Short-wave Envelope Research Articles

LONG WAVES FORCED BY SYMMETRIC AND ASYMMETRIC WAVE GROUPS

Two solutions have been proposed for forced waves by wave groups. Two interpretations are proposed by Longuet-Higgins and Stewart (1962), LHS1962, and Mei (1989) for steady groups. These indicate a forced wave oscillating either around or solely below the mean water level for the short waves, respectively. The existence of forced waves as either pure depressions or as oscillations about the still water level has significant implications in terms of shoaling behaviour and the generation of free long waves. With a bound wave with both positive and negative components then shoaling (an increase in amplitude in shallower water) does not necessarily require radiation of free waves to ensure continuity of volume, but with a bound wave of pure depression, then an increase in amplitude of that depression requires radiation of free waves to ensure continuity (Nagase and Mizuguchi, 1996, Nielsen and Baldock, 2010). Radiation of those free waves then changes the shape of the total infragravity wave during shoaling and may also influence the commonly observed lag between incident bound waves and the incoming short-wave envelope. Here, we show that pure long wave depression can be generated by Gaussian wave groups, and that asymmetric forced long waves, with a leading crest and following depression, are generated by triangular / Gaussian groups.

Near‐Bed Sediment Transport During Offshore Bar Migration in Large‐Scale Experiments

AbstractThis study presents novel insights into hydrodynamics and sediment fluxes in large‐scale laboratory experiments with bichromatic wave groups on a relatively steep initial beach slope (1:15). An Acoustic Concentration and Velocity Profiler provided detailed information of velocity and sand concentration near the bed from shoaling up to the outer breaking zone including suspended sediment and sheet flow transport. The morphological evolution was characterized by offshore migration of the outer breaker bar. Decomposition of the total net transport revealed a balance of onshore‐directed, short wave‐related and offshore‐directed, current‐related net transport. The short wave‐related transport mainly occurred as bedload over small vertical extents. It was linked to characteristic intrawave sheet flow layer expansions during short wave crests. The current‐related transport rate featured lower maximum flux magnitudes but occurred over larger vertical extents. As a result, it was larger than the short wave‐related transport rate in all but one cross‐shore position, driving the bar’s offshore migration. Net flux magnitudes of the infragravity component were comparatively low but played a nonnegligible role for total net transport rate in certain cross‐shore positions. Net infragravity flux profiles sometimes featured opposing directions over the vertical. The fluxes were linked to a standing infragravity wave pattern and to the correlation of the short wave envelope, controlling suspension, with the infragravity wave velocity.

Estimation of Irregular Wave Runup on Intermediate and Reflective Beaches Using a Phase-Resolving Numerical Model

Accurate wave runup estimations are of great interest for coastal risk assessment and engineering design. Phase-resolving depth-integrated numerical models offer a promising alternative to commonly used empirical formulae at relatively low computational cost. Several operational models are currently freely available and have been extensively used in recent years for the computation of nearshore wave transformations and runup. However, recommendations for best practices on how to correctly utilize these models in computations of runup processes are still sparse. In this work, the Boussinesq-type model BOSZ is applied to calculate runup from irregular waves on intermediate and reflective beaches. The results are compared to an extensive laboratory data set of LiDAR measurements from wave transformation and shoreline elevation oscillations. The physical processes within the surf and swash zones such as the transfer from gravity to infragravity energy and dissipation are accurately accounted for. In addition, time series of the shoreline oscillations are well captured by the model. Comparisons of statistical values such as R2% show relative errors of less than 6%. The sensitivity of the results to various model parameters is investigated to allow for recommendations of best practices for modeling runup with phase-resolving depth-integrated models. While the breaking index is not found to be a key parameter for the examined cases, the grid size and the threshold depth, at which the runup is computed, are found to have significant influence on the results. The use of a time series, which includes both amplitude and phase information, is required for an accurate modeling of swash processes, as shown by computations with different sets of random waves, displaying a high variability and decreasing the agreement between the experiment and the model results substantially. The infragravity swash SIG is found to be sensitive to the initial phase distribution, likely because it is related to the short wave envelope.

Interaction of complex short wave envelope and real long wave described by the coupled Schrödinger–Boussinesq equation with variable coefficients and beta space fractional evolution

This work deals with the interaction of complex short wave envelope and real long wave described by the coupled Schrödinger–Boussinesq equation with variable-coefficients and beta space fractional evolution. The above interacting wave phenomena are studied by evaluating new non-autonomous analytical solutions of the considered coupled equation. The solutions of the equations are obtained by modifying the auxiliary ordinary differential equation method and taking the conformable beta derivative properties into account. The behaviors of the resonance nonautonomous structures are discussed with physical insight for diverse consideration of the chance functions of time-dependent in the solutions.

Plasmon dromions in a metamaterial via plasmon-induced transparency

We propose a scheme to realize a giant Kerr nonlinearity and create stable high-dimensional nonlinear plasmon polaritons via plasmon-induced transparency (PIT) in a metamaterial, which is constructed by an array of unit cell consisting of a cut-wire and a pair of varactor-loaded split-ring resonators. We show that, due to the PIT effect and the nonlinearity contributed by the varactor, the system may possess very large second-order and third-order nonlinear susceptibilities. We further show that the system supports a resonant interaction between longwave and shortwave and hence effective third-order nonlinear susceptibility can be further enhanced one order of magnitude. Based on these peculiar properties, we derive Davey-Stewartson equations governing the evolution of longwave and shortwave envelope, and demonstrate that it possible to generate plasmon dromions [i.e., (2+1)-dimensional plasmon solitons with coupled longwave and shortwave components] with very low generation power. Our study raises the possibility for obtaining new, giant Kerr effect and stable high-dimensional nonlinear plasmon polaritons at very low radiation intensity by using nonlinear PIT metamaterials.

NUMERICAL MODELLING OF HYDRODYNAMICS AND SEDIMENT TRANSPORT IN THE SURF ZONE : A SENSITIVITY STUDY WITH DIFFERENT TYPES OF NUMERICAL MODELS

A comparison between two very different numerical models is presented: Delft3D and XBeach. Delft3D (Deltares) calculates non-steady flow and transport phenomena that result from tidal and meteorological forcing. The wave propagation is calculated in the frequency domain. XBeach (Unesco-IHE, Delft University and Deltares) consists of formulations for short wave envelope propagation (time-dependent wave action balance), non-stationary shallow water equations, sediment transport and bed update. The model is able to resolve the time dependent long waves, which are important in the surf zone. A number of simplified cases are defined beforehand taking into account actual features and conditions existing in chosen problem areas. The examination of these simplified cases allows for the identification of driving processes and the assessment of the sensitivity to certain relevant parameters, with the advantage of working in scenarios of limited complexity and without excessive computational load.

Infra-gravity wave generation by the shoaling wave groups over beaches

A physical parameter, µb, which was used to meet the forcing of primary short waves to be off-resonant before wave breaking, has been considered as an applicable parameter in the infra-gravity wave generation. Since a series of modulating wave groups for different wave conditions are performed to proceed with the resonant mechanism of infra-gravity waves prior to wave breaking, the amplitude growth of incident bound long wave is assumed to be simply controlled by the normalized bed slope, βb. The results appear a large dependence of the growth rate, α, of incident bound long wave, separated by the three-array method, on the normalized bed slope, βb. High spatial resolution of wave records enables identification of the cross-correlation between squared short-wave envelopes and infra-gravity waves. The cross-shore structure of infra-gravity waves over beaches presents the mechanics of incident bound- and outgoing free long waves with the formation of free standing long waves in the nearshore region. The wave run-up and amplification of infra-gravity waves in the swash zone appear that the additional long waves generated by the breaking process would modify the cross-shore structure of free standing long waves. Finally, this paper would further discuss the contribution of long wave breaking and bottom friction to the energy dissipation of infra-gravity waves based on different slope conditions.

Experimental and numerical study of the effect of variable bathymetry on the slow-drift wave response of floating bodies

An experimental campaign is reported on the slow-drift motion of a rectangular barge moored at different positions along an inclined beach, at waterdepths ranging from 54 cm to 21 cm, and submitted to irregular beam seas. The beach is achieved by inclining the 24 m long false bottom of the tank at a slope of 5%, from a depth of 1.05 m. The slow-drift component of the measured sway motion is first compared with state-of-the-art calculations based on Newman’s approximation. At 54 cm depth a good agreement is obtained between calculations and measurements. At 21 cm depth the Newman calculations exceed the measured values. When the flat bottom setdown contribution is added up, the calculated values become 2 to 3 times larger than the measured ones. A second-order model is proposed to predict the shoaling of a bichromatic sea-state propagating in varying water-depth. This model is validated through comparisons with an extension of Schäffer’s model for a straight beach [Schäffer HA. Infragravity waves induced by short-wave groups. J Fluid Mech 1993;247:551–88] and with a fully nonlinear Boussinesq model. It appears that the long wave amplitude is much less than predicted by the flat bottom model, and that its phase difference with the short wave envelope also deviates from the flat bottom model prediction. As a result of this phase shift the actual second-order wave loads can be lower than predicted by Newman’s approximation alone. Application of the shoaling model to the barge tests yields a notably better agreement between numerical and experimental values of its slow-drift sway motion.

И-Shaped surf beat understood in terms of transient forced long waves

The observed И-shape of the long wave generated by a single short wave pulse is explained qualitatively by including the free waves which are generated along with the well known bowl-shaped forced wave. Free waves of complicated shapes are generally formed as the wave group encounters gradual depth changes. However, the main qualitative feature, which has so far lacked a simple explanation, namely the long wave being И-shaped rather than a single bowl, can be explained through the simple free wave configuration which results from an abrupt onset of forcing at constant depth. И-shaped transient stages then occur between the initial flat surface and the final stage where free and forced waves have become completely separated from the forcing due to their different speeds. The И-shape is however also the shape of the gradually growing, resonant long wave due to a single short wave pulse, which travels with phase speed c and group velocity c g close to the long wave speed g h . The latter result is derived and compared with detailed laboratory experiments. It may explain the lack of observed H 2-dependence observed for some incident long waves, as well as the qualitative shapes of the long wave and of the cross correlation function between short wave envelope and long wave elevation.

Numerical simulation of “limiting” envelope solitons of gravity waves on deep water

The results of numerical simulation of “limiting” envelope solitons of gravity waves on deep water (i.e., long-lived nonlinear groups including waves close to breaking) are reported. The existence of such quasi-soliton structures was demonstrated by Dyachenko and Zakharov [JETP Let. 88(5), 307 (2008)]. Solitary propagation and various types of interaction of limiting envelope solitons are considered with the help of numerical solution of the equations of ideal potential hydrodynamics in conformal variables. The results are compared with the description based on the generalized weakly nonlinear envelope equation (modified Dysthe model). It is shown that the initial conditions in the form of exact solutions to the nonlinear Schrodinger equation taking into account asymptotic corrections of the three orders corresponding to bound waves correctly describe limiting envelope solitons. The effects associated with the strongly nonlinear envelope soliton dynamics (instability of overly steep groups, short-wave envelope soliton destruction by a longwave group, and formation of coupled groups of waves) are revealed.

Modeling and Statistical Properties of Long Period Waves in the Nearshore Zone

Long period waves are sometimes significantly large and cannot be neglected even for engineering purposes in the nearshore zone. The growth of amplitude of long waves and phase shift between short wave envelope and incoming long waves on sloping bottom have been studied but not clearly understood. We present a theoretical model for the long wave evolution on the basis of the linear long wave equation with forced term and the model of breakpoint forced long waves. The present model shows good agreement with experiment results of random wave field. Then this model was applied to field observation data, and we show that the amplitude of long waves inner surf zone could express with squared of significant wave height.

Long wave generation by the shoaling and breaking of transient wave groups on a beach

This paper presents new laboratory data on the generation of long waves by the shoaling and breaking of transient-focused short-wave groups. Direct offshore radiation of long waves from the breakpoint is shown experimentally for the first time. High spatial resolution enables identification of the relationship between the spatial gradients of the short-wave envelope and the long-wave surface. This relationship is consistent with radiation stress theory even well inside the surf zone and appears as a result of the strong nonlinear forcing associated with the transient group. In shallow water, the change in depth across the group leads to asymmetry in the forcing which generates significant dynamic setup in front of the group during shoaling. Strong amplification of the incident dynamic setup occurs after short-wave breaking. The data show the radiation of a transient long wave dominated by a pulse of positive elevation, preceded and followed by weaker trailing waves with negative elevation. The instantaneous cross-shore structure of the long wave shows the mechanics of the reflection process and the formation of a transient node in the inner surf zone. The wave run-up and relative amplitude of the radiated and incident long waves suggests significant modification of the incident bound wave in the inner surf zone and the dominance of long waves generated by the breaking process. It is proposed that these conditions occur when the primary short waves and bound wave are not shallow water waves at the breakpoint. A simple criterion is given to determine these conditions, which generally occur for the important case of storm waves.

Two-dimensional mechanisms of interaction between ultrasound and sound in bubbly liquids: Interaction equations

The interaction of long (sound) and short (ultrasound) waves propagating in a rarefied monodisperse mixture of a weakly compressible liquid with gas bubbles is considered. Using the multiscale method, the Davey-Stewartson system of equations is derived as a model of two-dimensional interaction. It is shown that, for some values of parameters, this system is reduced to an integrable form (the Davey-Stewartson I equations) and has localized solutions in the form of dromions (exponentially decaying waves of the short-wave envelope). One of the most important properties of dromions is their ability to move according to the law that governs the variations of the boundary conditions set at infinity for the long wave. It is suggested that these solutions be used for controlling the effects of ultrasound on bubbly liquids.

Long-wave-short-wave interaction in bubbly liquids

The interaction of long and short waves in a rarefied monodisperse mixture of a weakly compressible liquid containing bubbles of gas is considered. It is shown that the equations describing the dynamics of the perturbations in the bubbly liquid admit of the existence of short-wave-long-wave Benney-Zakharov resonance. A special modification of the multiple-scale method is employed to derive the interaction equations. In the non-resonant case, the interaction equations reduce to the non-linear Schrödinger equation in the form of the short-wave envelope while, in the resonance case, they reduce to the well-known system of Zakharov equations. The characteristics of long-wave-short-wave interaction in a bubbly liquid lie in the fact that, at certain values of the frequency of the short wave, the interaction coefficients vanish (“interaction degeneracy”). A class of new interaction models is constructed in the case of “degeneracy”. Degenerate resonance interaction in a bubbly liquid is investigated numerically using these models.

Bar formation due to wave groups and associated long waves

Field observations reveal that waves propagating onto a beach from deeper water have a grouped structure. Associated with this pattern of alternating high and low waves is a long wave forced by radiation stress variations due to the changing short wave height. As the wave groups propagate onshore, the relationship between the variation in the short wave height and the long wave motion is complicated by breaking of the short waves, reflection of the long wave and the generation of a free long wave by time variation of the breakpoint position. Nevertheless, there may still be a fixed phase relation between the short-wave envelope and the long wave motion at any particular location, resulting in a net transport of sediment over the timescale of the short wave groups. This transport arises from sediment grains mobilised by short waves and moved by the long wave motion. Model predictions of wave group propagation onto a beach and the associated long wave motions are presented which suggest that sediment transport due to short-wave/long-wave coupling may result in the formation of a bar within the inner surf-zone and the possibility of additional bars further offshore, outside the surf-zone. This mechanism for bar formation may provide a new framework for understanding bar morphodynamics.

Steady Trapped Solutions to Forced Long-Short Interaction Equation

In a two-layer fluid system, both wave generation by bottom topography and resonant interaction between two wave modes are examined simultaneously. An evolution equation is derived which describes not only the resonant interaction between long internal mode and the wave packet of short surface mode, but also the generation of internal mode due to the resonant motion of the fluid relative to a bottom unevenness of large horizontal scale. According to this equation, steady coupled waves trapped by a localized symmetric bottom unevenness of sech 2 -type are computed numerically both for positive and negative unevenness. Consequently, in addition to symmetric waves with one or two peaks in short-wave envelope, a variety of waves with many peaks, antisymmetric waves, and asymmetric waves are found.

Bragg scattering of surface waves by periodic bars: theory and experiment

We extend a recent linearized theory on Bragg scattering of surface waves by periodic sandbars to include second-order effects of the free surface and of the bars. New experiments are performed to verify the existence of the cutoff detuning frequency, the dispersive nature of the first-order wave envelope, and the radiation of second-order long waves. Measured transient and quasi-steady responses to incident wave packets and uniform wavetrains are compared with corresponding theoretical results. For quasi-steady incident waves of relatively small steepness it is found necessary to improve the theory to the second order in bar slope, in order that the calculated short-wave envelopes agree with those measured over the bars.

The resonant interaction among long and short waves

The resonant interaction among long and short waves is analyzed in this paper. The problem considered here consistent of two short wave envelopes with the same group speed resonating with a long wave. The phase speed of the long wave matches the group speed of the short waves. The problem is formulated by using perturbation expansions and stretched coordinates. The inverse scattering technique is applied to solve the evolution equations under some special conditions.