Wave Breaking Point Research Articles

Wave breaking, dispersive shock wave, and phase shift for the defocusing complex modified KdV equation.

We study the problem of wave breaking for a simple wave propagating to a quiescent medium in the framework of the defocusing complex modified KdV (cmKdV) equation. It is assumed that a cubic root singularity is formed at the wave-breaking point. The dispersive regularization of wave breaking leads to the generation of a dispersive shock wave (DSW). We describe the DSW as a modulated periodic wave in the framework of the Gurevich-Pitaevskii approach based on the Whitham modulation theory. The generalized hodograph method is used to solve the Whitham equations, and the boundaries of the DSW are found. Most importantly, we determine the correct phase shift for the DSW from the generalized phase relationships and the modified Gurevich-Pitaevskii matching conditions, so that a complete description of the DSW is obtained rather than just its envelope. All of our analytical predictions agree well with the numerical simulations.

Solitary wave-induced boulder transport on a beach

The influence of sediment on a slope on boulder transport is studied experimentally at the Hydrodynamics Laboratory of the University of Oslo. Experiments are conducted in a small 3 m long and 10 cm wide wave tank, filled with 5 cm of water. The transport of boulders is induced by breaking solitary waves with an amplitude normalized by water depth a/hw= 0.5. Generated solitary waves propagate towards a 1:10 beach. The experiments are conducted in two set-ups: (i) empty Polymethyl methacrylate (PMMA) bottom of the flume, (ii) the slope covered by a thin layer of 65μm sand. The boulders are represented by concrete blocks of different shape and size. They are placed alternately (one at a time) at different locations on the beach slope with respect to wave breaking point. Transport of boulders and their dynamics is studied with respect to boulder characteristics (size, orientation), their initial position and inclusion of sediment. It is shown that presence of sediment enhances boulder transport. In particular, in this set-up, the presence of sediment increased the boulder transport in 2–5 times. The maximum displacement increase was observed for boulders with the smallest length and height and the largest width initially located at the breaking position. Another interesting result regards the type of boulder motion. The boulders experienced either sliding or turning over. Boulders whose height was at least twice as large as their length exhibited turning-over. This held for boulders placed both on an empty PMMA slope and on a sedimentary slope. However, the largest boulder displacement on an empty PMMA slope occurred due to turning over, while on a sedimentary slope it occurred due to sliding.

Effect of open-area ratios on wave attenuation over a submerged reef breakwater: An experimental study

The reef-type submerged breakwater (RSB) has been implemented around the world to provide coastal protection and natural shelter and habitat for marine species at the same time. A series of laboratory experiments are conducted in a wave flume to study wave propagation and transmission over RSBs with the four open-area ratios of 0.047, 0.131, 0.257, and 0.560. The experimental results demonstrate that the RSB significantly alters the wave propagation characteristics. The RSBs with the open-area ratio of 0.047 and 0.131 reduce the wave height to ∼70%. The RSBs with the open-area ratio of 0.047 and 0.131 perform better in attenuating than those with the open-area ratio of 0.257, and 0.560. As the open-area ratio decreases, the transmission coefficient decreases, while the dissipation coefficient increases. An empirical formula is proposed to calculate the wave transmission coefficient as a function of the RSB open-area ratio and wave steepness under regular waves. A cubic empirical formula with one unknown is also proposed to relate the distance from RSB stoss face to wave breaking point and the open-area ratio. The present study is expected to optimize RSB design for coastal protection.

Non-intrusive measurements of wave nonlinearity over movable bed of sediment with different beach slopes

A series of laboratory experiments were undertaken to qualitatively investigate the evolution of wave nonlinearity over a movable bed of sediment with five different beach slopes under regular waves in a medium-scale wave flume. An innovative non-intrusive data collecting system, which mainly consists of three side-looking high-speed cameras, was developed to collect high-resolution and synchronous data on free-surface water elevation of waves and bed level changes without causing any disturbances to wave motions and the movable bed of sediment. On analyzing the collected experimental data, it is found that regular waves become nonlinear when they propagate to the shoaling zone and start breaking, and linear wave theory is quite accurate for calculating wave parameters such as orbital velocity with the correlation coefficient r2 = 0.8–0.95 before the waves break, but becomes less accurate after the waves break or are in the breaking zone with the smaller correlation coefficient r2 = 0.4–0.6. Four parameters, wave skewness Skη, asymmetry Ayη, Ursell number Ur, and Rocha number NP0, are introduced to describe the wave nonlinearity, of which Skη and Ayη are found to be of largest amplitudes at the wave breaking point and then start to decrease in the breaking zone and are almost unchanged for different beach slopes, while Ur further increases in the breaking zone and exceed the first larger value as waves approach to shoreline, but NP0 is almost linearly proportional to wave orbital velocity amplitude and quite sensitive to beach slope. The location of sandbar is found close to the wave breaking point in the wave flume and may be also considered as the point where wave nonlinearity becomes important for sediment transport in the surf zone, and the linear wave theory becomes less accurate.

DESIGN SCOUR LEVELS FOR DUNE REVETMENTS AND SEAWALLS

Particularly important for the design of dune revetments and seawalls subjected to breaking waves is the maximum depth of toe scour, the primary cause of failure of many coastal structures (Sutherland et al. 2006). Much of the published research on wave-induced toe scour has been under non-breaking waves with subaqueous seabed levels at the seawall/revetment toe. However, dune revetments and seawalls may become exposed to breaking waves for which this method has been derived. The method proposed herein assumes that the work done to excavate a scour hole is a function of the incident wave energy (Steetzel 1993), which incorporates wave period rather than wave height alone, and a formula for the toe scour level has been developed for a still water level datum at the wave breaking point. The formula has been calibrated with data derived from published laboratory studies covering a large range of scales, with some having been validated with prototype measurements.

Valuing beaches for beauty and recreation only? Uncovering perception bias through a hashtag analysis

Sandy beaches are ecosystems that hold great economic value at local, national, and international level. However, their perception as social-ecological systems is often limited by users and managers to the recreational asset, and their perception of “beauty”. Such a perception bias can hamper conservation management, negatively impact the biodiversity on sandy beaches and even threaten the ecosystem services they provide. To survey this perception bias, we applied the method of netnography to analyse Instagram posts with the hashtag #sandybeach, extracting perceptions and values for nature (intrinsic, instrumental, and relational) towards sandy beaches and their ecological features. We used the concept of littoral active zone (LAZ) to define the boundaries of this social-ecological system that spans the land and the sea. Our results show that Instagram users focus on their social media mainly on the recreational values of sandy beaches and prioritise ecological features fitting into an imaginary of a “pristine” beach, characterised by white sand and clear water – therewith supporting the hypothesis about a perception bias on sandy beaches. We identified a diverse set of relational values, especially around recreation and aesthetics, with a very marginal number of posts mentioning intrinsic or instrumental values. A recurring activity on LAZ was surfing, which emphasises the LAZ (wave break point, surf zone). To conserve the biodiversity and ecosystem services of sandy beaches, we see it as necessary to move away from a biased and narrow perception to one that, based on the LAZ as a system, is capable of including a diversity of biogeographical and ecological features, and the natural processes that maintain them.

The physical processes of sandy beach evolution under storm and non-storm wave conditions simulated in wave flume

Large waves and high water levels generated by coastal storms are predominant drivers for coastal beach erosion. This experimental study is undertaken to investigate the physical processes of beach profile evolution under storm waves by simulating step varying storm waves and variable water levels in a wave flume of 30 m long, 0.6 m wide and 1 m high and mild beach slope tanθ=1/10. The storm waves were simulated by following a conceptual storm hydrograph of three stages: at the first stage of 60 min both wave height and water level were gradually increased to peak storm wave height and water level, at the second stage of 60 min both maximum wave height and water level were kept for constant, and at the last stage of 60 min storm wave heights were gradually decreased to non-storm waves and water levels. Non-intrusive high-resolution cameras were used to record time-series of free surface water elevation, wave breaking point position, sandbar migration and location, and beach elevation profile in a spatial resolution of 1.4 mm and a temporal resolution of 0.1 s. In analyzing the collected experimental data, the physical processes of beach evolution under both non-storm and storm waves were found significantly different: under non-storm waves, the movement of sandbar always was found to be in the seaward direction, and the beach profile was eroded and accreted before and after the sandbar, respectively, but under storm waves, the sandbar location was found to migrate in either the landward or seaward direction, the beach profile was slowly accreted before the sandbar and eroded after the sandbar when the storm waves were gradually decreased to the non-storm waves. The surf zone is the main change area in beach morphology modification, and the sandbar is the important indicator for distinguishing the cross-shore sediment exchange. Under both non-storm and storm waves, large waves, high water levels, and especially great gradient of wave energy flux due to changes in wave height, period and water depth are the main factors controlling the formation and migration of sandbar, and resulting in beach erosion and accretion before and after the wave breaking point, respectively.

An experimental investigation into the evolving instability of a subaqueous mild silty slope under progressive waves

The silty sediments that are widely distributed in estuaries and deltas are susceptible to slope instability due to ocean storms. Flume experiments were carried out in this study to investigate the characteristics and failure process of a subaqueous mild silty slope subjected to ocean waves. The results show that the wave steepness varies accordingly to wave propagation over the mild silty slope due to shoaling effect. Where the wave steepness is up to the threshold value of 0.04, liquefaction may occur in silty slopes due to the substantial amount of residual pore water pressure development. In this process, the maximum wave pressure appears near the wave breaking point where it may be increased by about 40% compared with that at bottom of the slope; this means that liquefaction may primarily first occur in regions near the wave breaking point and become the predominant trigger for the overall slope instability. The maximum liquefaction and sliding depth of silty seabed may reach 0.1–0.15 times the wavelength for wave steepness of 0.04–0.10. Unlike the failures of a loose sandy slope that are characterized by a slumping stage along with a sudden rise in pore water pressure, wave induced failure of the mild silty slope progresses in three distinct stages, i.e., (i) initial erosion and scouring, (ii) coupled development of liquefaction and scour, and (iii) oscillating and sliding of sediments.

A Wave-Targeted Essentially Non-Oscillatory 3D Shock-Capturing Scheme for Breaking Wave Simulation

A new three-dimensional high-order shock-capturing model for the numerical simulation of breaking waves is proposed. The proposed model is based on an integral contravariant form of the Navier–Stokes equations in a time-dependent generalized curvilinear coordinate system. Such an integral contravariant form of the equations of motion is numerically integrated by a new conservative numerical scheme that is based on three elements of originality: the time evolution of the state of the system is carried out using a predictor–corrector method in which exclusively the conserved variables are used; the point values of the conserved variables on the cell face of the computational grid are obtained using an original high-order reconstruction procedure called a wave-targeted essentially non-oscillatory scheme; the time evolution of the discontinuity on the cell faces is calculated using an exact Riemann solver. The proposed model is validated by numerically reproducing several experimental tests of breaking waves on computational grids that are significantly coarser than those used in the literature to validate the existing 3D shock-capturing models. The results obtained with the proposed model are also compared with those obtained with a previously published model, which is based on second-order total variation diminishing reconstructions and an approximate Riemann solver usually adopted in the existing 3D shock-capturing models. Through the above comparison, the main drawbacks of the existing 3D shock-capturing models and the ability of the proposed model to simulate breaking waves and wave-induced currents are shown. The proposed 3D model is able to correctly simulate the wave height increase in the shoaling zone and to effectively predict the location of the wave breaking point, the maximum wave height, and the wave height decay in the surf zone. The validated model is applied to the simulation of the interaction between breaking waves and an emerged breakwater. The numerical results show that the proposed model is able to simulate both the large-scale circulation patterns downstream of the barrier and the onset of quasi-periodic vortex structures close to the edge of the barrier.

Characteristics of Breaking Wave Forces on Piles over a Permeable Seabed

Most offshore wind turbines are installed in shallow water and exposed to breaking waves. Previous numerical studies focusing on breaking wave forces generally ignored the seabed permeability. In this paper, a numerical model based on Volume-Averaged Reynolds Averaged Navier–Stokes equations (VARANS) is employed to reveal the process of a solitary wave interacting with a rigid pile over a permeable slope. Through applying the Forchheimer saturated drag equation, effects of seabed permeability on fluid motions are simulated. The reliability of the present model is verified by comparisons between experimentally obtained data and the numerical results. Further, 190 cases are simulated and the effects of different parameters on breaking wave forces on the pile are studied systematically. Results indicate that over a permeable seabed, the maximum breaking wave forces can occur not only when waves break just before the pile, but also when a “secondary wave wall” slams against the pile, after wave breaking. With the initial wave height increasing, breaking wave forces will increase, but the growth can decrease as the slope angle and permeability increase. For inclined piles around the wave breaking point, the maximum breaking wave force usually occurs with an inclination angle of α = −22.5° or 0°.

Laboratory and numerical investigations on characteristics of air bubbles in plunging breakers on beach

We present an experimental-numerical study on air entrainment and the statistics of air bubbles in plunging breaking waves in surf zones on a beach. In the experimental investigation, the evolution, numbers, and sizes of air bubbles in breaking waves are recorded. The relations are discussed between microscopic behaviors of these air bubbles and influential macrocopic factors including distance to the wave-breaking point, wave height, water depth, and depth of measurement. Empirical expressions for bubble sizes are formulated as functions of these factors. Base on three-dimensional, high-resolution, air-bubble-resolving numerical modeling, a further study is conducted on details of the fields of velocity, vorticity, and turbulence kinetic energy. It is conjectured with evidence that the number of bubbles is linearly correlated to turbulence kinetic energy within breaking waves. The multiscale physical phenomena and correlations between air bubbles behaviors and their influential factors revealed in this study shed light on understanding and quantifying air bubbles characteristics in breaking waves within surf zones.

Laboratory investigation of coastal beach erosion processes under storm waves of slowly varying height

Storm-induced large coastal waves are a predominant driver for coastal erosion problems, but the physical processes of storm erosion have not been fully understood due to some technical difficulty in simulating storm waves with timely varying heights in the laboratory and collecting storm beach profiles in the field. In this study, novel series of experiments are proposed to study sandy beach processes under coastal storm conditions in a laboratory wave flume of 60 m long, 3 m wide and 1.5 m high. Two water depths of h = 0.5 m and h = 0.7 m were used to investigate beach response under low and high tidal levels. Two initially uniform plane beach slopes of tanβ=1/15 for dissipation beach and tanβ=1/5 for reflection beach were used to study beach profile changes. Additionally, intermediate beach tanβ=1/10 was also experimentally investigated under regular waves in another wave flume of 30 m long, 0.6 m wide and 1 m high. Non-intrusive instrumentation of high-resolution cameras was used to record time-series photo images on beach profile evolution in spatial resolution of 0.2 cm and a temporal resolution of 0.1 s. Image analyzing techniques were also developed to transform the recorded photo images to digital data on both beach and water-surface profiles. In analyzing the collected data, it is found that the variation of storm wave height results in changes of wave breaking point and also sediment transport direction before and after the wave breaking point. Sediment transport direction was found to move onshore initially under small storm waves, and then a sand-bar was generated and a certain portion of sediment also moved offshore under large waves, but such monotonic relationship between sediment transport and wave forcing could transform as morphology feedback and phase lags between wave orbital fluid velocity and sediment concentration. The wave energy flux at the breaking point is shown to be suitable parameter in predicting beach mass change or storm erosion rate, while the dimensionless sediment settling velocity parameter Ω′ may not be a suitable criterion for predicting beach erosion or accretion under storm waves.

In-situ Observations of Infragravity Response during Extreme Storms on Sand and Gravel Beaches

Billson, O.J.; Russell, P.; Davidson, M.; Poate, T.; Amoudry, L.O., and Williams, M.E., 2020. In-situ observations of infragravity response during extreme storms on sand and gravel beaches. Global Coastal Issues of 2020. Journal of Coastal Research, Special Issue No. 95, pp. 382–386. Coconut Creek (Florida), ISSN 0749-0208.Infragravity waves (frequency = 0.005 – 0.05 Hz) play a key role in coastal storm impacts such as flooding and beach/dune erosion. They are known to dominate the inner surf zone on low-sloping sandy beaches during storms. However, in large wave conditions, their importance on different beach types, of variable swell and wind-waves dominance, is largely unknown. Here, a new dataset is presented comprising in-situ observations during storm wave conditions (significant wave height of 3.3 m, peak periods of 18 s and return periods up to 1 in 60 years) from two contrasting sites: a low-sloping sandy beach and a steep gravel beach. Wave measurements were collected seaward of the breakpoint by wave buoys and bed-mounted acoustic Doppler current profilers, and through the surf zone using arrays of pressure transducers. Wave spectra showed contrasting evolution from the shoaling zone to the inner surf zone at the two sites. At the sandy beach, gravity band energy dissipated gradually as depth reduced, while infragravity band energy simultaneously increased, resulting in strongly infragravity-dominated wave spectra in the inner surf zone. At the steep gravel site, a rapid drop in short wave energy was observed, with limited growth of infragravity energy so that inner surf zone spectra showed a low energy peak in the infragravity band. The normalized bed slope parameter indicated whether infragravity waves were generated by bound long wave release or breakpoint forcing, showing that the former (latter) was dominant on the sandy (gravel) beach. In spite of these differences, the shoreline wave spectra under storm wave conditions were infragravity-dominated on both the sandy and gravel beaches.

Numerical Simulations of Non-Breaking, Breaking and Broken Wave Interaction with Emerged Vegetation Using Navier-Stokes Equations

Coastal plants can significantly dissipate water wave energy and services as a part of shoreline protection. Using plants as a natural buffer from wave impacts remains an attractive possibility. In this paper, we present a numerical investigation on the effects of the emerged vegetation on non-breaking, breaking and broken wave propagation through vegetation over flat and sloping beds using the Reynolds-average Navier-Stokes (RANS) equations coupled with a volume of fluid (VOF) surface capturing method. The multiphase two-equation k-ω SST turbulence model is adopted to simulate wave breaking and takes into account the effects enhanced by vegetation. The numerical model is validated with existing data from several laboratory experiments. The sensitivities of wave height evolution due to wave conditions and vegetation characteristics with variable bathymetry have been investigated. The results show good agreement with measured data. For non-breaking waves, the wave reflection due to the vegetation can increase wave height in front of the vegetation. For breaking waves, it is shown that the wave breaking behavior can be different when the vegetation is in the surf zone. The wave breaking point is slightly earlier and the wave height at the breaking point is smaller with the vegetation. For broken waves, the vegetation has little effect on the wave height before the breaking point. Meanwhile, the inertia force is important within denser vegetation and is intended to decrease the wave damping of the vegetation. Overall, the present model has good performance in simulating non-breaking, breaking and broken wave interaction with the emerged vegetation and can achieve a better understanding of wave propagation over the emerged vegetation.

Computational Fluid Dynamics Investigations of Breaking Focused Wave-Induced Loads on a Monopile and the Effect of Breaker Location

Abstract The design of new offshore structures requires the calculation of the wave-induced loads. In this regard, the computational fluid dynamics (CFD) methodology has shown to be a reliable tool, in the case of breaking waves especially. In this paper, two CFD models are tested in the reproduction of the experimental spilling waves impacting a circular cylinder for four different wave impact scenarios for focused waves. The numerical and experimental free surface elevations at different locations around the cylinder are also compared to verify the both numerical models. The numerical results from the models are shown together with the experimental measurements. Both CFD models are able to model the impact forces with a reasonable accuracy. When the cylinder is placed at a distance of 0.7 m from the wave breaking point, the value of the measured wave impact forces is highest due to the overturning wave crest and air entrainment. The wave-induced impact forces decrease, when the monopile is placed at distances further away from the breaking location.

A Modified k – ε Turbulence Model for a Wave Breaking Simulation

We propose a two-equation turbulence model based on modification of the k − ε standard model, for simulation of a breaking wave. The proposed model is able to adequately simulate the energy dissipation due to the wave breaking and does not require any “a priori” criterion to locate the initial wave breaking point and the region in which the turbulence model has to be activated. In order to numerically simulate the wave propagation from deep water to the shoreline and the wave breaking, we use a model in which vector and tensor quantities are expressed in terms of Cartesian components, where only the vertical coordinate is expressed as a function of a time-dependent curvilinear coordinate that follows the free surface movements. A laboratory test is numerically reproduced with the aim of validating the turbulence modified k − ε model. The numerical results compared with the experimental measurements show that the proposed turbulence model is capable of correctly estimating the energy dissipation induced by the wave breaking, in order to avoid any underestimation of the wave height.

Self-trapping and acceleration of ions in laser-driven relativistically transparent plasma

Self-trapping and acceleration of ions in laser-driven relativistically transparent plasma are investigated with the help of particle-in-cell simulations. A theoretical model based on ion wave breaking is established in describing ion evolution and ion trapping. The threshold for ion trapping is identified. Near the threshold ion trapping is self-regulating and stops when the number of trapped ions is large enough. The model is applied to ion trapping in three-dimensional geometry. Longitudinal distributions of ions and the electric field near the wave breaking point are derived analytically in terms of power-law scalings. The areal density of trapped charge is obtained as a function of the strength of ion wave breaking, which scales with target density for fixed laser intensity. The results of the model are confirmed by the simulations.

Shell concentration dynamics driven by wave motion in flume experiments: Insights for coquina facies from lake-margin settings

Coquinas are important carbonate oil reservoirs in the sin-rift sequence of the Brazilian marginal basins, mainly in Campos and Santos basins. They are generated from shell accumulations in lake-margin environments in which waves and bottom currents, at storm conditions, were the main hydrodynamic agents of transport and reworking. In order to investigate the role of wave transformation processes in shallow waters – swash, breaking and wave shoaling – in shell concentrations, a physical modelling of the coastal environment was performed in laboratory. The experimental setup consisted of a 2D flume, a wave generator, and a sedimentary bottom composed by quartz sand and shells (bivalves, gastropods, and fragments). An underwater camera, wave gauges and an ADV were used to record the sedimentary processes and to perform a quantitative analysis of the hydrodynamics. The experimental scenario reproduced a rift-lake margin. A coastal sector of Lake Tanganyika, Tanzania, Africa, was used to set the model boundary conditions (in a 1/20 scale) as well as the coastal gradient and wave conditions. From 54 h of wave experiments which alternated from fair-weather to storm wave conditions, three shell concentration dynamics were observed: reworking, winnowing, and dynamic bypassing. The reworking (erosion) was characterized by traction, rolling, and saltation of bioclasts, together with the traction and suspension of sand. The occurrence of the latter was restricted to the breaking and swash zone in both fair-weather and storm conditions. The winnowing (erosion) was featured by sand suspension and the gentle traction and rotation of the bioclasts on the bottom. The winnowing occurred at the proximal area of the shoaling zone, just behind the first wave-breaking point in storm wave conditions. The dynamic bypassing (omission) was identified by the ripples migration over static shells on the bottom of the distal shoaling zone, which produced alternated burial and exhumation of bioclasts. The quantitative analysis of the oscillatory flow indicated higher velocities at the breaking zone (reworking domain) than at the shoaling zone (winnowing domain) mainly during storm conditions. This study provided new and more accurate insights to depositional models of lake-margin coquinas, suggesting that each wave transformation process control a specific dynamic of shell concentration. Also, the limit between the breaking zone and the shoaling zone in storm conditions (1st wave-breaking point), was identified as a better criterion to bound the reworking domain, where fragmented coquinas are produced, from the winnowing and dynamic bypassing domains, where non-fragmented coquinas occur, in contrast to the fair-weather wave base criterion.

Mountain-Wave Turbulence in the Presence of Directional Wind Shear over the Rocky Mountains

Abstract Mountain-wave turbulence in the presence of directional wind shear over the Rocky Mountains in Colorado is investigated. Pilot reports (PIREPs) are used to select cases in which moderate or severe turbulence encounters were reported in combination with significant directional wind shear in the upstream sounding from Grand Junction, Colorado (GJT). For a selected case, semi-idealized numerical simulations are carried out using the WRF-ARW atmospheric model, initialized with the GJT atmospheric sounding and a realistic but truncated orography profile. To isolate the role of directional wind shear in causing wave breaking, sensitivity tests are performed to exclude the variation of the atmospheric stability with height, the speed shear, and the mountain amplitude as dominant wave breaking mechanisms. Significant downwind transport of instabilities is detected in horizontal flow cross sections, resulting in mountain-wave-induced turbulence occurring at large horizontal distances from the first wave breaking point (and from the orography that generates the waves). The existence of an asymptotic wake, as predicted by Shutts for directional shear flows, is hypothesized to be responsible for this downwind transport. Critical levels induced by directional wind shear are further studied by taking 2D power spectra of the magnitude of the horizontal velocity perturbation field. In these spectra, a rotation of the most energetic wave modes with the background wind, as well as perpendicularity between the background wind vector and the wavenumber vector of those modes at critical levels, can be found, which is consistent with the mechanism expected to lead to wave breaking in directional shear flows.

Evolution of initial discontinuities in the DNLS equation theory

We present the full classification of wave patterns evolving from an initial step-like discontinuity for arbitrary choice of boundary conditions at the discontinuity location in the DNLS equation theory. In this non-convex dispersive hydrodynamics problem, solutions of the Whitham modulation equations are mapped to parameters of a modulated wave by two-valued functions what makes situation much richer than that for a convex case of the NLS equation type. In particular, new types of simple-wave-like structures, such as trigonometric shocks and combined shocks, appear as building elements of the whole wave pattern observable in the long-time evolution of pulses after the wave-breaking point. The developed theory can find applications to propagation of light pulses in fibers and to the theory of Alfvén dispersive shock waves.