The experiment was conducted at the Liubao Experimental Base of the Zhejiang Institute of Hydraulics & Estuary, using a wave flume measuring 70 m in length, 1.2 m in width, and 1.7 m in height. The experimental setup is shown in Figure 1 . The sandbar has a 1.0 m length, 0.5 m height, and a 1:2 slope on both the front and rear. The beach face is 0.8 m high with a 1:5 slope, supported by hard wooden boards at the back. To minimize the impact of the water flume width on test results, the water flume test section was divided, with the outer 0.5 m being used as the effective testing area. The flume floor consists of a concrete platform, 0.2 m high, 10 m long, and sloping at 1:20 at the front. The test area lies above this platform. Wave data were recorded using eight capacitive wave height gauges (W1-W8) at a sampling frequency of 50 Hz, with their layout shown in Figure 1 . The entire experimental process was recorded by four DSLR cameras, positioned near the sandbar, trough, and beach face. Contour measurements were taken through image captures. Before each test, the model was manually restored to its initial profile. The tests generally lasted for 120 minutes, except in cases where erosion and deposition occurred rapidly. Evolution of the model was driven by regular waves, with 20 different wave height-period combinations tested, as detailed in Table 1 . Based on marine hydrological data collected by Gong (2023) and Zhang (2021) in Zhejiang Province, the prototype wave’s effective wave height (H_1/3) ranged from 0.2 m to 1 m, and the effective wave period (T_1/3) ranged from 2 s to 8 s. Wave parameters for the experiments were designed accordingly ( Table 1 ). The model uses the following coordinate system ( Figure 1 ): the foot of the sandbar slope is the horizontal origin, wave propagation is along the positive X-axis, and the vertical upward direction is the positive Z-axis.

The model sand selected for this study is lightweight walnut sand. To ensure similarity in sediment initiation, settling, and hydrodynamics between the prototype and the physical model, it is essential to satisfy the Froude number, Shields number, and Ross number. Based on the modified Shields curve by Soulsby (1997) and the falling velocity of non-spherical particles according to Van Rijn (1993), a proportional relationship that ensures similarity in sediment initiation, sedimentation, and Froude criteria was derived (Van Rijn, 1993; Soulsby, 1997; Yin et al., 2017; Pan et al., 2019). The initial relationship is given by:

where ψ c is the critical Hertz number, D * is the size parameter of a dimensionless particle, D is the particle size of a sediment particle, ω S is the settling velocity of a sediment particle, ρ s is the dry density of sediment particles, ρ is the density of seawater, s = ρ s / ρ , v is the viscosity coefficient of water body motion, g is the gravitational acceleration, τ m c is the critical shear stress amplitude at the bottom, τ m c = 0.5 ρ f w U m c 2 , where f w is the wave friction coefficient and U m c is the critical trajectory velocity amplitude at the bottom.

Using Jonsson (1980) theory, the wave friction coefficient is calculated as:

where a m c represents the maximum motion amplitude of the bottom water particle.

For this experiment, the natural and model sand materials are within the sediment particle size range of 0.1-0.5 mm, and the dry density range of sediment is 1.2-2.65. Within these ranges, Equations 1, 4 were numerically fitted (Pan et al., 2019; Cong et al., 2023a), resulting in the following expressions:

According to the wave similarity criterion, λ U m c = λ h , λ a m c = λ L , where λ L represents the geometric scale.

By combining these formulas, the sediment initiation and settling ratio relationship for the sediment particle size range of 0.1 mm to 0.5 mm and dry density range of 1.2-2.65 is derived.

In the experimental simulation, Froude similarity must also be satisfied, which requires λ ω = λ L 0.5 .

Due to the limitations of the water tank test platform, the geometric scale used in this study is 8:1. Based on this, the value of λ_(s-1) is calculated to be 7.2:1. With a natural sand density of 2.65, the model sand density is calculated to be 1.23. Based on research on the particle size characteristics of beach sediments in Zhejiang (Gong, 2023; Zhang, 2021), the D₅₀ for the prototype sand is determined to be 0.2 mm, and the model sand particle size is calculated to be 0.27 mm. After multiple attempts, walnut shell sand with a particle size of 0.25 mm (s=1.2-1.3) was selected. After soaking and stabilizing, the dried walnut sand reaches an approximate particle size of 0.27 mm.

The images obtained are first preprocessed through cropping and correction Single-pixel marker points are then used to delineate the contours of the cross-section in each image. Origin data processing software is employed to extract the pixel coordinates of these marker points. Finally, these pixel coordinates are converted into the actual coordinate system and plotted as a graph.

In wave data analysis, each test utilized over 10,000 sampling points with a sampling frequency of 50 Hz. By analyzing the statistical characteristics of wave height and performing spectral analysis, the significant wave height H_s was determined. The formula used is as follows:

where m₀ represents the zero-order moment of the spectrum.

The wave reflection coefficient K r is derived by separating the amplitudes of the incident and reflected waves using the two-point method proposed by Goda and Suzuki (1977). This analysis is based on wave data collected from W3 and W4 wave height meters. The following formula was used for calculation:

where N is the total number of samples, and a i and a r denote the amplitudes of the incident wave and reflected wave, respectively.

As waves propagate towards the shore, the water depth decreases, leading to changes in waveforms and, in some cases, nonlinear phenomena such as wave breaking. The nonlinearity of the waves can be quantified by the skewness (Sk) and asymmetry (Asy) of the waveforms, which is as follows:

where η is the elevation of the free wave surface, η ¯ is the corresponding average value, and ℋ represents Hilbert transform.

Spectral analysis of the collected wave data was performed to examine the variation in effective wave height along the wave propagation path ( Figure 2 ). The figure illustrates that, as waves pass through the sandbar towards the seaward side, the effective wave height slightly decreases. Subsequently, due to the shallower water depth, wave shape deforms, and local wave heights increase. As the wave height reaches a critical level, it begins to decay again. This phenomenon is more pronounced for smaller input wave heights (H=0.025 m, H=0.05 m). In contrast, when the input wave height is larger (e.g., H=0.075 m, H=0.100 m), the fluctuation amplitude of the effective wave height in the area seaward of the sandbar is more significant, indicating stronger interactions between waves and the sandbar. Particularly for shorter periods, effective wave heights increase when waves reach the seaward side of the sandbar and decay rapidly after passing through it. This is due to the larger wave energy and the erosive effect of the sandbar, which leads to significant sediment deposition near the shore, causing a higher wave water level. Notably, Test19 (H=0.100 m, T=2.3 s) and Test20 (H=0.100 m, T=2.8 s) exhibit distinct wave characteristics, with a marked increase in wave height on the shoreward side of the sandbar. This may be attributed to wave reflection caused by the thicker sedimentation at the beach slope angle.

Figure 3 presents the wave reflection coefficient in front of the sandbar. Overall, the wave reflection coefficient is relatively low, with an average reflectivity of approximately 10%, consistent with the findings of Pan et al. (2022b) on the effects of submerged sandbar models. When the input wave height is small (H=0.025 m), the wave reflection coefficient tends to be higher than that in other cases. This is because wave reflection strength is closely linked to the shape of the sandbar profile. At H=0.025 m, the erosion of the seaward sandbar crest and the minimal sedimentation thickness of the shoreward sandbar crest result in higher wave reflection. In contrast, for groups 2, 3, and 4 (H ≥ 0.025 m), the irregular and undulating profile shapes (as observed in the following section) lead to partial dissipation of wave energy, thereby reducing the overall reflection.

Wave skewness and asymmetry are critical in characterizing wave shapes, playing a key role in understanding and predicting nonlinear wave propagation and coastal sediment transport. Figures 4 , 5 illustrate wave skewness and asymmetry, respectively. Figure 4 shows that at larger periods, the fluctuations in wave skewness amplitude become more pronounced. Notably, wave deviation at W6 is significant, likely caused by shallow deformation or wave breaking. The skewness curve remains stable from W1 to W5, then increases rapidly between W5 and W6, reaching a peak at W6 before sharply decreasing between W6 and W7, and fluctuating again from W7 to W8. These changes are especially evident when the incident wave height is small (H ≤ 0.050 m), particularly at H = 0.025 m. Experimental videos show that most experiments in groups 3 and 4 experienced wave overshoot. Waves climbed up the beach face, crossing its peak (0.8 m high at x = 8 m), and collided with a hard wooden board behind. This scenario resulted in both reflection and backflow, causing significant fluctuations in wave skewness. The wave asymmetry ( Figure 5 ) displays chaotic behavior, with smaller amplitudes compared to the skewness, resembling phenomena observed by Cong et al. (2023b) in the sandbar lagoon system.

To further investigate the impact of wave height and period on profile evolution, Figures 6 and 7 show the final profiles of the model under different conditions: the same wave height but varying periods, and the same period but varying wave heights. As wave height and period increase, more noticeable dynamic changes occur, including intensified erosion of sandbars and beach faces, along with continuous deposition in troughs.

Figures 6A–D display model profiles under constant wave height but different periods (a, H = 0.025 m; b, H = 0.050 m; c, H = 0.075 m; d, H = 0.100 m), illustrating how period influences profile evolution. The data show that, with constant wave height, longer periods significantly promote the evolution of submerged sandbars. Additionally, increasing the period makes the profile more tortuous and irregular. As wave height increases, the profile evolves more intensely and becomes increasingly irregular. At lower wave heights (H ≤ 0.050 m), the submerged sandbars maintain noticeable “symmetry,” with relatively minor erosion and sedimentation on the beach face, especially at H = 0.025 m. At higher wave heights (H ≥ 0.075 m), the top of the submerged sandbar shifts landward due to sediment transport towards the shore, causing the sandbar’s contour to become more asymmetric, with a steeper incline toward the shore and a gentler slope toward the sea. The sediment thickness at the beach slope angle also increases. During testing of the H = 0.100 m group, the high wave energy caused significant sediment transport from the submerged sandbar to the shore, filling the trough with sediment and creating a smoother, more continuous profile. Interestingly, numerous sand ripples appeared at the top of the submerged sandbars, indicating more complex hydrodynamic actions at the sandbar top. To investigate the formation of sand ripples, the frequency and size (length) of sand ripples were recorded for each test ( Table 2 ; Figure 7 ). Sand ripples appeared in nearly all tests except Test 11 and Test 17, concentrating at the sandbar. The formation of sand ripples depends heavily on wave breaking and water flow-induced transport of surface sediment (Ma, 2016). Waves over the sandbar are prone to shallow-water deformation and breakage, driving the laminar movement of surface sediments.

Figures 8A–E show the final profiles of the model under different wave height conditions with the same period (a, T=1.0 s; b, T=1.3 s; c, T=1.8 s; d, T=2.3 s; e, T=2.8 s), highlighting the influence of wave height on profile evolution when the period is constant. It is evident that increasing wave height primarily reduces the top height of the submerged sandbar. For periods up to T ≤ 1.3 s, the sandbar profile exhibits typical asymmetric changes. For periods T ≥ 1.8 s, the profile undergoes significantly larger changes, with greater amplitude in the submerged sandbar compared to those at T=1.0 s and 1.3 s. The increase in period significantly enhances sediment transport along the shore, resulting in a more tortuous profile. Notably, in some cases, such as Test5 and Test10, two peaks form at the sandbar crest, which may be attributed to rolling waves (Section 4.4 for further explanation).

To quantitatively assess the effects of wave height and period on profile topography evolution, the maximum sedimentation and erosion thicknesses of sandbars, troughs, and beach faces were calculated, as shown in Figure 9 . Overall, the maximum sedimentation thickness in sandbars exceeds the maximum erosion thickness. For groups 1 to 4, the maximum sedimentation thickness ranges from 0.08 to 0.38 m, while the maximum erosion thickness ranges from 0.11 to 0.28 m, with an increasing trend in sedimentation and a decreasing trend in erosion. Sedimentation is primarily concentrated on the shoreward side of the sandbar, with minor sedimentation observed at the seaward slope, where thickness remains small. Erosion predominantly occurs at the sandbar crest, particularly on the seaward side. This is due to increased wave height, which transfers more energy to the area above the sandbar, where waves break, causing energy dissipation and sediment transport to the trough. Continuous erosion of the sandbar crest leads to the deposition of suspended sediment on the seaward slope. In most cases, the trough is fully sedimented, with only a few instances of local erosion, particularly with shorter periods. Short-period waves are more likely to break, generating downward scouring currents that erode the sandbar shoreward. For the beach face, both maximum sedimentation and erosion thicknesses increase with wave height. This is because higher wave energy is transferred to the beach face, intensifying erosion and increasing sediment transport offshore to the trough, which also raises the sediment thickness at the beach face’s slope.

The single-width volume of each region, calculated and shown in Figure 9D , reveals a consistent trend: from Group 1 to Group 4, the single-width volume of the sandbar and beach face decreases, while that of the trough increases. This pattern corresponds to changes in the maximum erosion and sedimentation thickness across the three regions. The sediment deposited in the trough originates from the sandbars and beach face, with the sandbars being the primary source of sediment transport. The sediment transport volume (the difference between the volume per unit width of each region and the initial profile’s volume per unit width, in m³/m) shows that the sandbars contribute most of the sediment to the trough, while the contribution from the beach face is comparatively small.

To explore the relationship between wave parameters (height and period) and sandbar erosion, the best-fit lines for the average erosion thickness of the sandbar against wave height and period were plotted, as shown in Figures 10A–D . In these plots, the origin (0, 0) represents zero wave height and period. Figures 10A, B demonstrate a clear linear correlation between wave height and sandbar erosion thickness: as wave height increases, so does the erosion thickness. The T = 1.0s group ( Figure 10A ) has a coefficient of determination of 0.76, likely because the smaller period limits the influence of wave height on sandbar erosion, resulting in a weaker linear correlation. In contrast, the absolute coefficients for all other groups are above 0.92, with some reaching as high as 0.99.

In Figures 10C, D , the relationship between wave period and sandbar erosion thickness is modeled as a quadratic polynomial. Group 4 (H = 0.100 m) exhibits a coefficient of determination of 0.84, while the coefficients for other groups exceed 0.9, indicating a generally strong correlation. This leads to the following empirical formula:

where E S is the erosion thickness, and a * is the fitting coefficient, and * represents different subscripts.

Quadratic polynomial behavior suggests that there is a specific wave period at which sandbar erosion reaches its maximum for a given wave height. Understanding this relationship is crucial for engineers when selecting materials and designing structures to resist wave erosion.

In summary, sandbar erosion thickness is significantly influenced by wave parameters such as height and period. However, the interaction between waves and terrain is a complex process. The reflection, transmission, and breaking of waves significantly affect both the erosion thickness and the topographic evolution of sandbars. Therefore, further data collection is necessary to refine and calibrate more accurate empirical models.

To examine the relationship between wave parameters (height and period) and beach slope, the best-fit lines for average beach slope against wave height and period were plotted, as shown in Figures 11A–D . This study excludes areas where waves do not affect the top of the beach or where steep slopes are formed by landslides. From Figures 11A, B , a clear correlation is observed between wave height and beach slope, with the beach slope decreasing as incident wave height increases. Notably, the T=1.0 s group (indicated by the dashed line in Figure 11A ) exhibits a significantly lower rate of slope change compared to other groups. This may be due to the lower wave energy input in this group, where more energy is dissipated or reflected during wave propagation, limiting the energy that reaches and shapes the beach face. Figures 11C, D also show a linear correlation between the wave period and beach slope, with the beach slope decreasing as the incident wave period increases. Similarly, the H=0.025 m group (circled by the dashed line in Figure 11C ) shows a slower rate of slope change compared to other groups, likely due to the same energy dissipation and reflection mechanisms. In Figure 11D , two points (circled by dashed lines) deviate from the fitted line, corresponding to the minimum incident periods at H=0.075 m and 0.100 m, respectively. Removing these points increases the coefficient of determination of the fitted line from 0.69 and 0.57 to 0.79 and 0.92. This suggests the presence of a critical value where the wave period abruptly impacts the slope. Overall, the relationship between wave period and beach slope appears to be more complex than that between wave height and slope.

The beach equilibrium profile is influenced by various coastal factors, including local wave conditions, sediment characteristics, sediment supply, and coastal topography (Li and Chen, 2009). Human activities, such as beach maintenance and engineering construction, further complicate beach profiles, often deviating from naturally formed shapes. Several classic beach profile models, such as Dean (1991) power function, Bodge (1992) exponential formula, and Lee (1994) logarithmic model, have been proposed. However, these models are mainly applicable to natural beach profiles without sandbar maintenance. As shown in Figure 6 , the monotonicity and convergence of these models fail to accurately simulate profiles shaped by engineering activities. To address this, we propose a new empirical formula for calculating the profile morphology of submerged sandbar-beach systems.

The formula is a three-stage approach, similar to the segmented formula by Inman et al. (1993). For the x ≤ x 1 region, Dean’s power function formula is used. For the x 1 < x < x 2 region, polynomial fitting is applied. For x ≥ x 2 part, according to Pruszak et al. (1997), the profile equation is expanded around the segmentation point using Taylor’s formula. To enhance its practical application, a second-order approximation is adopted. The precise locations of the segment boundaries, x 1 and x 2 , are critical. x 1 corresponds to the crest on the shoreward side of the sandbar, while x 2 marks the start of the angular slope of the beach face ( Figure 12 ). In practical engineering applications, these points can be determined by analyzing the profile difference, specifically the abrupt change in the profile slope.

In equation z 3 ( x ) , the zero-order coefficient m x 1 and the first-order coefficient m x 2 are determined by the boundary conditions (Equation 17). The second-order coefficient m s , which reflects the average slope of the beach profile ( Figure 13E ), is obtained through fitting with profile data. Given the inherent variability of beach equilibrium profile morphology, the model’s findings are statistically significant only to a certain extent. Consequently, the physical meanings and implications of the other model parameters, which are derived through fitting, are not discussed in detail here.

Figures 13A–E present the fitting results, with recommended i values of 2. In most cases, the coefficient of determination is approximately 0.93, indicating a good model fit. However, when wave energy is high, sandbar erosion increases, significantly altering the structure and morphology of the sandbar. As a result, the “sandbar characteristics” of the entire profile become less distinct, leading to poor fit results for some tests, such as Test 14, 15, 19, and Test 20.

Following each experiment, the occurrence of sand ripples was recorded. The location of these ripples, as explained earlier, will not be reiterated here. However, the size of the sand ripples, formed by the lightweight model sand used in this experiment, cannot be conclusively attributed to natural sand ripple formation or experimental artifacts caused by the lightweight material or other model effects (Pan et al., 2022b). Thus, only a statistical analysis of the ripple size was performed. Further research is needed to elucidate the formation mechanism of sand ripples and investigate their role in natural environments, potentially through comparisons with different materials in future studies.

Additionally, in specific cases, such as Test 5 and Test 10, two peaks appeared on the sandbar crest. The formation of a bimodal sandbar crest may be attributed to rolling waves. This wave-breaking mode generates strong downward scouring currents, forming a groove on the sandbar crest. The sand washed from this groove accumulates on either side, gradually forming two elevated sand mounds or small peaks. As this process continues, a distinct bimodal shape emerges at the sandbar’s top. Notably, the peak on the offshore side further facilitates wave breaking at the same location, creating a feedback loop. In this loop, the scouring effect generated by wave breaking shapes the bimodal terrain, which, in turn, influences the wave-breaking mode.