Open AccessArticle

Impacts of an Artificial Sandbar on Wave Transformation and Runup over a Nourished Beach

Cuiping Kuang

Liyuan Chen

¹,

Xuejian Han

^2,*,

Dan Wang

¹,

Deping Cao

and

Qingping Zou

^3,*

College of Civil Engineering, Tongji University, Shanghai 200092, China

Nanjing Hydraulic Research Institute, Nanjing 210029, China

The Lyell Centre for Earth and Marine Science and Technology, Institute for Infrastructure and Environment, Heriot-Watt University, Edinburgh EH14 4AS, UK

Authors to whom correspondence should be addressed.

Geosciences 2024, 14(12), 337; https://doi.org/10.3390/geosciences14120337

Submission received: 25 October 2024 / Revised: 4 December 2024 / Accepted: 5 December 2024 / Published: 8 December 2024

(This article belongs to the Section Hydrogeology)

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Abstract

Due to increasing coastal flooding and erosion in changing climate and rising sea level, there is a growing need for coastal protection and ecological restoration. Artificial sandbars have become popular green coastal infrastructure to protect coasts from these natural hazards. To assess the effect of an artificial sandbar on wave transformation over a beach under normal and storm wave conditions, a high-resolution non-hydrostatic model based on XBeach is established at the laboratory scale. Under normal wave conditions, wave energy is mainly concentrated in short wave frequency bands. The wave setup is negligible on the shoreface but becomes more significant over the beach face, and wave nonlinearity increases with decreasing water depth. The artificial sandbar reduces the wave setup by 22% and causes considerable changes in wave skewness, wave asymmetry, and flow velocity. Under storm wave conditions, as the incident wave height increases, the wave energy in the long wave frequency bands rises, while it decreases in the short wave frequency bands. The wave dissipation coefficient of an artificial sandbar increases first and then decreases with increasing incident wave height, and the opposite is true with the transmission coefficient. It features that the effect of an artificial sandbar on wave energy dissipation strengthens first and then weakens with increasing incident wave height. Additionally, an empirical formula for the wave runup was proposed based on the model results of the wave runup for storm wave conditions. The study reveals the complex processes of wave–structure–coast interactions and provides scientific evidence for the design of an artificial sandbar in beach nourishment projects.

Keywords:

beach profile; wave dynamics; green infrastructure; artificial sandbar; coastal protection; wave runup; XBeach

1. Introduction

In recent years, global climate change and anthropogenic interventions have led to rising sea level and exacerbated coastal flooding and erosion, which pose significant threats to ecological environment, economic development, and infrastructure systems in coastal regions [1,2,3,4,5]. A growing number of soft coastal defense measures have been implemented for coastal protection and restoration [6,7]. One of the effective approaches for coastal defence is beach nourishment [8], during which an artificial sandbar is placed on a beach to attenuate waves and mitigate erosion by altering nearshore hydrodynamic conditions. It promotes natural recovery of beaches in wave shadow areas [9,10,11,12]. The design and implementation of the artificial sandbar are dependent on a variety of factors, such as local current and wave conditions [13], sediment transport mechanisms [14], geomorphological features [15], sediment characteristics [16], seepage, and wave asymmetry [15]. Moreover, design parameters, such as the distance to the shore, the crest elevation, and the crest width of the artificial sandbar [13,14] are critical for its performance in wave attenuation. Sandbars alter the wave propagation characteristics and influence the offshore sediment transport, ultimately changing the beach profile. Additionally, the performance of sandbars in wave attenuation and coastal protection is influenced by wave conditions and a variety of other factors.

Physical experiments are robust for studying the response and evolution of artificial sandbars under different wave conditions. Van Duin et al. [17] and Koster et al. [18] found that nearshore nourishment may cause offshore wave breaking and energy dissipation and water setup, which in turn generate a horizontal circulation that enhances onshore sediment transport. Hoyng [19] carried out experiments during normal and storm wave conditions and found shoreface nourishments may reduce offshore sediment transport and enhance onshore sediment transport. Through two-dimensional cross-shore flume experiments, Wu et al. [20] confirmed that artificial sandbars can reduce wave height, decrease littoral currents and sediment carrying capacity, and stabilize beach profiles. Di Risio et al. [21] carried out a series of laboratory experiments and verified that a hybrid system with a submerged breakwater and a nourished beach can actively reduce erosions and promote sedimentations by attenuating waves. Recently, a series of physical model experiments by Li et al. [22,23] indicated that contrary to the traditional understanding of sediment seaward transport under storm conditions, artificial sandbars of certain shapes can cause local sediment shoreward transport and onshore sandbar migration under large waves, thus changing the overall erosion and accretion state of the beach and the shoreline response. Pan et al. [24] carried out mobile bed flume experiments of evolutions of artificial underwater sandbars under low energy regular waves.

Overall, physical experiments provide an important theoretical basis and design guidance for beach nourishment projects. However, wave heights in physical experiments cannot reach storm wave magnitudes. Wave heights and beach profiles are measured only at limited points. Therefore, a high-resolution mathematical model is necessary to understand the mechanism of artificial sandbar evolutions.

The numerical models of nourished beaches with artificial sandbars have improved significantly. Douglass [25] conducted a numerical study on submerged sandbars based on Bagnold’s bed-load formula and Bailard’s sediment transport equation. The model compared well with the experiment in the direction and magnitude of sediment migration, except for deviation in the initial movement of sediment. Lee [26] established a model based on Bailard’s sediment transport equation that effectively predicted sandbar migrations and sediment transport volumes.

Van Duin et al. [17] combined the process-based beach profile model UNIBEST-TC and the coastal model DELFT3D-MOR to simulate the nearshore nourishment along the Egmond coast in the Netherlands. The key phenomena, such as sandbar movement, wave and longshore current attenuation, and sediment deposition in the sheltered area, was well captured. However, the predicted profile deviates slightly from the field measurement due to transverse sediment transport induced by wave asymmetry. Koster [18] used Delft3D to investigate sandbar nourishments under various conditions and found that the nourishment promotes onshore sediment transport and alleviates coastal erosion, with outcomes influenced by wave conditions, sandbar design, and coastline morphology.

Regional models cannot resolve small scale physical processes; therefore, it is difficult to capture free and forced motions [27]. Van Leeuwen et al. [28] used the linear stability analysis to determine the beach geomorphology change and found that the sandbars were eroded slowly, leading to onshore sediment transport. Through physical experiments and numerical simulation, Li et al. [29,30] investigated the lateral sandbar migrations and associated morphological evolution and sediment transport under variety wave conditions. Experiments by Li et al. [23] indicated that artificial sandbars play a crucial role in sediment retentions during storms and beach recovery after storms. Comparisons of XBeach model with laboratory tests by Liang et al. [31] verified that beach profiles approach the equilibrium state with seaward sandbar nourishment, which provides better storm protection and underwater profile recovery.

In addition, numerical models have advanced in designs and post-application evaluations of artificial sandbars. Kuang et al. [32] established the Delft3D wave–current coupled model to simulate the effect of artificial submerged sandbars on hydrodynamics for the Zhonghaitan beach in Beidaihe. The model results indicated that waves and hydrodynamic forces at the beach were reduced by the sandbars. Numerical simulations are powerful tools to investigate the mechanisms by which natural or man-made coastal structures influence sediment transport and beach profile evolution. The wave runup empirical formulas mainly developed for a natural beach under normal wave conditions may not be accurate in the presence of a structure. Therefore, there is an urgent need for a wave runup formula suitable for beaches with a structure during storm wave conditions for disaster preventions and mitigations.

In this study, a non-hydrostatic model based on XBeach is established to simulate the laboratory experiments on a nourished beach profile. The well-validated model is used to investigate the impacts of the artificial sandbar on local hydrodynamics under normal and storm wave conditions, including the wave setup, wave skewness, and wave asymmetry. Moreover, the wave transformation over the artificial sandbar is evaluated by reflection, transmission, and dissipation coefficients, respectively. This study systematically reviews the wave runup data under the storm wave conditions to derive an empirical formula for the 2% cumulative frequency wave runup as well as the mean water level of the wave runup, high-frequency significant surge wave height, and low-frequency significant surge wave height. The present study aims to reveal wave evolution on the nourished beach with an artificial sandbar, predict the wave runup, and provide theoretical and practical guidance for designing artificial sandbars in beach nourishment projects.

2. Methodology

2.1. Physical Model Setup

The numerical model is constructed to simulate the beach profile evolution experiments by Ma [33]. The experiment was designed based on a typical beach nourishment project on the Qinhuangdao beach; the nourished beach without and with an artificial sandbar are shown in Figure 1, respectively. The toe of beach slope was fixed in position, serving as the origin of the coordinate system. The model geometric scale and wave height scale of the experiment were 1:10, and the wave period scale was 1:

\sqrt{10}

. The wave flume was 50 m long, 0.8 m wide, and 1.2 m deep. The nourishment beach slope β was 1:10, the average offshore slope was 1:200, and the flume water depth was 0.6 m (see Figure 1).

The artificial sandbar was located 10.0 m from the intersection of still water level and the beach (shoreline), and the sandbar area was within 8 < X < 18 m. In Figure 1a, X and z are horizontal and vertical axes with an origin at beginning of the beach at the sea bed. The artificial sandbar was an isosceles trapezoid with a crest width of 3.0 m, a base width of 10.0 m, and foreslope and backslope of 1:10. Normal waves (JONSWAP spectrum, significant wave height of 0.1 m, spectral peak wave period of 1.57 s) were applied continuously for 90 min to beach profile with and without sandbar. The median grain diameter was 0.00017 m. Utilizing a one-dimensional grid system, two distinct types of beach profiles were established based on the experiment setup. The model domain extended 39.3 m in length, encompassing a 12 m buffer zone before the beach origin and followed by a study zone from X = 0 m to X = 27.3 m. The buffer zone had a 0.05 m resolution, with the sandbar zone grid size was refined to 0.02 m, resulting in a total of 1605 grid cells.

2.2. Non-Hydrostatic XBeach Model

To study the wave evolution on the nourished beach under the protection of an artificial sandbar, a high-precision non-hydrostatic dynamic XBeach model of beach profile on laboratory scale was established [34]. In the non-hydrostatic XBeach model, the derivation of the depth-averaged normalized dynamic pressure follows a similar approach to that of the one-layer version of the SWASH model [35]. Consequently, the one-dimensional non-linear shallow water equations are formulated as follows:

\frac{\partial η}{\partial t} + \frac{\partial h u}{\partial x} = 0

(1)

\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = - g \frac{\partial η}{\partial x} - \frac{1}{h} \int_{- d}^{η} \frac{\partial q}{\partial x} d z + ν_{t} \frac{\partial^{2} u}{\partial x^{2}} - c_{f} \frac{u |u|}{h}

(2)

where η is the free water surface elevation; d is the still water depth; h is the total water depth, h = η + d; u is the depth-averaged velocity in x-direction; g is the gravity constant, 9.81 m²/s; v_t is the horizontal eddy viscosity; p is the dynamic pressure; q is the normalized dynamic pressure, calculated as the ratio of the dynamic pressure p to the water density ρ, q = p/ρ; and c_f is the dimensionless friction coefficient. The first term on the right side of Equation (2) represents the hydrostatic pressure, while the second term denotes the non-hydrostatic pressure, also known as the dynamic water pressure. For details on the dynamic pressure term within the depth-averaged nonlinear shallow water equations of the non-hydrostatic XBeach model, one can refer to the previous research conducted by Kuang et al. [36].

The non-hydrostatic model utilizes the hydrostatic front approximation (HFA) to simulate the location and scale of wave breaking [37], assuming that the pressure distribution at the wave-breaking front is hydrostatic pressure. When employing the HFA, the water surface is treated as a single-valued function for time or spatial transverse coordinates, indicating that the wavefront is almost upright at the point of breakage.

To improve the capacity for simulations of wave dispersion phenomena, the number of layers of the vertical grid is typically increased [38]. However, the augmentation of vertical grid stratification (even double-layer grids) can lead to a marked escalation in computation time. To provide a detailed and nuanced understanding of how hydrodynamic and morphologic processes interplay within the surf zone, considering both phase and depth variables with significantly saving computation time, Deltares has introduced a new version of XBeach (reduced vertical two-layer grid technique), which splits the water depth into two layers, utilizing a fraction α (= h_down/h); more details can be found in the article of Elsayed et al. [39].

2.3. Mathematical Model Setup

The conventional model setup parameters in the XBeach model are shown in Table 1.

The variable ‘nx’ represents the local number of grid cells in the x-direction. The ‘CFL’ parameter denotes the maximum Courant–Friedrichs–Lewy number. The variable ‘tstop’ is the stop time of simulation in morphological time. The variable ‘depthscale’ is a factor used in order to set different cut-off values like ‘eps’ and ‘hswitch’. The variable ‘wbctype’ refers to the new type of wave boundary condition, which is essential for accurately simulating wave interactions at the model boundaries. The variable ‘tideloc’ indicates the number of corner points on which a tide time series is specified. The variable ‘zs0’ is the initial water level. The variable ‘bedfriction’ is the coefficient of bed friction, and ‘bedfriccoef’ defines the formulation used for calculating bed friction. The variable ‘nonhq3d’ controls the activation or deactivation of the reduced two-layer non-hydrostatic model, and the value of ‘nonhq3d’ being equal to 1 indicates that the reduced two-layer model is enabled. The variable ‘maxbrsteep’ is the maximum wave steepness criterium. The variable ‘tsfac’ is a coefficient that determines the value of ‘Ts’ in the sediment source term, calculated as ‘Ts = tsfac + h/ws’. The variable ’por’ stands for porosity. The variable ‘sedcal’ is the sediment transport calibration coefficient per grain type. The variable ‘morfac’, the morphological acceleration factor, allows users to decouple the hydrodynamic and morphological time scales. The variables ‘dryslp’ and ‘wetslp’ define the critical avalanching slope above and below water, respectively.

2.4. Validation

The non-hydrostatic model for the wave frequency spectrums and significant wave heights on the nourished beach with and without the artificial sandbar under the action of normal waves were validated against the experimental results by Ma [33].

2.4.1. Beach Profile Without Sandbar

The validation of the BF profile (beach profile without sandbar) under normal wave action focused primarily on the wave spectra and significant wave height by 13 wave gauges (Figure 2a). Model performance can be evaluated with the relative mean absolute error RMAE [40,41]. The range of RMAE values ranges from 0 to infinity. The evaluation criteria were as follows: RMAE < 0.05 indicates excellent model performance; 0.05 < RMAE < 0.1 indicates good; 0.1 < RMAE < 0.2 indicates reasonable; 0.2 < RMAE < 0.3 indicates poor; and RMAE > 0.3 indicates bad. The formula for RMAE is given by

R M A E = \frac{\frac{1}{n} \sum_{i = 1}^{n} |O_{i} - P_{i}|}{\frac{1}{n} \sum_{i = 1}^{n} |O_{i}|}

(3)

where O represents the measured value, P represents the simulated value, i represents the serial number of data points, and n represents the number of data points.

Figure 2a–m show the validation of wave spectra at the 13 wave gauge locations, and it can be observed that the simulations are in good agreement with the measurements. Figure 2n shows the validation of wave height. Overall, the simulations were in good agreement with the measurements, with RMAE of 0.0292. Therefore, the simulation performance of the model was excellent (Table 2).

2.4.2. Beach Profile with Sandbar

The validation of the beach profile with sandbar under normal wave action focuses primarily on the wave spectra and significant wave height by 13 wave gauges (Figure 2b). Figure 3a–m show the simulated wave spectra generally agree with the measurements by the 13 wave gauges. However, at the front slope and crest of the sandbar (W7–W11), the model did not capture the measured wave spectra with a peak in the low-frequency band. Similar discrepancy is also observed near the waterline (W1, W2). According to bispectral analysis of numerical and experimental data by Zou and Peng [42] and Peng et al. [43], the sum interactions among waves dominate above the front slope and on the structure crest, but the difference interaction dominates above the rear slope of a low crest structure, such as sandbar, reef, and breakwaters. The former enhances the harmonics, and the latter enhances the subharmonics at the low frequency band, where the XBeach model underestimates the wave height amplification over the front slope of the sandbar and the submerged beach face. As shown in Figure 3n, the simulated wave heights at W2 and W9 were lower than the measured values. This underestimation led to a reduced wave-breaking intensity; therefore, weakened low-frequency waves were generated by wave breaking. Additionally, the XBeach model used a hydrostatic front approximation theory to simulate the physical process of wave breaking, underestimating the energy loss during wave breaking, therefore being the low-frequency waves. Figure 3n shows the validation of the cross-shore significant wave height. Except at W2 and W9, the overall agreement between the measured and simulated wave height was good, with RMAE of 0.0372. Therefore, the model performance was excellent (Table 2).

2.5. Data Analysis Methods

2.5.1. Wave Height and Energy

For the cross-shore variation in wave height, the total H_s was derived from the wave spectrum through Fourier transform of the wave surface time series. The H_s of short wave and long wave (low-frequency wave or infragravity wave) were the H_s for the 0.05–1.00 Hz and 0.004–0.05 Hz frequency band in the field scale [44], corresponding to 0.16–3.16 Hz and 0.01–0.16 Hz, respectively, according to the time scale of

1 : \sqrt{10}

[33].

There is a direct relationship between wave height and wave energy. Wave energy refers to the energy possessed by a wave, which is proportional to the square of the wave height and can be estimated through measurements of the wave height [12].

2.5.2. Wave Skewness and Asymmetry

Wave skewness (Sk) and asymmetry (Asy) are indicators of wave deformation and nonlinearity. Skewness represents the asymmetry of the wave profile to the horizontal axis, corresponding to the asymmetry in wave velocity along the horizontal axis [45]. Positive skewness indicates a sharper crest and broader trough (i.e., Stokes wave or cnoidal wave), with greater velocity at the crest. Wave asymmetry reflects the asymmetry of the wave profile to the vertical axis, quantifying the skewness of acceleration [46]. A wave asymmetry of zero indicates that the wave crest is symmetrical to the vertical axis, and the acceleration magnitude ahead of the wave crest is equal to that behind the wave crest. Negative asymmetry means the crest moves faster than the trough, resulting in a sawtooth-shaped wave profile leaning shoreward. In finite water depth, waves usually manifest as Stokes waves or cnoidal waves with non-zero skewness values, and when waves propagate into the surf zone, they are usually skewed asymmetric waves with non-zero skewness and asymmetry. The velocity skewness [47,48] and the acceleration skewness [46,49] can both drive sediment transport toward the shore, such as the onshore movement of sandbars.

The formulas for wave skewness and wave asymmetry [50] are as follows:

S k = \frac{〈{(η - \bar{η})}^{3}〉}{{〈{(η - \bar{η})}^{2}〉}^{3 / 2}}

(4)

A s y = \frac{〈H^{3} (η - \bar{η})〉}{{〈{(η - \bar{η})}^{2}〉}^{3 / 2}}

(5)

where η represents the time series of free surface elevation;

\bar{η}

is the time average of η, aimed at filtering out the effects of the wave setup; 〈 〉 is the time averaging operator; and

H

represents the Hilbert transform.

2.5.3. Wave Reflection, Transmission, and Dissipation Coefficient

To estimate the reflection coefficient K_R, transmission coefficient K_T, and dissipation coefficient K_L, two wave gauges in front of the artificial sandbar and one behind it were installed to measure the wave surface time series. The reflection coefficient K_R, the incident significant wave height H_I, and the reflected significant wave height H_R are typically calculated by Goda’s two-point method [51].

The transmission coefficient K_T is calculated by

K_{T} = H_{T} / H_{I}

, where H_T is the transmitted wave height derived from the wave surface time series measured by the single wave gauge behind the sandbar. The dissipation coefficient K_L is derived from

K_{L} = \sqrt{1 - K_{R}^{2} - K_{T}^{2}}

, a formula based on the conservation of energy.

2.5.4. Wave Runup

As an important dynamic factor in the swash zone, wave runup represents the vertical distance between the maximum water level elevation and the still water level on the foreshore. It can generally be estimated by the incident wave height, wave period, and nearshore topography, such as the slope at the wet–dry zone in the swash zone [52,53,54]. Based on a large amount of observational data for offshore wave conditions and wave runup at the same time, numerous parametric empirical formulas have been proposed to predict wave runup [52,55,56,57,58]. Stockdon et al. [52] proposed the runup parameterizations (hereinafter referred to as S2006) based on field experiments on natural beaches by fitting a large amount of wave runup observational data with observed offshore significant wave height, wave period, and beach slope, as follows.

R_{2 %} = 1.1 (〈η〉 + S / 2)

(6)

S = \sqrt{{(S_{short})}^{2} + {(S_{long})}^{2}}

(7)

〈η〉 = 0.35 β {(H_{0} L_{0})}^{1 / 2}

(8)

S_{short} = 0.75 β {(H_{0} L_{0})}^{1 / 2}

(9)

S_{l o n g} = 0.06 {(H_{0} L_{0})}^{1 / 2}

(10)

where R_2% is the 2% cumulative frequency wave runup;

〈η〉

is the mean water level at the shoreline (maximum wave setup). S represents the significant swash height, where Fourier transform is utilized to obtain the wave spectrum from the time series of waterline edge elevation, and then, the significant wave height of the swash can be obtained from the wave spectrum. S_short and S_long represent the high- and low-frequency significant wave height of swash, respectively, and their calculation methods are similar to those of swash significant wave height. β is the foreshore slope, H₀ is the deep-water equivalent wave height, and L₀ is the deep-water equivalent wave length.

3. Results and Discussion

Based on the XBeach model results, this section analyzes the cross-shore change of wave height, mean water level, wave skewness and asymmetry, and depth-averaged flow over the nourished beach with and without the artificial sandbar.

3.1. Wave Transformation Under Normal Waves

3.1.1. Wave Height

The cross-shore variation of the total, short wave and long wave significant wave heights under normal waves are shown in Figure 4. On both profiles with and without the sandbar, the variation in the total H_s (hereafter referred to as H_{s_total}) and the short wave H_s (hereafter referred to as H_{s_short}) are almost the same, indicating that wave energy is predominantly in the form of short waves, with relatively small energy in the long wave band, i.e., short and long waves constitute 98.45% and 1.55% of the incident wave energy. Without the sandbar (Figure 4a), the H_{s_total} and the H_{s_short} decrease uniformly in the shoreface zone (0 < X < 20.55 m) due to the seabed friction at 0.15%. In the beach face zone (X > 20.55 m), H_{s_short} increases slightly due to wave shoaling and then decrease rapidly because of wave breaking. Long wave H_s (hereafter referred to as H_{s_long}) remains stable in the shoreface zone and then increases sharply in the beach face zone due to energy transfer from breaking waves before gradually decreasing as the breaking subsides. The proportion of total incident significant wave height (incident significant wave height hereafter referred to as H₀) for the H_{s_short} at the shore has decreased by 56.10%, while the proportion of H₀ for the H_{s_long} has increased by 2.10%. At this time, the H_{s_total} reduced by 57.71% compared to the total H₀.

The sandbar (Figure 4b) shows a similar initial attenuation of H_{s_total} and H_{s_short} on the seaward side of the sandbar (0 < X < 8 m). Within the sandbar area (8 < X < 18 m), there is a slight increase due to wave shoaling on the fore slope of the sandbar, followed by a rapid decrease due to wave breaking on the crest, with the cross-shore attenuation rate increasing to 0.41%. Behind the sandbar (18 < X < 20.55 m), the attenuation rate gradually decreases to 0.10%. In the beach face zone (X > 20.55 m), the trend is similar to the beach profile without a sandbar. H_{s_long} remains nearly constant in front of the sandbar, increases due to the enhanced wave breaking in the sandbar area, and then decreases as breaking weakens. It slowly increases behind the sandbar and rapidly increases again due to intensified wave breaking in the beach face zone, reaching 1.47 times the incident long wave height. In both scenarios, the proportion of H_{s_long} at the shore significantly increases, which may be due to the phase differences created by nonlinear interactions during the propagation of wave groups, leading to the growth of bound long waves [59]. With the sandbar, the proportion of total H₀ for the H_{s_short} at the shore has decreased by 62.32%, while the proportion of H₀ for the H_{s_long} has increased by 3.19%, with the H_{s_total} reduced by 61.94% compared to the total H₀, the H_{s_total} at the shore is 89.72% of that without artificial sandbar.

Compared to the case without a sandbar, a similar total and the H_{s_short} at the shoreline but a larger H_{s_long} are observed for the beach profile with the sandbar, indicating the artificial sandbar plays a significant role in wave attenuation by reducing the short wave height reaching the shore. Note that measurements of H_{s_long} reaching the shore may overestimate the true proportion due to flume resonance and long wave reflection.

3.1.2. Wave Setup

When the still water level is subtracted from the mean water level at each cross-shore location, the difference between these two is the wave setup. The maximum wave setup can be identified by locating the point where the mean water level reaches its peak, usually near the shoreline.

The cross-shore variation in the time mean water level on the beach without a sandbar is shown in Figure 5a. In the shoreface zone (0 < X < 20.55 m), the mean water level remains almost constant, coinciding with the still water level. In the beach face zone, the mean water level increases due to the wave radiation stress gradient caused by wave breaking, which is known as the wave setup. The maximum value of the wave setup at the shoreline is 0.134 m. As shown in Figure 5b, the variation characteristics on the beach with the sandbar are similar to those on the beach without a sandbar, with the maximum wave setup at the shoreline at 0.104 m. Therefore, the artificial sandbar can reduce the wave setup by 22.39%.

3.1.3. Wave Skewness and Wave Asymmetry

Figure 6a shows the cross-shore variation in wave skewness and asymmetry on the beach profile without a sandbar. In the shoreface area (0 < X < 20.55 m), both wave skewness and wave asymmetry are close to zero, suggesting minimal wave nonlinearity and consistent waveform. As the wave propagates into the beach face zone (X > 20.55 m), the wave skewness rises to a peak value of 0.94 at the submerged terrace with the decrease in water depth and then reduces slightly. In contrast, wave asymmetry becomes negative (indicating pitch shoreward wave face). The absolute value of wave asymmetry increases with the decreasing water depth and reaches a peak value of 1.27 at the submerged terrace and then declines slightly. Therefore, the wave skewness and wave asymmetry are dominated by the water depth through the change in relative submergence (relative free board), indicating that shallow water induces deformation and intensifies the nonlinearity of waves. In the presence of the artificial sandbar (Figure 6b), wave skewness and asymmetry on the seaward side mirror the BF profile. As waves propagate towards the shore, wave skewness experiences an increase and a subsequent decrease twice for the decreasing water depth in the artificial sandbar area (8 < X < 18 m) and the beach face zone (X > 20.55 m), respectively. The peak wave skewness is 0.91 at the sandbar and becomes 0.83 at the shore. The variation trend of wave asymmetry is opposite to that of wave skewness. This result aligns with the numerical and experimental data on wave transformation over natural or man-made low crest structures, as presented by Zou and Peng [42] and Peng et al. [43] as well as the numerical model results of wave evolution over an artificial reef as detailed by Kuang et al. [36].

The sum interactions among waves dominate above the front slope and on the structure crest, but the difference interaction dominates above the rear slope of a low crest structure, such as a sandbar, a reef, and breakwaters. The peak wave asymmetry is −0.20 at the sandbar and −1.21 at the shore. Due to the foreslope and backslope of the artificial sandbar being 1:10, the variation in wave skewness and wave asymmetry is relatively moderate. The artificial sandbar can reduce the wave skewness and asymmetry by 11.70% and 4.89% upon reaching the shore. The correlation between water depth, wave skewness, and wave asymmetry on the beach with the sandbar also demonstrates that water depth and relative submergence of the sandbar determines the degree of wave deformation or wave nonlinearity.

3.2. Upper, Lower, and Depth-Averaged Velocity

Figure 7a shows the cross-shore variation in the depth-averaged current velocity, upper and lower layer velocities, and their difference for the case without a sandbar. In the shoreface zone (0 < X < 20.55 m), these velocities are small and negative (seawards), indicating the minimal wave nonlinearity over the shoreface, resulting in a small undertow that balances the onshore Stokes mass drift current. In the beach face zone (X > 20.55 m), velocities are more negative, indicating stronger seaward bottom currents, also known as undertow, which are generated due to the mass transport driven by the breaking waves. The undertow occurs as the broken waves push water shoreward in the upper layer, and the water returns seaward in the lower layer to maintain mass balance. Additionally, the depth-averaged current velocity and the upper and lower layer velocities increase first and then decrease in the beach face zone, consistent with the occurrence of wave breaking. The upper layer velocity is greater than the depth-averaged current velocity across the beach profile, while the lower layer velocity is lower. This difference highlights the effect of wave breaking, where water pushed shoreward in the upper layer generates an offshore-directed undertow in the lower layer. Under this circumstance, the upper and lower layer flows display a shoreward and seaward velocity relative to the depth-averaged current velocity, respectively.

As shown in Figure 7b, on the seaward side of the artificial sandbar, the depth-averaged current velocity and the velocities of the upper and lower layers are small and all negative (seawards), reflecting minimal wave radiation stress gradient and wave nonlinearity. Both the depth-averaged velocity and the upper and lower velocities are negative (seawards), and they increase first and then decrease over the sandbar (8 < X < 18 m). On the shoreward side of the artificial sandbar, these velocities are small and all negative (seawards). In the beach face zone (X > 20.55 m), the depth-averaged velocity and the upper and lower velocities are all negative, similar to the case without a sandbar shown in Figure 7a but with smaller magnitudes. This result indicates that the artificial sandbar can reduce the depth-averaged velocity.

The comparison between the numerical results for beach with and without an artificial sandbar under normal wave conditions, indicates that the variations in wave skewness, wave asymmetry, and current velocity are also relatively smooth because of the gentle foreslope and backslope of the 1:10 ratio of the sandbar. By inducing wave shoaling and wave breaking in the sandbar area, the artificial sandbar contributes to a considerable decrease in lower velocity over the beach face by 10% and average velocity by 9.5%.

Artificial sandbar significantly impacts wave propagation and hydrodynamics by reducing the H_{s_short} and increasing the H_{s_long} near the shoreline. It also decreases the wave skewness and asymmetry at the shore and reduces the wave setup by 22%. The sandbars, through wave shoaling and breaking, notably lower velocities in the beach face (X > 20.55 m). These findings are consistent with the cross-shore physical model test of the Beidaihe submerged sandbar beach project by Wu et al. [60], that is, the nearshore artificial sandbar can reduce the wave by more than 30%. Han et al. [61] also confirmed that the sandbars weaken the nearshore current and therefore hydrodynamic forces in the coastal region adjacent to the beach.

3.3. Wave Responses to the Artificial Sandbar Under Storm Wave Conditions

To simulate storm wave conditions, the incident wave conditions in the following modeling are based on the maximum significant wave height in the offshore area under field conditions. According to the wave data by field observation from 2017 to 2019 in the offshore area of Qinhuangdao, the maximum H_s ranged between 1.6 and 2.2 m [33]. The wave height scale and the wave period scale of the experiment are set as 1:10 and 1:

\sqrt{10}

, respectively. Then a series of H₀ are set in the laboratory experiments for investigations on the responses of the artificial sandbar to various incident wave conditions. The H₀ is set as 0.14 m, 0.16 m, 0.18 m, 0.20 m, 0.22 m, and 0.24 m, respectively, with the spectral peak wave period T₀ by 1.57 s uniformly, and deep-water wave length L₀ is 3.85 m. Both the cases with and without an artificial sandbar are included to point to the influence mechanisms of the artificial sandbar on wave evolution during storm conditions.

Figure 8 shows the H_{s_total}, H_{s_short}, and H_{s_long} on the beach profile without a sandbar under storm wave conditions. To obtain more accurate results, the H_{s_total}, H_{s_short}, and H_{s_long} at different positions of the beach profile are compared with the H₀. The ratios between them are shown in Table 3. According to Figure 8 and Table 3, at the offshore position (X = 0 m), the H_{s_short} from Case 1 to Case 3 is almost same as the H₀, while the H_{s_long} is minimal (around 10% of H₀₎. As H₀ increases, the ratio of H_{s_short} decreases; when the H₀ is 0.24 m, the ratio of H_{s_short} to H₀ is at its minimum of 85%, and H_{s_long} becomes more significant and attains the peak value of 50% of H₀, although the short wave remains dominant over long waves. This means that the seaward part of the beach is directly affected by the incident short waves, while the impact of the incident long waves is small but increases with increasing H₀. In the shoreface zone, the H_{s_total} decreases with the attenuation rate that varies with H₀ increases. For H₀ under 0.18 m (Cases 1 to 3), the shore sees a 30% reduction in H_{s_total} and H_{s_short}. For larger H₀ (Cases 4 to 6), attenuation exceeds 40%, with H_{s_short} experiencing the greatest decrease, reaching over 57%. H_{s_long} varies slightly across the shore and its proportion in total wave energy increasing as H_{s_total} and H_{s_short} decrease. This increase is gradual with rising H₀, is sharp between Cases 3 and 4, and is comparable to the short wave energy in Case 6. The rise in long wave energy is attributed to enhanced wave-breaking effects, which are more pronounced from Case 4 to Case 6. The variation trend of long waves resembles the prediction by the unified infragravity wave solution by Liao et al. [62].

Correspondingly, the H_{s_long} increases with increasing H₀. The H_{s_long} increases from Case 1 to Case 3, then sharply between Case 3 and Case 4, and the increase remains consistent from Case 4 to Case 6. H_{s_long} experiences minimal attenuation across the shore under storm wave conditions. The long wave energy of the total wave energy grows while the short wave energy declines from Case 1 to Case 6, attaining similar magnitude in the beach face in Case 6. This shift is due to the seaward expansion of the breaking wave and enhanced wave breaking with rising wave heights, which slightly increase from Case 1 to Case 3 and sharply increase from Case 4 to Case 6. Consequently, the long wave energy from wave breaking rises slightly initially and then sharply, causing a gradual then sharp increase in its proportion of the total wave energy between Case 3 and Case 4.

The H_{s_total}, H_{s_short}, and H_{s_long} of the beach profile with the sandbar are presented in Table 4 and Figure 9. In the shoreface area in front of the sandbar, the magnitude and variation characteristics of the H_{s_total}, H_{s_short}, and H_{s_long} are consistent with those of the beach without a sandbar. In the sandbar area, due to the shallow water effect on the front slope of the sandbar and the wave dissipation effect on the top of the sandbar, the cross-shore variation of the H_{s_total}, H_{s_long}, and H_{s_short} all exhibit a trend of slightly increasing first and then decreasing. After passing the sandbar, the H_{s_total} and H_{s_short} decrease rapidly. When reaching the shore, the H_{s_total} decays by about 50%, and the attenuation H_{s_short} increases with increasing H₀, reaching a maximum reduction of 65.12% for H₀ of 0.24 m. Behind the sandbar, the H_{s_long} is slightly lower than the case without a sandbar with a slight cross-shore variation. Similarly, the proportion of long wave energy in the total wave energy gradually increases towards the shore.

Comparing the scenarios with and without a sandbar, the wave characteristics in the shoreface area and seaward of the sandbar do not change significantly. However, due to the depth limit effect and bottom friction resistance on the foreslope of the sandbar and the wave dissipation at the top of the sandbar, the H_{s_total} and the H_{s_short} are greatly attenuated by about 12% more than the same position of the beach profile without a sandbar at Case 1 to Case 3 and about 8% more than the beach profile without a sandbar at Case 4 to Case 6. The sandbar significantly attenuates H_{s_total} and H_{s_short} for H₀ from 0.14 to 0.18 m, and the sandbar causes an additional 15% attenuation in the H_{s_total} reaching the shore and about 10% for H₀ from 0.20 to 0.24 m. The attenuation of the H_{s_short} is similar to that of the total wave height, and H_{s_long} attenuation is similar in both scenarios, indicating the sandbar primarily protects the beach by damping the short wave energy, significantly reducing the total wave height that ultimately reaches the shoreline.

3.4. Relationship Between Wave Coefficients and Mean Water Level

The reflection coefficient K_R, transmission coefficient K_T, and dissipation coefficient K_L under 6 storm wave conditions are listed in Table 5.

As depicted in Figure 10, the reflection coefficient remains low and almost unchanged from Case 1 to Case 3. It increases rapidly when the H₀ exceeds 0.18 m. The reflection coefficient in Case 4 is 1.92 times that of Case 3, likely due to the increased wave action depth with a larger H₀, which enhances the reflective effect of the front slope of the sandbar on the waves. The transmission coefficient follows a similar trend to the reflection coefficient, initially showing a downward trend as wave breaking gradually increases. When the H₀ exceeds 0.18 m, the transmission coefficient changes from a decreasing to an increasing trend, indicating an increase in energy transmitted to the sandbar at this point. The dissipation coefficient, derived from the principle of energy conservation, shows an opposite trend to the transmission coefficient, first increasing and then decreasing; the turning point also occurs at the H₀ of 0.18 m. To further analyze the cause of this turning point, the mean water level behind the sandbar in each case was calculated, as shown in Figure 10. When the H₀ exceeds 0.18 m, the mean water level also increases by a large margin, which corresponds to the turning point of the wave characteristic coefficient. It is consistent with the conclusion by Cong et al. [63] that the mean water level in the lagoon behind the sandbar corresponds to the turning point of the reflection coefficient.

3.5. Wave Runup for Nourished Beaches Under Storm Wave Conditions

Storm waves can cause intensive uprush and downwash in the swash zone and determine local geomorphologic evolution. The field data used in the derivation of the Stocken et al. [52] formula S2006 are limited, and the measurements during storms are not included [64], so the accuracy of the wave runup prediction according to the XBeach simulation results is not enough. Moreover, the data for S2006 all derive from natural beaches and exclude the effects of hard or soft protection in beach nourishment projects. However, the introduction of beach nourishment projects may also cause actual wave runup to differ from S2006. Therefore, this study employs the non-hydrostatic mode of XBeach based on model experiments of the beach nourishment project to simulate the wave propagation under the storm waves. The wave runup calculated from the XBeach simulation results is compared with S2006, and the formulas for wave runup under the storm wave conditions are proposed ultimately, as follows.

R_{2 %} = (〈η〉 + S / 2) + 0.0022

(11)

〈η〉 = 2.4376 β {(H_{0} L_{0})}^{1 / 2} - 0.3879

(12)

S_{s h o r t} = 0.2763 β {(H_{0} L_{0})}^{1 / 2} + 0.0204

(13)

S_{l o n g} = 0.5145 {(H_{0} L_{0})}^{1 / 2} - 0.3879

(14)

The relationship between the simulated R_2% and

〈η〉

+ S/2 is shown in Figure 11a. Compared to S2006, the slope of the linear relationship between R_2% and

〈η〉

+ S/2 under storm wave conditions is slightly smaller. Figure 11b illustrates the simulated correlation between

〈η〉

and β(H₀L₀)^1/2, which compares well with the S2006 formula under slight storm waves, but as the waves intensify, the slope increases sharply and becomes significantly larger than that of S2006. This discrepancy may be due to the fact that S2006 was derived from normal wave conditions. Under storm wave conditions, the wave-breaking effect is significantly enhanced, leading to a sharp increase in the wave setup; the maximum wave setup (i.e., the time-average water level

〈η〉

at the shoreline) becomes substantially higher in storm wave conditions than the prediction by S2006. Figure 11c shows the simulated relationship between S_short and β(H₀L₀)^1/2. The slope matches S2006 under mild storm waves but decreases as the wave height increases. This change is attributed to the decrease in the short wave energy relative to the total wave energy as waves grow larger, with a particularly sharp decline under storm wave conditions. Figure 11d presents the simulated relationship between S_long and β(H₀L₀)^1/2. The slope becomes significantly larger than the S2006 as the waves increase. This is due to the proportion of long wave energy in the total wave energy increasing as wave heights gradually increase, with a particularly sharp rise in this proportion under storm wave conditions, which is in contrast to what is shown in Figure 11c.

Although the non-hydrostatic model captures the wave propagation and wave runup well, it tends to underestimate the long wave heights. It may be due to the resonance of long wave in the flume and the model sensitivities to the maximum wave steepness criterium (maxbrsteep) for wave breaking onset. A one- and two-phase Reynolds Averaged Navier–Stokes solver (RANS) and a Volume of Fluid (VOF) surface capturing scheme may directly resolve the breaking point, breaking wave profile deformation, turbulence, and morphological changes with and without a reef (Lin and Liu [65], Lara et al. [66], Wang et al. [67], Xin et al. [68], Peng et al. [69]). A future XBeach model should incorporate vertical multi-layer grids and account for the cross-shore varying bed friction coefficients in order to be a more reliable tool for beach protection and management.

4. Conclusions

To investigate the impact of artificial sandbars on wave dynamics of a nourished beach, a non-hydrostatic XBeach model is established to simulate the previous experiment of Ma [33], the beach with or without the artificial sandbar. The well-validated non-hydrostatic model of XBeach is used to explore the wave dynamic and wave nonlinearity response to an artificial sandbar over the beach under different storm waves. Based on the non-hydrostatic XBeach model results, the empirical formula is extracted for the 2% cumulative frequency wave runup as well as the mean water level at the shoreline, high-frequency short wave height, and low-frequency long wave height.

The conclusions are as follows:

Under normal wave action, the wave energy is predominantly short-wave energy, with minimal long-wave energy. The wave setup is negligible near the shoreline but significant on the beach face, where shallower depths increase wave nonlinearity, therefore increasing skewness and asymmetry.
Under normal wave conditions, the artificial sandbar enhances the wave nonlinearity, causing dramatic changes in wave skewness and asymmetry across the sandbar and reducing the wave setup by 22%; causing significant wave height, wave skewness, and wave asymmetry; and causing the flow velocity to have a significant downward trend over the artificial sandbar.
Under storm wave conditions, the increase in incident wave height leads to an expansion of the wave-breaking seaward, enhanced wave-breaking, which in turn increases the long-wave energy at the expense of short wave energy in the total wave energy.
Under storm wave conditions, the wave dissipation coefficient over the artificial sandbar remains above 0.65 and changes slightly within a range less than 0.105. The artificial sandbar shows good performances in protecting beaches from excessive erosion under storm wave conditions.
Under storm wave conditions, wave breaking is intensified rapidly by the sandbar, which increased the wave runup and the proportion of long wave energy significantly. The mean water level at the shoreline, high-frequency surge wave height, and low-frequency surge wave height are greater than, less than, and greater than the empirical formula prediction, respectively, under normal wave conditions.

A non-hydrostatic hydrodynamic–morphological Xbeach model was applied to simulate the laboratory results of wave transformation and runup over a nourished beach with artificial sandbars. Based on the Xbeach model results and the wave runup empirical formula S2006 by Stockdon et al. [52], the wave runup formula under storm wave conditions has been proposed as a new theoretical tool for the construction and management of natural and artificial coastal defenses. Future studies can be dedicated to enhancing the accuracy and reliability of non-hydrostatic Xbeach models, especially in simulating complex coastal dynamic processes to achieve more precise predictions. The physical experiment may overlook the effects of flume resonance and wave reflection on the proportion of long and short waves reaching the shore; these aspects should be addressed in future studies.

For the simplicity of experimentation and model execution, symmetric sandbars are used in the present experiments and model. However, in natural conditions, sandbars are often asymmetric. In subsequent research, it is necessary to consider asymmetric sandbars to improve the applicability of the model.

Author Contributions

Conceptualization: C.K., L.C., and X.H.; methodology: C.K., L.C., X.H., and Q.Z.; simulations: L.C. and X.H.; validation L.C. and X.H.; Data analysis: X.H., D.W., and D.C.; writing—original draft preparation, C.K., L.C., and X.H.; writing—review and editing, C.K. and Q.Z.; funding acquisition: C.K. and Q.Z. All authors have read and agreed to the published version of the manuscript.

Funding

The research was financially supported by the National Key Research and Development Program of China (No. 2022YFC3106205) and the National Natural Science Foundation of China (No. 41976159). Professor Qingping Zou has been supported by the Natural Environment Research Council of UK (Grant No. NE/V006088/1).

Data Availability Statement

The data that has been used are confidential.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Experimental layout and beach profile types: (a) without and (b) with an artificial sandbar (W1 to W13 indicates wave gauges). The blue horizontal line represents the still water level, and the yellow part is the beach profile.

Figure 2. Comparisons of predicted and observed (a–m) wave spectra and (n) significant wave height over the beach profile without an artificial sandbar under normal wave conditions.

Figure 3. Comparisons of predicted and observed (a–m) wave spectra and (n) significant wave height over the beach profile with an artificial sandbar from X = 10 m to X = 22 m under normal wave conditions.

Figure 4. H_{s_total}, H_{s_short}, and H_{s_long} under normal wave action on the beach (a) without and (b) with an artificial sandbar from X = 8 m to X = 18 m.

Figure 5. Mean water level under normal wave conditions on a nourished beach (a) without and (b) with an artificial sandbar from X = 8 m to X = 18 m.

Figure 6. Wave skewness and wave asymmetry under normal wave conditions over a beach (a) without and (b) with an artificial sandbar (8 < X < 18 m).

Figure 7. Depth-averaged velocity, upper and lower layer velocities, and their difference under normal wave conditions over a beach (a) without and (b) with an artificial sandbar (8 < x < 18 m).

Figure 8. The (a) H_{s_total}, (b) H_{s_short}, and (c) H_{s_long} on the beach profile without an artificial sandbar under 6 storm wave conditions.

Figure 9. The (a) H_{s_total}, (b) H_{s_short}, and (c) H_{s_long} on the beach profile with an artificial sandbar (8 < x < 18 m) under 6 storm wave conditions.

Figure 10. Reflection, transmission, and dissipation coefficients of the artificial sandbar and the mean water level behind the sandbar (W3, W4, and W5) under 6 storm wave conditions.

Figure 11. Simulated values of (a) wave runup of 2% cumulative frequency, (b) mean water level of wave runup, (c) high-frequency significant swash height, and (d) low-frequency significant swash height under storm wave conditions.

Table 1. Model setup of validated non-hydrostatic XBeach morphodynamic model.

Wave Model	Non-Hydrostatic Mode (Random Wave) Wavemodel = Nonh Input Parameter
Grid numbers	nx = 1605
Time step	CFL = 0.2 tstop = 5400 s
Physical constants	rho = 1000 kg/m³ depthscale = 10–100
Wave boundary condition	wbctype = parametric
Flow boundary condition	front = abs_1d back = abs_1d left = wall right = wall
Tide boundary condition	tideloc = 0 zs0 = 0.6 m
Bed friction	bedfriction = cf bedfriccoef = 0.05–0.14
Non-hydrostatic correction	nonhq3d = 1 maxbrsteep = 0.4
Sediment transport	form = vanrijn1993 tsfac = 0.3
Bed composition	por = 0.4 D₅₀ = 0.00017 m D₉₀ = 0.00018 m rhos = 1430 kg/m³ sedcal = 0.2–0.7
Morphology	morfac = 1 wetslp = 0.2 dryslp = 30

Table 2. Model performance: the measures and evaluation criteria of significant wave height on the beach profile without and with a sandbar under normal wave action.

Significant Wave Height	RMAE	RMAE Evaluation Criteria
The beach without sandbar	0.0292	Excellent
The beach with sandbar	0.0327	Excellent

Table 3. The ratio of H_{s_total}, H_{s_short}, and H_{s_long} to the H₀ on the beach profile without an artificial sandbar at the offshore (X = 0 m), the sandbar position (X = 11.5 m, X = 14.5 m) corresponding to the beach with an artificial sandbar, and the shoreline (X = 20.55 m).

	H₀ (m)	Offshore Position			Without Artificial Sandbar						Shoreline
		X = 0 m			X = 11.5 m			X = 14.5 m			X = 20.55 m
		$\frac{H_{s_t o t a l}}{H_{0}}$	$\frac{H_{s_s h o r t}}{H_{0}}$	$\frac{H_{s_l o n g}}{H_{0}}$	$\frac{H_{s_t o t a l}}{H_{0}}$	$\frac{H_{s_s h o r t}}{H_{0}}$	$\frac{H_{s_l o n g}}{H_{0}}$	$\frac{H_{s_t o t a l}}{H_{0}}$	$\frac{H_{s_s h o r t}}{H_{0}}$	$\frac{H_{s_l o n g}}{H_{0}}$	$\frac{H_{s_t o t a l}}{H_{0}}$	$\frac{H_{s_s h o r t}}{H_{0}}$	$\frac{H_{s_l o n g}}{H_{0}}$
Case 1	0.14	100%	99.21%	7.43%	83.93%	83.71%	6.43%	80.14%	79.93%	6.29%	72.64%	72.35%	6.57%
Case 2	0.16	100%	98.56%	10.31%	81.88%	81.44%	8.31%	77.75%	77.31%	8.25%	69.88%	69.38%	8.25%
Case 3	0.18	100%	98.06%	11.78%	78.78%	78.17%	9.67%	74.39%	73.78%	9.56%	66.44%	65.78%	9.61%
Case 4	0.20	100%	95.65%	25.85%	70.40%	67.60%	19.65%	65.75%	62.90%	19.25%	58.65%	54.85%	20.85%
Case 5	0.22	100%	91.18%	39.68%	69.68%	61.45%	32.77%	65.32%	56.82%	32.23%	59.41%	48.50%	34.32%
Case 6	0.24	100%	85.21%	50.63%	70.67%	54.96%	44.42%	66.75%	50.29%	43.88%	62.88%	42.58%	46.29%

Table 4. The ratio of H_{s_total}, H_{s_short}, and H_{s_long} to H₀ on the beach profile with an artificial sandbar at the offshore position (X = 0 m), the sandbar position (X = 11.5 m, X = 14.5 m), and the shoreline position (X = 20.55 m).

	H₀ (m)	Offshore Position			Artificial Sandbar						Shoreline
		X = 0 m			X = 11.5 m			X = 14.5 m			X = 20.55 m
		$\frac{H_{s_t o t a l}}{H_{0}}$	$\frac{H_{s_s h o r t}}{H_{0}}$	$\frac{H_{s_l o n g}}{H_{0}}$	$\frac{H_{s_t o t a l}}{H_{0}}$	$\frac{H_{s_s h o r t}}{H_{0}}$	$\frac{H_{s_l o n g}}{H_{0}}$	$\frac{H_{s_t o t a l}}{H_{0}}$	$\frac{H_{s_s h o r t}}{H_{0}}$	$\frac{H_{s_l o n g}}{H_{0}}$	$\frac{H_{s_t o t a l}}{H_{0}}$	$\frac{H_{s_s h o r t}}{H_{0}}$	$\frac{H_{s_l o n g}}{H_{0}}$
Case 1	0.14	100%	97.07%	7.14%	82.21%	81.86%	7.86%	70.50%	70.14%	6.93%	57.50%	57.14%	6.07%
Case 2	0.16	100%	96.19%	10.19%	78.31%	78.31%	10.00%	65.69%	65.31%	7.13%	52.75%	52.38%	6.25%
Case 3	0.18	100%	96.39%	11.56%	75.72%	74.89%	11.28%	61.50%	60.89%	8.56%	48.67%	48.11%	7.39%
Case 4	0.20	100%	94.55%	25.70%	69.45%	66.10%	21.20%	56.85%	54.25%	16.95%	45.85%	42.85%	16.30%
Case 5	0.22	100%	89.91%	39.32%	69.18%	60.64%	33.27%	57.14%	49.14%	29.09%	48.50%	38.77%	29.09%
Case 6	0.24	100%	84.83%	49.08%	70.67%	54.79%	44.58%	59.96%	44.54%	40.08%	53.46%	34.88%	40.50%

Table 5. Reflection, transmission, and dissipation coefficients of the artificial sandbar under 6 storm wave conditions.

Profile	With Sandbar
Case No.	K_R	K_T	K_L
1	0.079	0.684	0.725
2	0.078	0.644	0.761
3	0.076	0.610	0.789
4	0.145	0.622	0.769
5	0.177	0.652	0.737
6	0.187	0.705	0.684

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Kuang, C.; Chen, L.; Han, X.; Wang, D.; Cao, D.; Zou, Q. Impacts of an Artificial Sandbar on Wave Transformation and Runup over a Nourished Beach. Geosciences 2024, 14, 337. https://doi.org/10.3390/geosciences14120337

AMA Style

Kuang C, Chen L, Han X, Wang D, Cao D, Zou Q. Impacts of an Artificial Sandbar on Wave Transformation and Runup over a Nourished Beach. Geosciences. 2024; 14(12):337. https://doi.org/10.3390/geosciences14120337

Chicago/Turabian Style

Kuang, Cuiping, Liyuan Chen, Xuejian Han, Dan Wang, Deping Cao, and Qingping Zou. 2024. "Impacts of an Artificial Sandbar on Wave Transformation and Runup over a Nourished Beach" Geosciences 14, no. 12: 337. https://doi.org/10.3390/geosciences14120337

APA Style

Kuang, C., Chen, L., Han, X., Wang, D., Cao, D., & Zou, Q. (2024). Impacts of an Artificial Sandbar on Wave Transformation and Runup over a Nourished Beach. Geosciences, 14(12), 337. https://doi.org/10.3390/geosciences14120337

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