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)).

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.

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.

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.

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.

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).

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.

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.

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.

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.

The design of crest walls is often based on empirical formulations, physical model tests, numerical models and a fair amount of expert judgement. The present work validates the prediction of wave induced forces on the front face of crest walls on top of composite breakwaters in the numerical model OpenFOAM. The results show that OpenFOAM is able to capture the shape and order of magnitude of the force events caused by non-breaking and heavily breaking waves. In addition, a calibrated model predicts the highest wave induces forces caused by breaking waves with errors lower than 20%.

During storms, ensuring the protection of people, vehicles and infrastructure on the crest of coastal structures from wave overtopping hazards is crucial. The thickness of the wave overtopping layer is a key variable used for assessing safety and maintaining a secure design. Traditionally, this parameter is associated with the height difference between the fictitious wave run-up level exceeded by 2% of waves and the crest freeboard of coastal structures. This study aims to investigate the wave overtopping layer thickness on the crest of rubble mound seawalls. To achieve this, a series of 125 small-scale 2D physical model tests were conducted on a two-layer rubble mound seawall with an impermeable core and slopes of 1:1.5 and 1:2. The obtained results indicated that the existing empirical formulas, originally developed for dikes, underestimate the overtopping layer thickness on the studied seawall. Therefore, modifications were made to the formulas found in the literature specifically tailored for rubble mound seawalls. The newly proposed formulas for estimating overtopping layer thickness at both the seaward edge and the middle of the crest showed improvements compared to the existing formulas.

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.