Wave Celerity Research Articles

Experimental research on self-amplifying density waves in horizontal pipelines of two phase granular slurries: Measurements on the effect of particle diameter and concentration

Self-amplifying density waves in hydraulic transport pipelines is a scarcely researched topic. Density waves are in essence the result of a spatial redistributing effect and clustering of solids in hydraulic transport pipelines. Self-amplifying density waves are very undesirable for practical applications, as these waves increasing the risk of pipeline blockages. The two available experimental studies (Talmon et al., 2007; Matoušek and Krupička, 2013) report conflicting properties of the density waves, such as wave length and wave celerity. This new experimental research aims to shed light on the reported differences, by broadly varying particle size and concentration in a new dedicated experiment. The main highlight of this research is that two separate mechanisms were identified that can cause density waves, and Talmon et al. (2007) and Matoušek and Krupička (2013) in hindsight were studying the two different mechanism respectively. Both wave type mechanisms come into effect at mixture velocities close to the deposit limit velocity, and require a stationary bed layer to initiate. The first mechanism is caused by an imbalance of erosion and sedimentation of the bed layer, which is predominant for fine sand particles (∼242μm and ∼308μm in this research). The second mechanism occurs when the bed layer starts sliding, instead of being eroded, and is specific for larger sand sizes (∼617μm and ∼1.08mm in this research). These two mechanisms are clearly distinguishable, having different wave lengths, celerity, amplitudes and amplification rates. The results also show a clear relationship between the mean concentration of a density wave, the wave amplitude and wave celerity specific for each of the two mechanisms.

Physical Modeling of Tsunamis Generated by Submarine Volcanic Eruptions

AbstractSubmarine volcanic eruptions can induce major local and regional tsunami hazards through varied source mechanisms. Large‐scale experiments of tsunamis generated by submarine volcanic eruptions are conducted to study the cylindrical wave generation and propagation in a three‐dimensional wave basin. A unique volcanic tsunami generator (VTG) was deployed at the bottom of the wave basin to generate volcanic tsunamis with repeatable and controlled source parameters. The physical modeling is based on the generalized Froude number defined with the VTG velocity and the near source water depth. The pneumatically driven vertical stroke motion generates leading elevation waves in the wave basin. The tsunami waves generated by the VTG are measured with a wave gauge array. The wave maker performance is characterized by the dimensionless leading wave amplitudes and periods. The experimental data show the variations of the leading wave amplitude, period, and celerity along the radial propagation distance. The generated cylindrical waves belong to the weakly nonlinear wave regime in the near field with decaying wave amplitude along radial propagation. The attenuation rate of the leading wave exceeds the range from the linear wave theory in runs with higher Froude numbers. The dimensionless leading wave period increases with the dimensionless propagation distance due to the dispersion relation. Empirical equations for the characteristic wave parameters such as amplitudes and periods are derived. The experimental results contribute to the understanding of volcanic tsunami generation processes and may serve rapid volcanic tsunami hazard assessments as well as the advancement and validation of numerical models.

Granular Cooling of Uni‐Sized Inelastic Particles in an Obstructed Chute

AbstractThis research aims to experimentally characterize cooling, clogging and jamming of a dry granular flow in a chute partially obstructed by a stopping wall with two slits adjacent to the side walls. We ensemble‐average velocities, determined with Particle Tracking Velocimetry, and its fluctuations, to compute mean flow and granular temperature fields. Full chute‐wide jamming is triggered by the formation of stable arch‐like clogging structures in front of the slits. The statistical distribution of the clogging instant is not heavy‐tailed, which indicates that clogging occurs only when the flow through the slits is liquid‐like. An upstream‐progressing jamming wave eventually forms, similar to that observed in fully obstructed chutes. There is no triple point anywhere, since the flow cools down to a granular liquid before jamming. We identified three main stages of jamming wave propagation. The initial buildup is characterized by high values of the upstream Froude number, slow progression, and transformation of kinetic into potential energy. This occurs with significant granular head losses, as particles attempt to flow over the jam. In the second stage, accretion becomes dominant, characterized by smaller head losses and, consequently, higher values of the jamming wave celerity. In the third stage, the jamming wave propagates against a cooler but faster flow that pushes against the jam, slightly increasing the wave strength. Accretion is the main mechanism of jammed mass increase, justifying a further increase of wave acceleration. The macroscopic aspects of the jamming wave dynamics can be described, as a first approximation, by shallow‐water theory.

Generative domain-adapted adversarial auto-encoder model for enhanced ultrasonic imaging applications

In this study, we propose a class-conditioned Generative Adversarial Autoencoder (cGAAE) to improve the realism of simulated ultrasonic imaging techniques, in particular the Multi-modal Total Focusing Method (M-TFM), based on the availability of both simulated and experimental TFM images. In particular, this work studied the case of the inspection of a complex geometry block representative of weld-inspection problem based on ultrasonic multi-elements probe. The cGAAE is represented by a tailored learning schema, trained in a semi-supervised fashion on a labeled mixture of synthetic (class 0) and experimental (class 1) M-TFM images, obtained under different meaningful inspection set-ups parameters (i.e., the celerity of the transverse ultrasonic wave, the specimen back-wall slope and height, the flaw tilt and heigh). That is, the cGAAE schema consists in a combination of learning stages involving class-conditioned spatial-transformers and arbitrary style transfer endows the cGAAE of powerful generative features, such as quasi real-time generation of M-TFM images by sweep of the inspection parameters. We exploited the cGAAE model to improve the realism of simulated M-TFM images and enhance the accuracy of the inverse problem, aiming at estimating the inspection parameters based on experimental acquisitions.

Predictive capabilities, robustness and limitations of two event-based approaches for lag time estimation in heterogeneous watersheds

Several studies have demonstrated that response times in natural catchments decrease with increasing rainfall intensity. Consequently, event-based estimations of catchment response times are of paramount importance in applied hydrology. Specifically, they have the potential to address a major inconsistency in the use of empirical formulas. These formulas often assume response times as constant parameters, regardless of whether extreme or frequent flood events are considered, thus neglecting the role of flow velocities.In this paper, built upon previous approaches developed and/or analyzed by the authors, two different recent methods for event-based estimations of catchment response times are critically reviewed, and their predictive performances are compared. First, four “physically-based” formulas, calibrated using synthetic rainfalls in three small Italian watersheds to reproduce the results of a two-dimensional hydrodynamic-based rainfall/runoff model and, consequently, the simulated wave celerities, are considered. Then, the detrending moving-average cross-correlation analysis (DMCA) has been applied to assess the average time elapsed between the centroids of precipitation and discharge time series.The soundness of these two approaches is initially assessed based on their ability to reproduce estimated lag times from observations. Their robustness is further evaluated by analyzing the magnitude and basin scale dependence of the inferred velocities compared to observed values, following a recent approach proposed in the literature. These issues are discussed with reference to 60 rainfall-runoff events occurring across 27 watersheds in Hungary and Italy, which possess substantially different geomorphic and climatic features, highlighting both the potential and the need for further improvements. Both approaches give error rates of around 37% for the proposed dataset.

Wave transformation across impermeable and porous artificial reefs

Porous artificial reefs can function as nature-based solutions for coastal protection, due to their ability to dissipate wave energy while providing habitat for marine species. However, due to the lack of quantitative understanding of wave interactions with porous artificial reefs, there is uncertainty in how to optimize design for coastal protection applications. To address this gap, physical modelling experiments were conducted in a wave flume to investigate wave transformation across both porous and impermeable artificial reefs exposed to a range of regular wave conditions and submergence depths. The results highlight how key changes to wave kinematics (e.g., wave celerity, wave breaking) and wave energy (through reflection, dissipation and transmission) differ considerably between impermeable and porous artificial reefs. These differences in wave kinematic properties and energy balances across each reef can be well characterized by consideration of the effective crest depth (i.e., the total height of water at a point on the reef, including that in the voids) that accounts for the porosity of the reef structure, rather than the actual depth of the reef crest. The ratio of the effective crest depth to the incident wave height, termed the ‘relative effective crest depth’, was used to develop robust formulations to accurately predict wave reflection and transmission coefficients across both reefs over the range of wave and water depth conditions. The new formulations were also compared to other datasets reported in the literature, and were found to provide accurate predictions of both wave reflection and transmission for the broad range of submerged porous coastal structures considered (including artificial reefs and submerged breakwaters).

Dimensionless Groups by Entropic Similarity: II-Wave Phenomena and Information-Theoretic Flow Regimes.

The aim of this study is to explore the insights of the information-theoretic definition of similarity for a multitude of flow systems with wave propagation. This provides dimensionless groups of the form Πinfo=U/c, where U is a characteristic flow velocity and c is a signal velocity or wave celerity, to distinguish different information-theoretic flow regimes. Traditionally, dimensionless groups in science and engineering are defined by geometric similarity, based on ratios of length scales; kinematic similarity, based on ratios of velocities or accelerations; and dynamic similarity, based on ratios of forces. In Part I, an additional category of entropic similarity was proposed based on ratios of (i) entropy production terms; (ii) entropy flow rates or fluxes; or (iii) information flow rates or fluxes. In this Part II, the information-theoretic definition is applied to a number of flow systems with wave phenomena, including acoustic waves, blast waves, pressure waves, surface or internal gravity waves, capillary waves, inertial waves and electromagnetic waves. These are used to define the appropriate Mach, Euler, Froude, Rossby or other dimensionless number(s)-including new groups for internal gravity, inertial and electromagnetic waves-to classify their flow regimes. For flows with wave dispersion, the coexistence of different celerities for individual waves and wave groups-each with a distinct information-theoretic group-is shown to imply the existence of more than two information-theoretic flow regimes, including for some acoustic wave systems (subsonic/mesosonic/supersonic flow) and most systems with gravity, capillary or inertial waves (subcritical/mesocritical/supercritical flow). For electromagnetic wave systems, the additional vacuum celerity implies the existence of four regimes (subluminal/mesoluminal/transluminal/superluminal flow). In addition, entropic analyses are shown to provide a more complete understanding of frictional behavior and sharp transitions in compressible and open channel flows, as well as the transport of entropy by electromagnetic radiation. The analyses significantly extend the applications of entropic similarity for the analysis of flow systems with wave propagation.

Dimensional analysis of literature formulas to estimate the characteristic flood response time in ungauged basins: A velocity-based approach

When investigating the wide literature dealing with the assessment of the critical time scale for basin hydrological response, several aspects still to be clarified can be acknowledged. Despite the high sensitivity of design flood peaks to the estimated time parameter value, there is still no agreement on the conceptual and operational definitions of the basin response time, resulting in several different approaches and formulations available.In our work, we suggest a conceptual approach to either reject or recommend formulas of the characteristic basin response time for ungauged basins, with the aim of devising some practical steps in the choice of a robust formulation to be used in hydrological modelling and flood hydrograph design. To this end, 29 empirical and semi-empirical formulas, all containing a basin length and slope, have been carefully selected and their structure compared in dimensional terms, using a simple hydraulic reasoning (e.g., the Chezy formula) as indicator of hydraulic consistency. 13 hydraulically consistent formulas have been identified.Starting from wave celerities and using the river network morphology of 135 watersheds in north-western Italy, we have then investigated and compared the variability of the average flow velocities estimated using all formulas whose input data are available within the study area. By comparing the magnitude and basin scale dependence of the inferred velocities with the values observed in the literature, which generally increase with basin size, some formulas are considered not reasonable, while 5 of them are identified as more robust, i.e. consistent with the observations. These are the formulas of Chow (1962), NERC (1975), SCS (1954), McEnroe and Zhao (1999) and Watt and Chow (1985).Our findings lead to identify analytically the relationships between the exponents of each formula and those of the scaling law linking the length and slope of the basin. These relationships, driving the increase or decrease in the velocity values with basin size, allow us to identify the range of length and slope exponents in the characteristic time formulas for which velocity increases with basin area, as literature suggests. Based on the same relationships, one of the 5 formulas above can be adopted in practical applications and a guideline for calibrating new formulations can be followed.

Detection of decadal time‐series changes in flow hydrology in eastern Australia: Considerations for river recovery and flood management

AbstractSince European colonisation, many coastal rivers of New South Wales (NSW) have been highly modified by channelisation, flow regulation, sediment mining, intensive grazing and agriculture, deforestation, and riparian vegetation and wood removal. Since the late 1980s, nearly 55% of these rivers have undergone a significant ‘re‐greening’ and geomorphic recovery with changes to instream and riparian roughness. However, little research has been undertaken to quantify if there have been any coeval changes in flow hydrology. Approximately 7000 flow hydrographs with one‐hour time‐steps from 117 gauges on 45 study rivers (17 of 20 coastal catchments of NSW) were used to assess changes to in‐channel and overbank hydrology over decadal timeframes. Hydrograph shape and attenuation characteristics for three morphologically defined flow stages were analysed: in‐channel fresh, high flow and overbank flood. Time‐series analysis was used to quantify changes in flow hydrology from the early‐20th century to present and identify trends occurring between known degradation phases (1910s–1940s and 1940s–1980s) and a recovery phase (1990s to present) for these rivers. Our findings indicate that, on average, and for some key examples, changes in flow hydrology are occurring. Flows are slowing and attenuating as indicated by decreases in flood wave celerity and hydrograph kurtosis, skewness and rate of rise and increases in peak‐to‐peak travel time, flood peak attenuation and flood wave attenuation index. The most significant changes have occurred since the 1980s with the most noticeable effects on the behaviour of high flows (around the bankfull stage). However, there is not a consistent direction change in these indicators in all places. We use these findings to categorise the flow mitigation signal of the study regions and catchments and discuss the implication for river and flood management.

MODELLING WAVE OVERTOPPING AND WAVE IMPACTS BY MEANS OF IMAGE CLUSTERING TECHNIQUES

The experimental modelling of the wave-structure interaction processes is a very complicated issue, due to the highly non-linear turbulent flow conditions, including wave breaking and air entrainment (Stringari et al., 2021). Recently, Formentin et al. (2021) demonstrated that videography can provide a reliable representation of the wave breaking, of the wave overtopping and of the wave impact at walls. They filmed laboratory tests of wave overtopping at smooth dikes with crown walls (Formentin et al., 2018; Palma et al., 2020) with a low-cost full-HD camera and compared the results of the video-cluster analysis applied to its records with the measurements obtained through traditional techniques. They derived accurate quantitative estimations of free-surface elevation, wave overtopping discharge, volumes, velocities and celerities. The aim of this paper is to further investigate the possibilities of the videography by focusing on the analysis of the wave impacts at the walls and on the detection of the amount of air entrapped during the impacts themselves.

Comparison study of fluid thermal boundary-bulk behaviors in the close-to-critical region under different property trends

The high-efficiency energy system based on the utilization of supercritical CO2 has been widely developed in recent years to meet the growing demands of clean energy and the elimination of CO2 emissions. Within this field, the design of the corresponding equipment requires an understanding of the thermodynamic behavior of supercritical fluid affected by the singular thermophysical properties. The analysis of this study is based on the asymptotic expansion of hydrodynamic equations and the thermophysical properties characterized by the exponential trends. The asymptotic model finds the wave and diffusion modes valid within the bulk region in second-order and within the boundary layer region (BLR) in first-order, respectively. Main findings include: (1) The wave mode is found in the bulk region with the wave celerity Γ. When the critical point is approached, Γ decreases from (γ0Mac0/κT0)0.5 = 1 to [(γ0–1)Mac0/κT0]0.5, which; (2) The diffusion mode found in BLR is characterized by non-dimensional diffusion coefficient ζ. This coefficient has the same behavior as thermal diffusivity, which is enlarged by thermal conductivity but weakened by isobaric specific heat. (3) Mass transport from BLR to bulk leads to the generation of the thermal wave, which can be measured by mass transport coefficient Mb=ζβp. Mb also characterizes the magnitude of the thermal wave as u2=Mb(Tb1)z|z=0 (that is, 0.17 mm/s when ϕ=10−4). The behavior of Mb is similar to one of ζ. However, the effect of cp is limited; (4) Two different modes of viscosity are identified. The increase in viscosity leads to the change of viscous stress mode from the second order to the first order, which also leads to a decrease in mass transport.

Simultaneous mapping of nearshore bathymetry and waves based on physics-informed deep learning

This paper uses physics-informed neural networks (PINNs) to simultaneously determine nearshore water depths and wave height fields based on remote sensing of the ocean surface with limited or sparse measurements. Two methods that integrate the knowledge of water wave mechanics and fully connected neural networks are introduced. The first method utilizes observed wave celerity fields and scarce measurements of wave height as training data. The model performance was examined with linear waves over an alongshore varying barred beach and nonlinear waves over an alongshore uniform barred beach. The second method uses scarce wave height and water depth measurements as training points, and the model performance was investigated with water waves over a circular shoal and the alongshore varying barred beach. One advantage of applying PINNs to solve bathymetry inversion problems is that wave height and bathymetry can be simultaneously estimated by PINN models. Thus, the impact of wave amplitude dispersion on depth inversion in nonlinear wave systems can be considered without measuring the entire wave height field. Overall, this study demonstrates the potential of the inverse PINN model as a promising tool for estimating nearshore bathymetry and reconstructing wave fields using observations from different remote sensing platforms.

A Comparison of Numerical Schemes for Simulating Reflected Wave on Dry and Enclosed Domains

This paper is to investigate the capability of six numerical schemes to simulate reflected wave over a dry and closed domain with and without building, namely: (a) two proposed 2D numerical models solving the conservation form of 2D Shallow Water Equations (2D-SWEs) by Finite Volume Method (FVM) with Roe and HLLC schemes are invoked to approximate Reimann solver; (b) three options of shallow models in the commercial software Flow 3D based on a non-conservation form of 2D-SWEs and (c) the Flow 3D with turbulence modules. By analyzing flooding maps, the area of the reflected wave, and water level profiles on a dry and closed domain, two proposed models give reasonable solutions, while three options of the shallow module of Flow 3D originate result less accurately when initial wave celerity (c0) is small. The accuracy level will be increased if c0 value increases. The 3D model presented the best performance of the complex flow pattern in the dry and enclosed domain in both cases without and with building.

In situ ultrasonic interface tracking for photovoltaic silicon directional solidification

This paper presents an experimental validation of an in situ ultrasound technique for tracking the solid–liquid interface during directional solidification of photovoltaic silicon. Ultrasound bursts are introduced from the top into the melt via the tip of a carbon glass waveguide directly plunged into the liquid. Several runs of solidification have been conducted using the same waveguide thus demonstrating its reusability, its mechanical resilience and its appropriate acoustical and chemical compatibility with the melt. We present the times of flight analysis to track the location of the solid–liquid interface. As part of the signal crosses the solidifying ingot, we also measure cs, the average celerity of sound waves in the solid. Since cs exhibits a significant dependance on crystal orientation, we consider the possibility to use the echoes to extract some information on the crystalline orientation in the ingot. By quantifying the uncertainties and their dependence with the duration of the experiment in particular, we outline the conditions necessary to detect accurately the average orientation in the solid.

Serre Equations in Channels and Rivers of Arbitrary Cross Section

A variational approach was used to derive a set of Serre equations for fully nonlinear, dispersive waves in channels of arbitrary cross section. A family of travelling waves was found, as well as the relation between amplitude and celerity of solitary waves. An upper bound is proposed for the solitary wave amplitude as a function of the Froude number in trapezoidal cross-sectional canals, and it showed good agreement with existing theory. For waves of moderate amplitude, cnoidal waves result with a soliton limit; these waves and their properties (celerity and wave number) are written as functions of the channel bank slope and channel bank curvature. The theoretical findings are in agreement with well-established results in the literature, in particular with more-recent Boussinesq-type theories. A validation is proposed against existing experimental data.

Directing Shallow-Water Waves Using Fixed Varying Bathymetry Designed by Recurrent Neural Networks

Directing shallow-water waves and their energy is highly desired in many ocean engineering applications. Coastal infrastructures can be protected by reflecting shallow-water waves to deep water. Wave energy harvesting efficiency can be improved by focusing shallow-water waves on wave energy converters. Changing water depth can effectively affect wave celerity and therefore the propagation of shallow-water waves. However, determining spatially varying bathymetry that can direct shallow-water waves to a designed location is not trivial. In this paper, we propose a novel machine learning method to design and optimize spatially varying bathymetry for directing shallow-water waves, in which the bathymetry is assumed fixed in time without considering morphodynamics. Shallow-water wave theory was applied to establish the mapping between water wave mechanics and recurrent neural networks (RNNs). Two wave-equivalent RNNs were developed to model shallow-water waves over fixed varying bathymetry. The resulting RNNs were trained to optimize bathymetry for wave energy focusing. We demonstrate that the bathymetry optimized by the wave-equivalent RNNs can effectively reflect and refract wave energy to various designed locations. We also foresee the potential that new engineering tools can be similarly developed based on the mathematical equivalence between wave mechanics and recurrent neural networks.

Natural flood management: Lessons and opportunities from the catastrophic 2021–2022 floods in eastern Australia

AbstractNatural flood management (NFM), a nature‐based solution to flood mitigation where hydrological and biophysical processes are harnessed to reduce flow velocity, erosive energy and flood risk, is an emerging global theme of water and river management. The catastrophic 2021 and 2022 floods in eastern Australia are used to assess the hydrological properties of discrete events and to start an investigation of whether widespread changes in flood hydrology are occurring. We find that most coastal rivers in New South Wales (NSW) had a noticeable decrease in flood wave celerity (increase in flood travel time) when the 2021 and 2022 floods are contrasted with equivalent floods since the 1970s and that several also exhibit increases in flood peak attenuation. For some rivers, there is a coincident trend between these changing flood properties, riparian vegetation regrowth (regreening) and geomorphic recovery over the last +30 years. These may be the first signals that passive riparian management is counteracting some of the more severe hydrological effects of floods on these rivers and that some degree of NFM is possible. Lessons from this work are that there is an immediate opportunity to implement large‐scale NFM in coastal catchments of NSW, but this will only occur if we re‐examine current flood mitigation and adaption strategies and recognise and prioritise nature‐based solutions that include space‐to‐flood and corridors of river recovery, as essential parts of the modern flood mitigation toolkit.

Thermodynamic Concepts in Civil Engineering

Thermodynamics is not always done well or usefully brought to bear in civil engineering. This paper addresses historical aspects of misunderstandings of thermodynamic concepts; applies the Second Law of Thermodynamics to applications including shock waves in compressible fluid flow, the tidal bore, spillway flow , and junction flow. Additional applications of thermodynamics in civil engineering are discussed. These include deriving hydraulic transient wave celerities for waterhammer analyses; the First Law of Thermodynamics for a closed system and for a control volume; one-dimensional flow, energy loss due to friction; parallel incompressible flow; application of the control volume to a pressure conduit; the modified Bernoulli equation; tees with small inflow and outflow branches; isentropic flow of a perfect gas and its application to flow metering, determination of choked flow conditions, and determination of the shaft power required to drive a blower.

Non-spectral linear depth inversion using drone-acquired wave field imagery

We present an efficient approach for video-based bathymetry survey. The proposed algorithm consists of preprocessing for drone-acquired video, extracting incident gravity wave, estimating celerity vector field, operating linear water depth inversion, and mapping bathymetry. The image preprocessing includes resizing, masking, barrel distortion correction, ortho-rectification, and georeferencing. The challenges posed by noisy, non-stationary, and irregular wave field data was resolved using ensemble empirical mode decomposition (EEMD), which is capable of extracting incident gravity wave structure without arbitrary initial guessing of frequencies. The extracted wave structures were used to estimate their celerity vector field using large-scale particle image velocimetry (LSPIV) technique. Given that wave celerity depends on water depth in nearshore based on linear dispersion relation, spatial distribution of water depths was determined from wave celerity estimates. The proposed algorithm was tested using the wave field videos and ground truth data, collected during the field experiment at Jangsa beach, located in the east coast of South Korea, and good agreement was achieved.

HyWaves: Hybrid downscaling of multimodal wave spectra to nearshore areas

Long-term and accurate wave hindcast databases are often required in different coastal engineering projects. The assessment of the nearshore wave climate is often accomplished by using downscaling techniques to translate offshore waves to coastal areas. However, dynamical downscaling approaches may incur huge computational cost. Additionally, the common use of bulk parameterizations are often not accurate for multidimensional waves. To overcome these limitations, we present a hybrid downscaling approach that combines mathematical algorithms (statistical downscaling) and numerical modeling (dynamical downscaling) over the individual spectral partitions. Every wave partition is downscaled and aggregated afterward by using principles of wave linear theory. By assuming linearity in the propagation of the wave celerity, the application of the method is limited from offshore to intermediate water depths. In addition, the method proposed uses a technique to simplify the spectral boundary conditions in complex domains. The methodology has been applied and validated in the island states of Samoa, American Samoa, Majuro, and Kwajalein, showing good skill at reproducing the spectral hourly time series of significant wave height, peak period, and peak direction. Moreover, an accurate representation of the observed energy spectrum was achieved. This study provides insight into the numerical approximation of the combined sea-swell states while improving the quality of fast spectral forecasting and early warning systems.