Physical model experiments are conducted in a wave basin to investigate the influence of directional spreading on wave overtopping in shallow water. Offshore wave steepness, wave height, water depth, and directional spreading are systematically varied to assess their impact on the non-dimensional mean overtopping discharge (q∗). Additional tests with oblique wave attack are performed to examine the role of the wave direction. To better understand the underlying hydrodynamics, the dependence of low-frequency wave energy on directional spreading is analyzed. Results confirm that low-frequency wave energy strongly depends on directional spreading, consistent with previous studies. An empirical formulation is introduced to predict the ratio of low-frequency wave height to total incident wave height using the relative water depth, the offshore wave steepness, and offshore directional spreading, achieving an R ²=0.91. Excluding directional spreading from the formulation decreases the R ² to 0.38, highlighting its importance. Variations in low-frequency wave energy also affect q∗, as low-frequency waves temporarily raise the water level, leading to larger overtopping volumes and thus higher q∗. Consequently, directional spreading influences q∗ primarily through its effect on low-frequency energy, particularly in shallow water. To evaluate how existing prediction tools perform under these conditions, several formulations for q∗ are assessed. Their performance ranges from poor to reasonable, with the best results using the formulation in De Ridder et al. (2024) that were based on 2DV tests in shallow water rather than 3D tests including directional spreading. The tests with oblique waves show that the existing formulation captures the trends found in shallow water. Therefore, the existing formulation for the influence of oblique wave attack is also recommended for shallow water. To incorporate directional spreading effects into overtopping prediction, the relative crest height was adjusted by including the contribution of the low-frequency wave height as done in De Ridder et al. (2024). Due to reduced correlation between the short-wave steepness and the low-frequency height in this new dataset, coefficients could be estimated more reliably. The revised equations are validated against (long-crested) wave flume and new datasets with both short- and long-crested conditions and oblique attack. The expression including the low-frequency wave height results in the highest accuracy (R ²=0.87, Equation (32)) and is recommended, while a relatively simple expression with only the relative crest and short-wave steepness also performs well (R ²=0.83, Equation (28)).

Sea level rise can compromise the safety of coastal flood defences, as wave overtopping events are becoming more frequent and severe. This increasing threat emphasizes the need for accurate assessment of wave overtopping hydrodynamics over dikes, which is essential for evaluating flood safety. The currently available methods do not combine computational efficiency, detailed results and general applicability, which limits their use in modelling wave overtopping and the resulting dike erosion. To address these limitations, this study introduces the Wave Overtopping Surrogate Model (WOSM), a novel method for rapidly generating high-quality two-dimensional simulations of wave overtopping over the dike crest and landward slope. The foundation of the WOSM is the Vision Transformer Image to Image (ViTI2I), a new deep learning model that combines an adapted Vision Transformer with a convolutional decoder for next-frame prediction. Trained on CFD wave overtopping simulations, the WOSM accurately reproduces the overtopping hydrodynamics such as flow velocities, water depths, overtopping duration and vertical velocity profiles, including both spatial and temporal variations. The scope of the training data limits the applicability of the WOSM and its ability to consistently capture complex phenomena such as flow separation and reattachment, both of which could be improved by enriching the dataset. Its low computational demand makes it suitable for exploring additional applications, such as probabilistic design or simulating wave overtopping with evolving dike profiles for erosion assessment. Additionally, this study serves as a proof of concept that the WOSM framework could benefit other fields encountering comparable modelling constraints.

The present research aims to investigate the uncertainties in the evaluation of stone armor stability. Data synthesis was achieved by collecting and homogenizing data from 4 distinct studies, considering the inherent variability of the original data. Established stability equations are then applied to the synthetized database to assess both the strengths and limitations of different approaches across deep, shallow, and very shallow water. The results indicate that while nearly all formulations perform well in deep water, some inadequacies emerge in shallow and very shallow water. To address these limitations, the stability equations were recalibrated using the new database, with a focus on error and uncertainty quantification. The refitted Etemad-Shahidi et al. (ES, 2020) and Modified ES (Scaravaglione et al., 2025) equations consistently demonstrate better predictive capability across all water depths. However, damage assessment reveals persistent uncertainties across all formulations, rendering the selection of a single equation inconclusive, mainly due to the high uncertainty of the available laboratory data. Further synthesizing and homogenizing require additional modeling given the varying modeling approaches, the non-homogenous nature of the parametric data, and the limited understanding possible of the detailed laboratory techniques and data analysis carried out.

Individual overtopping events are important variables when designing a coastal structure as they can deviate significantly from the mean overtopping discharge. Thus, in this study, extreme overtopping events at rubble mound structures with a smooth crest in shallow water have been studied. Both the water layer thickness (flow depth), front velocity and individual overtopping volumes are measured in a wave flume for typical coastal structures with a smooth crest in shallow water for a large range of hydraulic conditions and three different foreshore slopes. An analysis of the individual overtopping volumes shows that the largest individual overtopping volumes arise from short waves that travel on the crest of a low-frequency wave in shallow water and short waves that travel on top of the trough in deep water. Due to the temporal water level variation caused by the low-frequency waves in shallow water, there are fewer overtopping events compared to deep water conditions with the same non-dimensional overtopping discharge. However, the individual overtopping volumes of these events are larger. To quantify the extreme overtopping variables, an empirical formulation based on the relative crest height and short-wave steepness is proposed for the non-dimensional 2 % exceedance water layer thickness, front velocity and individual overtopping volume in terms of incident waves with an R² of 0.84, R² of 0.55 and R² of 0.85 respectively. A further small improvement is found when the low-frequency wave height and 2% exceedance wave height are included, but the added value of this expression does not outweigh the additional wave variables needed for the expression. A log-normal distribution with a constant shape and an expression for the scale of the distribution is proposed to describe the distribution of the individual overtopping volumes in shallow water which accurately captures the distribution (R² of 0.90). Compared to most of the current design approach which is based on a cascade of empirical formulations, this is a significant improvement. In addition, the reasonable results for a distribution with a constant shape parameter show that the shape of the distribution does not change significantly for shallow water conditions.

Mean wave overtopping discharges at rubble mound breakwaters were measured in a wave flume for various rock-armoured slopes. In the physical model tests, the structure slope was varied: 1:1.5, 1:2, 1:4, 1:6 and 1:8 slopes were studied. The mean wave overtopping discharges appeared to be strongly dependent on the structure slope for both “breaking waves” and for “non-breaking waves”. Existing expressions that also account for friction, a berm (if present), a protruding crest wall (if present), and the angle of wave attack, were extended by incorporating both the slope angle and the wave steepness of the incident waves at the toe. The match between the empirical equations and the data (new and earlier tests) is good. The guidelines to estimate wave overtopping discharges as presented here perform much better than existing guidelines since most existing guidelines for rubble mound breakwaters ignore the explicit influence of the structure slope and the wave steepness. Both for “breaking waves” and for “non-breaking waves”, the structure slope and the wave steepness clearly affect wave overtopping discharges, and therefore expressions that ignore these effects should not be used if accurate estimates of mean wave overtopping discharges are required.

Curved concrete crownwalls are commonly installed on vertical breakwaters in deep water to mitigate wave overtopping. This study compares the hydraulic and structural performance of fully curved and recurved crownwalls under impulsive wave loads induced by non-breaking waves, known as Confined-Crest Impact. Using one-way coupled numerical simulations in OpenFOAM and structural analyses in DIANA FEA, we assess the pressure fields and structural responses of the two geometries. Results reveal that while the fully curved crownwall significantly reduces overtopping, it experiences wave forces up to 2.5 times greater than the recurved crownwall, along with longer pressure impulse durations, leading to amplified tensile stresses and higher risk of cracking. In contrast, the recurved crownwall, despite localized peak pressures, benefits from a broader cross-section and linear stress distribution, resulting in better structural performance. These findings underscore the importance of integrating dynamic structural analysis in crownwall design to balance hydraulic efficiency with structural resilience.

Rising sea levels caused by climate change are increasing the risk of overtopping on coastal structures. Moreover, there is a growing societal concern about the visual impact of these structures, which leads to the lowering of their crest freeboards. In previous studies, safety during overtopping events was assessed considering the overtopping layer thickness (h_c), the overtopping flow velocity (u_c) and the individual wave overtopping volume (V). Existing models in the literature to estimate h_c, u_c and V on mound breakwater crests are mainly deterministic, involve a chain of successive estimations leading to accumulated errors and/or do not account for the dependencies between h_c, u_c and V. This study proposes a model to describe the joint probability distribution of h_c, u_c and V based on bivariate copulas. Experimental data from small-scale 2D physical tests conducted on mound breakwaters with three armor layers (single-layer Cubipod®, and double-layer cubes and rocks) in depth-limited breaking wave conditions on two mild bottom slopes and dimensionless crest freeboards between 0.33 and 3.20 is used. Lognormal distribution functions are proposed for each variable and a multivariate dependence model is developed through a one-tree vine-copula. The parameters of this model are quantified directly using wave characteristics and the structure geometry minimizing the accumulated errors in the final predictions. The application of the model is illustrated by computing the probability of not fulfilling at least a tolerability limit for one of the studied variables (OR probability). The OR probability is computed both considering the dependence and assuming independence between the variables and a significant difference is obtained. It is concluded that by accounting for the multivariate dependence between the variables, it is possible to reduce the crest freeboard and, thus, achieve a more economic design within the required safety level.

Physical model tests have been performed to study static stability of rock-armoured mild slopes. Current stability design formulae for steeper rock-armoured slopes focus on plunging and surging waves. Slopes of 1:6 and milder usually have more spilling breakers which decreases the load. Also, on mild slopes displaced rocks more often remain present in the wave attack zone, which increases the strength. These aspects lead to an overdesigned structure when existing formulae for steep rock-armoured slopes are used. The present wave flume tests were used to understand the processes and develop a design formula for rock-armoured mild slopes with an impermeable core. These tests were performed for statically stable rock-armoured slopes of 1:6 to 1:10. The tests confirmed that not all existing damage parameters are able to accurately describe the static stability on milder slopes. For mild slopes it is more accurate to describe the damage based on the eroded depth rather than on the eroded area or number of moved stones. In this study, a design formula and guidelines are provided for practicing engineers that design or evaluate the stability of mild rock-armoured slopes.

Climate adaptation of coastal structures has become more important due to climate change, resulting in sea level rise and increased wave loading for coastal structures with depth-limited wave conditions. If sea level rise causes wave loading that becomes too severe, one of the options is to reduce the wave loading before the waves reach the existing coastal structure (see for instance Van Gent, 2019, and Van Gent and Teng, 2023). This can be achieved by increasing the foreshore (e.g. sand nourishment) or by constructing a low-crested structure in front of the coastal structure. In this study the climate adaptation measure to add a submerged low-crested structure in front of an existing (emerged) coastal structure has been studied. Between the two structures, structure-induced wave set-up occurs. This structure-induced wave set-up has been studied based on wave flume tests. The effects of structure-induced wave set-up on wave transmission at the low-crested structures and the effects on wave overtopping at the emerged coastal structure were also measured and analyzed.

To evaluate the performance of submerged low-crested structures Van Gent et al (2023) performed wave flume tests to examine wave transmission at various types of submerged low-crested structures, without an emerged structure behind the low-crested structures. For a submerged low-crested structure in front of an emerged coastal structure, the transmitted waves can be used as incident waves for estimates of wave overtopping at a rubble mound breakwater, using wave overtopping expressions described in Van Gent et al (2022). However, to verify whether the expressions for wave transmission and wave overtopping can be applied for submerged low-crested structures in front of rubble mound breakwaters, new physical model tests have been performed at Deltares. The new wave flume tests were performed with impermeable and permeable low-crested structures in front of impermeable and permeable emerged structures (see Figure 1 for permeable structure).

Initial damage, caused by previous wave loading or other events, might affect the hydraulic stability of pattern-placed revetments. Three common types of damage are considered in this study. The effect of this assumed initial damage on the hydraulic stability and failure probability of revetments is quantified using a FEM model. This model is developed using data from large-scale flume and field experiments. Using results from the FEM model, surrogate models are created to predict the effect of each type of initial damage on the hydraulic stability and failure probability. Through the use of these surrogate models, it is demonstrated that S-shaped deformation caused by filter migration around the wave impact zone has the largest effect on the hydraulic stability decreasing up to 30%, and failure probability per year increasing up to 10,000 times. When the granular filling between the joints of the columns is washed-out, the stability decreases up to 29% and the failure probability increases up to 700 times. A missing column has a limited effect on the hydraulic stability and failure probability when there is no other (structural) damage. However, if it originates from underlying damage, it might be an initial sign of total failure of the revetment. This study demonstrates the effectiveness of finite element modeling for studying (damaged) revetments, which can be used to complement flume experiments. The results can be used to prioritize maintenance efforts in risk-based maintenance of pattern-placed revetments.

Climate adaptation of coastal structures has become more important due to climate change, resulting in sea level rise and increased wave loading on coastal structures with depth-limited wave conditions. One of the climate adaptation measures to ensure that existing coastal structures continue to perform their function after unforeseen sea level rise, is to reduce the wave loading before the waves reach the existing coastal structure. This can be achieved by constructing a low-crested structure in front of the existing structure. Between the two structures, structure-induced wave set-up occurs. This structure-induced wave set-up has been studied based on wave flume tests. The effects of structure-induced wave set-up on wave transmission at the low-crested structures and the effects on wave overtopping at the emerged coastal structures were also measured and analyzed.

The structure-induced wave set-up depends on the freeboard, wave steepness, and permeability of the low-crested structure. For configurations with impermeable low-crested structures, this wave-set-up does not depend on the distance between the two structures. Empirical expressions to estimate structure-induced wave set-up are derived for impermeable and permeable low-crested structures.

The measurements indicate that the effect of structure-induced wave set-up on the wave transmission coefficients is negligibly small.

The structure-induced wave set-up increases the wave overtopping discharges at the emerged coastal structure. This effect can be taken into account in wave overtopping estimates by reducing the freeboard with the structure-induced wave set-up.

Wave overtopping of coastal structures has been studied using physical model experiments with rubble mound breakwaters in shallow water. The mean overtopping discharge is determined for three different foreshore slopes and various hydrodynamic conditions. The hydrodynamic results confirm that energy is transferred to low-frequency waves in very shallow water and that the short waves are in phase with the lower-frequency waves in very shallow water. As a result, the extreme waves (e.g. 2% exceedance wave height) become relatively large in very shallow water due to the energy of the low-frequency waves affecting thereby the wave overtopping. To estimate the amount of energy at the low-frequency waves, an expression is derived which reasonably accurately predicts the low-frequency wave energy (RMSE of 0.06). Considering the non-dimensional overtopping discharge, the existing formulations for the non-dimensional mean wave overtopping discharge perform poorly to reasonably in shallow water with RMSLE ranging from 1.04 to 2.92. A parameter sensitivity study shows that the short-wave steepness, relative crest height and the low-frequency wave height are the most important parameters when predicting the mean overtopping discharge in shallow water. When including the short-wave steepness and relative crest height in an empirical formulation the RMSLE for the current dataset reduces to 0.69. A further increase in accuracy is found when the low-frequency wave height and 2% exceedance wave height are included (RMSLE 0.64).

Single layer randomly placed armour units are used in many rubble mound breakwaters around the world. For these armour layers, breakage of armour units due to rocking could be a major damage mechanism, but no good methods exist to evaluate and quantify rocking. The aim of the study is to quantify the rocking impact velocities for irregularly placed single layer armour units. This study utilizes embedded Rocking Sensors to obtain the first measurements of rocking impact velocities of single layer armour units. More generally, the paper (Hofland et al. 2023) shows how novel measurement techniques can be used for the investigating the stability of single layer armour layers.

The hydraulic stability of rock armour layers has been extensively discussed in the literature, with numerous formulae proposed for design purposes. However, limited attention has been given to armour stability under shallow water conditions, largely due to the scarcity of experimental data. This research aims to address this gap by providing new insights into the stability of rock armour layers with rubble mound breakwaters in shallow water. Hydraulic stability was determined for four different structure slopes and various hydrodynamic conditions, spanning from deep to extremely shallow water in presence of a 1V:30H foreshore. Newly experimental data were compared with existing stability formulae valid in shallow water, specifically those by van Gent et al. (2003, VG), Eldrup and Andersen (2019, EA), and Etemad-Shahidi et al. (2020, ES). Initially, the data were used to evaluate the accuracy of the original formulae. Following this, the formulae were recalibrated to account for the influence of shallow water, with data grouped according to water levels. Finally, modified versions of VG and ES formulae were developed to fit the experimental data, incorporating the effects of wave steepness to better capture shallow water dynamics.

The crest level of seawalls is often based on estimates of the amount of wave overtopping. Methods to estimate the mean overtopping discharge have been provided in several guidelines. One of the important parameters affecting wave overtopping is the wind. However, the effects of wind have not been accounted for in detail in present design guidelines although some guidance for coastal structures with crest elements is provided in literature. For onshore wind the expected wave overtopping discharge at coastal structures with a crest element can be up to a factor 5 larger than for situations without wind. In the present study the maximum influence of wind on wave overtopping at impermeable seawalls with crest elements has been studied based on physical model tests. The result of the study is a guideline to estimate the maximum influence of wind on wave overtopping at seawalls with crest elements.

Wave overtopping at rubble mound structures is one of the most important phenomena affecting the hydraulic performance of these coastal structures. In addition to the design of coastal structures, also the climate adaptation of coastal structures has become more important due to sea level rise. Adding a crest wall to an existing structure, increasing the height of a crest wall, adding a berm, or increasing the width or height of a berm, can be effective measures to account for effects of sea level rise. For this purpose, the individual effects of a crest walls and a berm need to be predicted, but also the combination of both (see for instance Van Gent, 2019, and Van Gent and Teng, 2023).

Wave overtopping estimates are generally based on physical modelling in wave flumes and wave basins. Numerical modelling of wave overtopping provides additional opportunities to examine wave overtopping for a wide variety of structure geometries. The combination of physical modelling with numerical modelling is referred to as hybrid modelling. To provide design guidelines for rubble mound structures with a crest wall and for structures with a berm in the seaward slope, Van Gent et al (2022) provides design guidelines based on physical model tests. Numerical modelling provides opportunities to examine wave overtopping at structures with a crest wall and a berm to further extend guidelines for the design and (climate) adaptation of rubble mound structures. In Irías Mata and Van Gent (2023) guidelines based on physical modelling have been extended based on numerical modeling with OpenFOAM to examine the influence of several aspects such as the wave steepness, crest wall and recurved parapet, berm, and structure slope on wave overtopping at rubble mound breakwaters. Although the present work focusses on wave overtopping, also forces on crest walls have been examined using the applied numerical model, see for instance Jacobsen et al, 2018, and Irías Mata et al, 2023.

Hard stabilization methods have traditionally been employed to mitigate coastal erosion. Concrete armour is widely used due to its high level of dependence, robustness, ease of production and cost effectiveness (Cooke et al., 2020; Pikey and Cooper, 2012). It is inevitable that coastline ‘armouring’ will continue to rise because of the growing human population and urbanization, desire for and value of coastal property, opposed to predicted climate change (Chapman and Underwood, 2011). The environmental impact of such 'armouring' on coastal systems can be detrimental, resulting in a degradation or destruction of habitats and the loss of ecologically trivial species (Gittman et al., 2015).

The CoastalockTM, a single-layer armour unit, aims to blend coastal protection with marine habitat creation. This armour unit is designed to mimic inter- and sub-tidal habitats, with chemical composition of substrate and micro and macro features that provide niches for various species. The key feature of CoastalockTM is the cavity that is integrated into the design, that caters to diverse marine life needs depending on its orientation (ECOncrete Tech Ltd., 2019). CoastalockTM's hydraulic performance is under research. Preliminary tests conducted in the Hydraulic Engineering Laboratory (HEL) of the Technical University of Delft (TUD) on a 2V:3H impermeable slope in deep water conditions highlighted that with tight placement of the units significant pressure gradients across the top layer led to damage. The introduction of spacings between units for enhanced permeability improved stability significantly (Gutiérrez et al., 2023). A redesign of the unit was proposed incorporating protrusions to enforce the spacings between the blocks (Molenkamp, 2022).

This research focuses on evaluating the influence of a porous core on the hydraulic performance of a CoastalockTM armour layer, specifically assessing its stability, overtopping, and reflection on a 2V:3H breakwater slope in deep water conditions—from the toe to just below the crest. A pivotal aspect of this research is the investigation of the impact of protrusions on the hydraulic performance. Furthermore, the study explores the influence of different toe configurations, aiming to comprehend the vulnerability of the armour layer to sliding. Toe scour falls outside the scope of this study.

Conventional rubble mound structures such as breakwaters, seawalls, and revetments are the most common type of coastal structures around the world used to protect harbour basins and embankments from wave action. To have a safe and economic design, two aspects need to be considered. The first one is the structural stability where the required armor size (weight) must be determined. The second aspect is the safety, where the crest freeboard of the structure is usually determined based on the allowable mean wave overtopping rate. Several semi-empirical formulas have been developed for these purposes. These formulas, which have evolved over time, are generally semi-empirical and based on the small-scale laboratory experiments where both incident wave characteristics and the structure configuration are considered. This paper aims to provide a comprehensive overview of the performance of existing formulas developed for the assessing the stability and mean overtopping rate of conventional rubble mound structures, while also introducing the recent ones. The Rock Manual formulas for the slope stability and EurOtop formula for estimating the mean overtopping rate will be discussed, and their performances will be compared with those of more recent and comprehensive ones using both lab and field data. It will be shown that the recent formulas that utilize the spectral energy mean period for stability analysis and run-up for the mean overtopping rate are more robust and physically sound. Finally, design formulas and uncertainty estimates are presented, along with guidance for practitioners.

Single layer randomly placed armour units are used in many rubble mound breakwaters around the world. For these armour layers first extraction of units starts at high loads and can then progress quickly. Before the first extraction of a unit, typically no quantitative description of damage can be given. But additional to extraction, breakage of armour units due to rocking could be a major damage mechanism. This paper treat novel embedded Rocking Sensors. The technique is used to obtain the first measurements of rocking-impact velocities of single-layer units. They are also the first tests where the instrumented units can naturally move with the compacting layer during storm build-up. Physical model tests were performed on an armour layer with XBloc units. With 8 to 10 instrumented units per test run, in total 640 single measurements of the rocking motion of a unit during a 1200 wave test run were obtained, for three water levels and five wave heights. From the Rocking Sensors the number of impacts and rotational impact velocities were obtained. From an image analysis the along-slope settlement of the units during the tests was quantified. The rotational motion expressed by was found to be most convenient to express the motion. It can be seen that the units in the armour layer rock much more often than visually observed. Settlement seems to be a continuous progress, with most units rocking intermittently. Highest impact velocities are seen to occur around the water line, and in the uprush phase of the waves. A maximum impact velocity for all tests of 0.34 m/s (model scale) was measured. A preliminary design expression for rocking impact velocities of single layer units (Xblocs) is given. The paper shows that novel measurement techniques like the Rocking Sensors and vision techniques can and should be used to quantify damage mechanisms to rubble mound single-layer armour, additional to counting the extracted number of intact units.

Wave transmission at low-crested coastal structures has been studied, based on physical model tests with trapezoidal impermeable, permeable and perforated structures. The differences between wave transmission at impermeable and permeable structures are relatively limited. For a perforated hollow structure with an impermeable vertical screen in the middle, the wave transmission is significantly less than for perforated structures without an impermeable vertical screen; the blocking of the orbital motion by the screen significantly reduces wave transmission. The effectiveness of an impermeable vertical screen to block the orbital motion and consequently reduce wave transmission assists designers of artificial reefs to design structures that reduce wave transmission. Empirical expressions based on a hyperbolic tangent function have been derived to describe the test results. For permeable structures also available data for emerged structures has been used in the analysis, and the newly introduced expression appears to be accurate for both submerged and emerged permeable structures.