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 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.

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.

scholarly journals A UNIFIED RUNUP FORMULA FOR BREAKING SOLITARY WAVES ON A UNIFORM BEACH

2018 ◽  
pp. 3
Author(s):  
Yun-Ta Wu ◽  
Philip Li-Fan Liu ◽  
Philip Li-Fan Liu ◽  
Kao-Shu Hwang ◽  
Kao-Shu Hwang ◽  
...  
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Large Scale ◽  
Storm Surges ◽  
Research Area ◽  
Easy Access ◽  
Wave Runup ◽  
Wave Flume ◽  
Long Wave ◽  
Analytical Approaches

For coastal management, it is of great importance to understand long-wave induced runup processes and predict maximum runup heights. Long-wave in nature could take different forms, such as swells, storm surges and tsunamis. One of the fundamental waveforms is solitary wave, which has a permanent form in a constant depth. Thus, the issue of solitary wave propagation, shoaling, breaking and runup has been an active research area in coastal engineering community, using experimental, numerical and analytical approaches. Among existing runup experiments, only limited numbers of experiments were conducted in large-scale wave flume facilities because of the lack of easy access. To enhance the range of surf parameters for breaking solitary waves, new laboratory experiments were carried out in a large-scale wave flume with a 1/100 slope. Several wave conditions in the experiments were on the borderline of plunging and spilling breakers. The main objective of this paper is twofold. The first aim is to present a new dataset for solitary wave runup. The second objective aims to develop a unified empirical formula, based on the available runup data in the literature and the present new data, for the runup of breaking solitary waves on a uniform slope.

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.

Oblique runup of non-breaking solitary waves on an inclined plane

2011 ◽  
Vol 668 ◽  
pp. 582-606 ◽  
Author(s):  
GEIR K. PEDERSEN
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Evolution Equations ◽  
Boussinesq Equations ◽  
Normal Incidence ◽  
Finite Depth ◽  
Breaking Waves ◽  
Wave Pattern ◽  
Angle Of Incidence ◽  
Inclined Plane

When a wave of permanent form is obliquely incident on an inclined plane, the wave pattern becomes stationary in a frame of reference which moves along the shore. This enables a simplified mathematical description of the problem which is used herein as a basis for efficient and accurate numerical simulations. First, a nonlinear and weakly dispersive set of Boussinesq equations for the downstream evolution of such stationary patterns is derived. In the hydrostatic approximation, streamline-based Lagrangian versions of the evolution equations are developed for automatic tracing of the shoreline. Both equation sets are, in their present form, developed for non-breaking waves only. Finite difference models for both equation sets are designed. These methods are then coupled dynamically to obtain a single nonlinear model with dispersive wave propagation in finite depth and an accurate runup representation. The models are tested by runup of waves at normal incidence and comparison with a more general model for the refraction of a solitary wave on a slope. Finally, a set of runup computations for oblique solitary waves is performed and compared with estimates of oblique runup heights obtained from a combination of an analytic solution for normal incidence and optics. We find that the runup heights decrease in proportion to the square of the angle of incidence for angles up to 45°, for which the height is reduced by around 12% relative to that of normal incidence. In Appendix A, the validity of the downstream formulation is discussed in the light of solitary wave optics and wave jumps.

Integral properties of periodic gravity waves of finite amplitude

1975 ◽  
Vol 342 (1629) ◽  
pp. 157-174 ◽  
Keyword(s):  
Solitary Wave ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Wave Theory ◽  
Finite Amplitude ◽  
Cnoidal Wave ◽  
The Mean ◽  
Exact Relations ◽  
Kinetic And Potential Energies

A number of exact relations are proved for periodic water waves of finite amplitude in water of uniform depth. Thus in deep water the mean fluxes of mass, momentum and energy are shown to be equal to 2T(4T—3F) and (3T—2V) crespectively, where T and V denote the kinetic and potential energies and c is the phase velocity. Some parametric properties of the solitary wave are here generalized, and some particularly simple relations are proved for variations of the Lagrangian The integral properties of the wave are related to the constants Q, R and S which occur in cnoidal wave theory. The speed, momentum and energy of deep-water waves are calculated numerically by a method employing a new expansion parameter. With the aid of Padé approximants, convergence is obtained for waves having amplitudes up to and including the highest. For the highest wave, the computed speed and amplitude are in agreement with independent calculations by Yamada and Schwartz. At the same time the computations suggest that the speed and energy, for waves of a given length, are greatest when the height is less than the maximum. In this respect the present results tend to confirm previous computations on solitary waves.

scholarly journals AN ELECTROMAGNETIC ANALOGY FOR LONG WATER WAVES

1980 ◽  
Vol 1 (17) ◽  
pp. 48
Author(s):  
G.W. Jackson ◽  
D.L. Wilkinson
Keyword(s):  
Electromagnetic Radiation ◽  
Gravity Waves ◽  
Wave Energy ◽  
Water Waves ◽  
Wave Length ◽  
Alternative Methods ◽  
Linear Wave ◽  
Microwave Range ◽  
Nodal Lines ◽  
Long Wave

Seiching must be considered when designing mooring, berthing or navigational facilities in semi enclosed basins and harbours. The considerable oscillatory currents which may be generated along nodal lines of a seiche can result in serious surging of a vessel moored in such areas. Hydraulic modelling of the long gravity waves which form the seiche poses particular problems for the hydraulic modeller. In the prototype situation, a balance is achieved between the long wave energy entering a basin, the energy dissipation within the basin, and the wave energy which is radiated back into the ocean. The presence of the wave generator in the model may result in much of the radiated energy being reflected back into the basin, thereby distorting the seiche observed in the model. The relatively low Reynolds numbers present in the model leads to exaggerated frlctional damping of the waves and possible suppression of certain resonant modes. Scaling of long wave amplitudes in resonance' situations cannot be determined from simple Froude laws but must be based on equivalent dissipation rates in the model and the prototype (Ippen, 1966). Alternative methods of physically reproducing long wave behaviour have been studied, such- as acoustic modelling (Nakamura, 1977). However, these models also have their limitations and are largely confined to constant depth situations. It Is suggested that electromagnetic radiation in the microwave range may be used to model long gravity waves and that this method has many advantages over the various techniques which have previously been used. It can be rigorously shown that the laws which govern the propagation, reflection and dissipation of electromagnetic radiation (Maxwell's equations) are identical to the linearised equations describing the motion of long gravity waves, (Jackson and McKee, 1980). The linear wave equation gives an accurate description of long wave oscillation in basins as the wave length is of the same order as the basin dimensions and non-linear effects do not have time to develop.

On the existence of solitary waves in rotating fluids

1995 ◽  
Vol 125 (5) ◽  
pp. 1105-1129
Author(s):  
S. M. Sun
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Solitary Wave Solutions ◽  
Rotating Fluid ◽  
Rotating Fluids ◽  
First Order ◽  
Wave Solutions ◽  
Long Wave ◽  
Long Wave Approximation ◽  
Conditions At Infinity

This paper considers the existence of axisymmetric solitary waves in an inviscid and incompressible rotating fluid bounded by a rigid cylinder. It has been obtained by many experiments and formal derivations that this flow has internal solitary waves in the fluid when equilibrium state at infinity satisfies certain conditions. This paper gives a rigorous proof of the existence of solitary wave solutions for the exact equations governing the flow under such conditions at infinity, and shows that the first-order approximations of the solitary wave solutions for the exact equations are solitary wave solutions derived formally using long-wave approximation. The ideas in the proof of the existence of solitary waves in two-dimensional stratified fluids are used and a main difficulty from the singularity at axis of rotation is overcome.

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.

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