Jump conditions for Boussinesq equations due to an abrupt depth transition (Q6598451)

The manuscript investigates the behaviour of water waves as they encounter a sudden change in depth, a scenario that significantly influences the dynamics of wave propagation in shallow or finite-depth water. The study is also motivated by the role of abrupt depth transitions in the generation of extreme waves, including so-called rogue waves, and seeks to understand the nonlinear and dispersive effects that arise from such transitions.\N\NThe problem is approached using a sophisticated asymptotic analysis up to the third order in terms of the shallowness parameter, capturing both the nonlinear and dispersive contributions. The authors adopt a three-scale analysis framework to handle the complexity of the depth transition, which differs significantly from cases of smooth bathymetry. They develop distinct mathematical models for different regions: far from the step, near the free surface, and near the abrupt depth transition. This enables the derivation of modified Boussinesq equations, supplemented by jump conditions that take into account the discontinuity of the bathymetry at the depth transition. These jump conditions are crucial to accurately describe the impact of the abrupt depth change on wave propagation.\N\NThe methodology employed involves deriving the governing equations for water wave motion in two dimensions, starting from the assumptions of an inviscid and incompressible fluid in irrotational motion. The equations are then reduced to a one-dimensional form in regions away from the depth discontinuity, leading to a set of Boussinesq equations. The core contribution of the paper lies in the derivation of the jump conditions, which provide a novel mathematical framework to describe the abrupt changes in wave motion at the depth transition. The authors employ boundary layer analysis to establish the effective parameters entering these jump conditions, which depend solely on the ratio of depths before and after the transition. The jump conditions are presented as explicit equations, incorporating both the velocity field and surface elevation at the discontinuity, and are validated through a detailed asymptotic procedure.\N\NThe main findings of the study highlight that the derived jump conditions are critical for accurately modelling wave propagation over abrupt depth changes. These conditions are shown to have a significant influence on the transmission and reflection of waves, including nonlinear contributions that could be overlooked in simpler models. The paper's results suggest that these jump conditions are essential for predicting the occurrence of extreme wave events in real-world scenarios, such as those involving undersea topography with sharp depth variations.