hsu

Keyword(s):

Gravity Waves ◽

Particle Motion ◽

Mass Conservation ◽

Perturbation Parameter ◽

Third Order ◽

The Third ◽

Asymptotic Expressions ◽

Lagrangian Form ◽

Standing Gravity Waves ◽

The Given

This study reports the transformations between the third-order Eulerian and Lagrangian solutions for the standing gravity waves on the uniform depth. Regarding the motion of a marked fluid particle, the instantaneous velocity, the mass conservation and the free surface must be the same for either Eulerian or Lagrangian methods. We impose the assumption that the Lagrangian wave frequency is a function of wave steepness. Expanding the unknown function in a small perturbation parameter and using a successive expansion in a Taylor series for the water particle path and the period of a particle motion, the third order asymptotic expressions for the particle trajectories and the period of particle motion can be derived directly in Lagrangian form. It shows that the given Eulerian solutions are capable of being transformed into the completely unknown Lagrangian solutions and the reversible process is also identified.

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Nonlinear probability distributions of waves in bimodal following and crossing seas generated in laboratory experiments

Natural Hazards and Earth System Sciences Discussions ◽

10.5194/nhessd-1-5403-2013 ◽

2013 ◽

Vol 1 (5) ◽

pp. 5403-5452

Author(s):

P. G. Petrova ◽

C. Guedes Soares

Keyword(s):

Water Waves ◽

Probability Distributions ◽

Spectral Characteristics ◽

Order Effects ◽

Wave Interactions ◽

Third Order ◽

Empirical Distributions ◽

Wave Effects ◽

Wave Trains ◽

Large Coefficient

Abstract. This paper presents an analysis of the nonlinear distributions of crests, troughs and heights of deep water waves from mixed following sea states generated mechanically in an offshore basin and compares with previous results for mixed crossing seas from the same experiment. The random signals at the wavemaker in both types of mixed seas are characterized by bimodal spectra following the model of Guedes Soares (1984). In agreement with the Benjamin–Feir mechanism, the high-frequency spectrum shows decrease of the peak magnitude and downshift of the peak with the distance, as well as reduction of the tail. The observed statistics and probabilistic distributions exhibit, in general, increasing effects of third-order nonlinearity with the distance from the wavemaker. However, this effect is less pronounced in the wave systems with two following wave trains than in the crossing seas with identical initial spectral characteristics. The relevance of third-order effects due to free modes only is demonstrated and assessed by excluding the vertically asymmetric distortions induced by bound-wave effects of second and third order. The fact that for records characterized by relatively large coefficient of kurtosis, the empirical distributions for the non-skewed profiles continue deviating from the linear predictions, corroborate the relevance of free-wave interactions and thus the need of using higher-order models for the description of wave data.

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