scholarly journals Interction of a short-wave field with a dominant long wave in deep water: derivation form Zakharov's spectral formulation

1988 ◽  
Vol 29 (4) ◽  
pp. 430-439 ◽  
Author(s):  
A. D. D. Craik
Keyword(s):  
Wave Packet ◽  
Wave Field ◽  
Gravity Waves ◽  
Deep Water ◽  
Short Wave ◽  
Wave Action ◽  
Leading Order ◽  
Order Interaction ◽  
Short Waves ◽  
Long Wave

AbstractThe leading-order interaction of short gravity waves with a dominant long-wave swell is calculated by means of Zakharov's [7] spectral formulation. Results are obtained both for a discrete train of short waves and for a localised wave packet comprising a spectrum of short waves.The results for a discrete wavetrain agree with previous work of Longuet-Higgins & Stewart [5], and general agreement is found with parallel work of Grimshaw [4] which employed a very different wave-action approach.

Radiation of short waves from the resonantly excited capillary–gravity waves

2016 ◽  
Vol 810 ◽  
pp. 5-24 ◽  
Author(s):  
M. Hirata ◽  
S. Okino ◽  
H. Hanazaki
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Gravity Waves ◽  
Wave Pattern ◽  
Short Wave ◽  
Bond Number ◽  
Linear Wave ◽  
Cnoidal Wave ◽  
Short Waves ◽  
Long Wave

Capillary–gravity waves resonantly excited by an obstacle (Froude number: $Fr=1$) are investigated by the numerical solution of the Euler equations. The radiation of short waves from the long nonlinear waves is observed when the capillary effects are weak (Bond number: $Bo<1/3$). The upstream-advancing solitary wave radiates a short linear wave whose phase velocity is equal to the solitary waves and group velocity is faster than the solitary wave (soliton radiation). Therefore, the short wave is observed upstream of the foremost solitary wave. The downstream cnoidal wave also radiates a short wave which propagates upstream in the depression region between the obstacle and the cnoidal wave. The short wave interacts with the long wave above the obstacle, and generates a second short wave which propagates downstream. These generation processes will be repeated, and the number of wavenumber components in the depression region increases with time to generate a complicated wave pattern. The upstream soliton radiation can be predicted qualitatively by the fifth-order forced Korteweg–de Vries equation, but the equation overestimates the wavelength since it is based on a long-wave approximation. At a large Bond number of $Bo=2/3$, the wave pattern has the rotation symmetry against the pattern at $Bo=0$, and the depression solitary waves propagate downstream.

scholarly journals The suppression of short waves by a train of long waves

1996 ◽  
Vol 315 ◽  
pp. 139-150 ◽  
Author(s):  
A. M. Balk
Keyword(s):  
Diffusion Equation ◽  
Wave Field ◽  
Short Wave ◽  
Wave Action ◽  
Long Waves ◽  
Fourier Space ◽  
Short Waves ◽  
Wave Resonance ◽  
Resonance Interactions

It is shown that a train of long waves can suppress a short-wave field due to four-wave resonance interactions. These interactions lead to the diffusion (in Fourier space) of the wave action of the short-wave field, so that the wave action is transported to the regions of higher wavenumbers, where it dissipates more effectively. The diffusion equation is derived.

Infragravity waves induced by short-wave groups

1993 ◽  
Vol 247 ◽  
pp. 551-588 ◽  
Author(s):  
Hemming A. Schäffer
Keyword(s):  
Surf Zone ◽  
Break Point ◽  
Short Wave ◽  
Radiation Stress ◽  
Wave Groups ◽  
Infragravity Waves ◽  
Short Waves ◽  
Long Wave ◽  
Integrated Conservation ◽  
Infragravity Wave

A theoretical model for infragravity waves generated by incident short-wave groups is developed. Both normal and oblique short-wave incidence is considered. The depth-integrated conservation equations for mass and momentum averaged over a short-wave period are equivalent to the nonlinear shallow-water equations with a forcing term. In linearized form these equations combine to a second-order long-wave equation including forcing, and this is the equation we solve. The forcing term is expressed in terms of the short-wave radiation stress, and the modelling of these short waves in regard to their breaking and dynamic surf zone behaviour is essential. The model takes into account the time-varying position of the initial break point as well as a (partial) transmission of grouping into the surf zone. The former produces a dynamic set-up, while the latter is equivalent to the short-wave forcing that takes place outside the surf zone. These two effects have a mutual dependence which is modelled by a parameter K, and their relative strength is estimated. Before the waves break, the standard assumption of energy conservation leads to a variation of the radiation stress, which causes a bound, long wave, and the shoaling bottom results in a modification of the solution known for constant depth. The respective effects of this incident bound, long wave and of oscillations of the break-point position are shown to be of the same order of magnitude, and they oppose each other to some extent. The transfer of energy from the short waves to waves at infragravity frequencies is analysed using the depth-integrated conservation equation of energy. For the case of normally incident groups a semi-analytical steady-state solution for the infragravity wave motion is given for a plane beach and small primary-wave modulations. Examples of the resulting surface elevation as well as the corresponding particle velocity and mean infragravity-wave energy flux are presented. Also the sensitivity to the variation of input parameters is analysed. The model results are compared with laboratory experiments from the literature. The qualitative agreement is good, but quantitatively the model overestimates the infragravity wave activity. This can, in part, be attributed to the neglect of frictional effects.

A stochastic model of sea-surface roughness. II

1991 ◽  
Vol 435 (1894) ◽  
pp. 405-422 ◽  
Keyword(s):  
Wave Energy ◽  
Short Wave ◽  
Long Waves ◽  
Scale Model ◽  
Short Waves ◽  
Long Wave ◽  
Proportional Rate ◽  
The Mean ◽  
Sheltering Effect ◽  
Increasing Function

A two-scale model of a wind-ruffled surface is developed which includes (1) modulation of the short waves by orbital straining in the long waves, (2) dissipation of short-wave energy by breaking, and (3) regeneration of the short-wave energy by the wind. For simplicity the long waves are at first assumed to be uniform. It is shown that the character of the surface is governed by the parameter Ω = (β/σγKA ), where β is the proportional rate of short-wave growth due to the wind, σ , K and A are the long-wave frequency wavenumber and amplitude, and γ = 2.08. When Ω < 1 the short waves break over only part of the long-wave surface. When Ω ≽ 1 they break everywhere. The mean-square steepness s 2 ¯ of the short waves is an increasing function of β/σ , but a decreasing function of the long-wave steepness AK . The phase angle between s 2 ¯ and the long-wave elevation η is an increasing function of Ω . The correlation between s 2 ¯ and η is largest when Ω ≪1, but tends to 0 as Ω → 1. The simple model is extended to the case when the long-wave amplitude A has a Rayleigh probability density. To take account of the ‘sheltering ’ effect of high waves we compute the case when any two successive waves have a bivariate Rayleigh density. The application of the model to laboratory and field data is discussed.

Interaction of the variable-coefficient long-wave–short-wave resonance interaction equations

2019 ◽  
Vol 33 (01) ◽  
pp. 1850426
Author(s):  
Hui-Xian Jia ◽  
Da-Wei Zuo
Keyword(s):  
Gravity Waves ◽  
Acoustic Waves ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Short Wave ◽  
Resonance Interaction ◽  
Bilinear Method ◽  
Ion Acoustic Waves ◽  
Long Wave ◽  
Wave Resonance

Long-wave–short-wave resonance interaction (LSRI) equations have been studied in the plasmas, gravity waves, nonlinear electron-plasma and ion-acoustic waves. By virtue of the bilinear method, two soliton solutions of the variable-coefficient LSRI equations are attained. Interaction of the solitons are studied when the coefficients are taken as the generalized Gauss functions. New types of the soliton interaction are exhibited. Position and width of the disturbances can be controlled.

The stability of short waves on a straight vortex filament in a weak externally imposed strain field

1976 ◽  
Vol 73 (4) ◽  
pp. 721-733 ◽  
Author(s):  
Chon-Yin Tsai ◽  
Sheila E. Widnall
Keyword(s):  
Strain Field ◽  
Circular Cross Section ◽  
Short Wave ◽  
Vortex Filament ◽  
Linear Stability Theory ◽  
Short Waves ◽  
Constant Vorticity ◽  
Long Wave ◽  
Intersection Points ◽  
The Stability

The stability of short-wave displacement perturbations on a vortex filament of constant vorticity in a weak externally imposed strain field is considered. The circular cross-section of the vortex filament in this straining flow field becomes elliptical. It is found that instability of short waves on this strained vortex can occur only for wavelengths and frequencies at the intersection points of the dispersion curves for an isolated vortex. Numerical results show that the vortex is stable at some of these points and unstable at others. The vortex is unstable at wavelengths for which ω = 0, thus giving some support to the instability mechanism for the vortex ring proposed recently by Widnall, Bliss & Tsai (1974). The growth rate is calculated by linear stability theory. The previous work of Crow (1970) and Moore & Saffman (1971) dealing with long-wave instabilities is discussed as is the very recent work of Moore & Saffman (1975).

scholarly journals Rogue waves for a long wave–short wave resonance model with multiple short waves

2016 ◽  
Vol 85 (4) ◽  
pp. 2827-2841 ◽  
Author(s):  
Hiu Ning Chan ◽  
Edwin Ding ◽  
David Jacob Kedziora ◽  
Roger Grimshaw ◽  
Kwok Wing Chow
Keyword(s):  
Rogue Waves ◽  
Short Wave ◽  
Resonance Model ◽  
Short Waves ◽  
Long Wave ◽  
Wave Resonance

scholarly journals Spatial Form of a Hamiltonian Dysthe Equation for Deep-Water Gravity Waves

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 103
Author(s):  
Philippe Guyenne ◽  
Adilbek Kairzhan ◽  
Catherine Sulem ◽  
Boyang Xu
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Short Wave ◽  
Birkhoff Normal Form ◽  
Spatial Form ◽  
Periodic Groups ◽  
Dimensional Gravity ◽  
Nonlinear Modulation ◽  
Dysthe Equation

An overview of a Hamiltonian framework for the description of nonlinear modulation of surface water waves is presented. The main result is the derivation of a Hamiltonian version of Dysthe’s equation for two-dimensional gravity waves on deep water. The reduced problem is obtained via a Birkhoff normal form transformation which not only helps eliminate all non-resonant cubic terms but also yields a non-perturbative procedure for surface reconstruction. The free surface is reconstructed from the wave envelope by solving an inviscid Burgers’ equation with an initial condition given by the modulational Ansatz. Particular attention is paid to the spatial form of this model, which is simulated numerically and tested against laboratory experiments on periodic groups and short-wave packets. Satisfactory agreement is found in all these cases.

scholarly journals Satellite observations of interaction between tropics and mid-latitudes

MAUSAM ◽  
2021 ◽  
Vol 43 (1) ◽  
pp. 59-64
Author(s):  
S.R. KALSI ◽  
S. R. HALDER
Keyword(s):  
Satellite Data ◽  
Circulation Pattern ◽  
Short Wave ◽  
Satellite Observations ◽  
Low Latitude ◽  
Short Waves ◽  
Long Wave ◽  
Tropical Areas ◽  
Northwest India

In certain seasons and over certain locations, the mid-latitude westerlies invade subtropical and tropical areas. Short wave perturbations moving in the broad mid-latitude westerlies amplify the. long wave troughs creating new baroclinic zones in relatively southern latitudes. These. baroclinic zones Interact .with the low-latitude circulations thus leading to development of new circulation pattern .In which low level easterlies extend northward over the Peninsula, central and northwest .India. The paper describes the role of short waves in the interaction between tropics and mid-latitudes and presents satellite data of a few sequences In which such Interactions have actually taken place.

Laboratory measurements of modulation of short-wave slopes by long surface waves

1991 ◽  
Vol 233 ◽  
pp. 389-404 ◽  
Author(s):  
Sarah J. Miller ◽  
Omar H. Shemdin ◽  
Michael S. Longuet-Higgins
Keyword(s):  
Surface Waves ◽  
Gravity Waves ◽  
Wind Waves ◽  
Short Wave ◽  
Wave Interactions ◽  
Wind Speeds ◽  
Short Waves ◽  
Hydrodynamic Modulation ◽  
Work Done ◽  
Good Agreement

Hydrodynamic modulation of wind waves by long surface waves in a wave tank is investigated, at wind speeds ranging from 1.5 to 10 m s−1. The results are compared with the linear, non-dissipative, theory of Longuet-Higgins & Stewart (1960), which describes the modulation of a group of short gravity waves due to straining of the surface by currents produced by the orbital motions of the long wave, and work done against the radiation stresses of the short waves. In most cases the theory is in good agreement with the experimental results when the short waves are not too steep, and the rate of growth due to the wind is relatively small. At the higher wind speeds, the effects of wind-wave growth, dissipation and wave-wave interactions are dominant.

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