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.

Measurements of third-order resonant wave interactions

1966 ◽  
Vol 25 (3) ◽  
pp. 437-456 ◽  
Author(s):  
L. F. Mcgoldrick ◽  
O. M. Phillips ◽  
N. E. Huang ◽  
T. H. Hodgson
Keyword(s):  
Gravity Waves ◽  
Resonant Interaction ◽  
Experimental Demonstration ◽  
Wave Interactions ◽  
Third Order ◽  
The Third ◽  
Resonant Wave ◽  
Interaction Distance ◽  
At Resonance ◽  
Wave Trains

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.

An experiment on second-order capillary gravity resonant wave interactions

1970 ◽  
Vol 40 (2) ◽  
pp. 251-271 ◽  
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.

Resonant wave interactions in a stratified shear flow

1988 ◽  
Vol 190 ◽  
pp. 357-374 ◽  
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.

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

1994 ◽  
Vol 269 ◽  
pp. 1-22 ◽  
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.

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

2007 ◽  
Vol 64 (6) ◽  
pp. 1977-1994 ◽  
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.

scholarly journals Third-order resonant interaction of atmospheric gravity waves

2013 ◽  
Vol 118 (5) ◽  
pp. 2197-2206 ◽  
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

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.

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

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.

scholarly journals On two approaches to the third-order solution of surface gravity waves

2021 ◽  
Vol 33 (9) ◽  
pp. 097101
Author(s):  
Zhe Gao ◽  
Z. C. Sun ◽  
S. X. Liang
Keyword(s):  
Gravity Waves ◽  
Surface Gravity ◽  
Third Order ◽  
Surface Gravity Waves ◽  
The Third ◽  
Order Solution

scholarly journals Third-order resonant wave interactions

1977 ◽  
Vol 20 (11) ◽  
pp. 1793 ◽  
Author(s):  
Spyridon G. Lekoudis ◽  
Ali H. Nayfeh ◽  
William S. Saric
Keyword(s):  
Wave Interactions ◽  
Third Order ◽  
Resonant Wave
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