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.

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).

A 2-layer non-hydrostatic model with improved dispersive behaviour is presented. Due to the assumption of a constant non-hydrostatic pressure distribution in the lower layer, the dispersive behaviour is improved without much additional computational time. A comparison with linear wave theory showed that this 2-layer model gives a better result for the dispersion relation and shoaling of waves in intermediate water. This means that the 2-layer model is applicable in shallow and intermediate water depths (up to relative depths kh equals 4), whereas the 1-layer model is only applicable in shallow water depths (kh smaller than 1). Three laboratory experiments, including a fringing reef and a barred beach, were used to validate the presented mode for different hydrodynamic conditions. Based on these results, it can be concluded that the 2-layer model can be applied to accurately simulate the bulk wave height and spectral properties. The low frequency wave height, the setup and in particular the second order statistics contain more scatter, but the model accurately captured the general trend. Furthermore, the model showed good results for complex bathymetries in shallow to intermediate water.

Quantification of human impact on water, sediment, and nutrient fluxes at the global scale demands characterization of reservoirs with an accuracy that is presently unavailable. This letter presents a new method, based on virtual dam placement, to make accurate estimations of area-volume relationships of large reservoirs, using solely readily available elevation data. The new method is based on regional similarity of area-volume relationships. The essence of the method is that virtual reservoirs are created in the vicinity of an existing reservoir to derive area-volume relationships for the existing reservoir. The derived area-volume relationships reproduced in situ bathymetric data well. An intercomparison for twelve reservoirs resulted in an average R² = 0.93. This is a significant improvement on estimates using the best existing global regression model, which gives R² = 0.54 for the same set of reservoirs.