The passage of ships in confined waterways creates a stern wave that can overflow bank protection structures such as groins. This overflow, due to the long-period primary ship-induced waves, can be high in velocity, especially at the lee-side slope of groins, potentially causing significant damage to the structure. This study derives an equation to express overflow velocities, intended as a design tool for groins exposed to these types of waves. A detailed experimental investigation was performed on four physical models of groins with different slopes and stone sizes in the armor layer under the influence of different hydraulic heads. Particle image velocimetry (PIV) was used to measure the flow velocities at the crest and lee sides of the structure. All PIV measurements were performed thrice under free-flow conditions with no initial water level at the lee side of the structure. The depth- and time-averaged flow velocities (Uavg) were extracted from four positions along the lee-side slope and accelerated from 0.7 to 2.2 m/s. A dimensionless equation of the overflow velocities was obtained as a function of the hydraulic head (h), slope (θ), freeboard (Rc), and nominal rock diameter (dn50).

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During the last two decades, increasing vessel size in major German estuaries has led to the significant change of the local loading regime i.e. increased importance of ship-induced waves and currents. As a consequence, the intensity of ship-induced loads has increased considerably, resulting in damage to rock structures such as revetments, training walls, and groynes. Research into the causes of rock structure deterioration by the Federal Waterways Engineering and Research Institute (BAW) has shown that for large ships in relatively narrow waterways, the long-period primary ship wave loading has become the most prescient factor for rock structure damage. Looking into the future, it can be expected that the increase in the vessel dimensions will lead to an increase in the ship-wave loading. For this reason, analysing long-term changing trends of long-period ship waves and vessel speed to understand the wave-structure interaction is of significant importance. In this study, the stochastic characterization of long-period primary wave height, drawdown, and speed of the vessel through the water at Juelssand in the Lower Elbe Estuary was analysed via extreme value analysis and copula modeling, and the bivariate return periods were calculated. The one-parameter bivariate copula was utilized to analyse the data. The dependence pattern between the variables was investigated using five parametric copula families: Gaussian, Gumbel, Clayton, Frank, and student's t. @en