Propagation of nonlinear surface waves over non-periodic oscillatory bottom profiles (Q6548380)

The manuscript presents a thorough examination of how surface and internal waves propagate over irregular bottom profiles. The authors focus on the interaction of waves with three specific types of non-periodic oscillatory profiles: monotonically decreasing oscillatory, exponentially decaying oscillatory, and Gaussian oscillatory profiles, within a two-layer stratified fluid system. These profiles serve as models for natural seabed undulations and are crucial for understanding the dynamics of wave scattering and energy transmission in real-world scenarios, such as coastal engineering and tsunami dynamics.\N\NThe scientific problem addressed in the paper pertains to how stratified fluid systems, commonly found in oceanographic settings, interact with non-periodic bottom topographies. Specifically, the study aims to determine the impact of such topographies on the reflection and transmission of surface and internal waves. Traditional studies in this area have often considered periodic or sinusoidal bottom profiles, which simplify the mathematics but may not fully capture the complexity of real-world seabed conditions. The present work contributes to the literature by extending the analysis to more realistic, non-periodic profiles, thereby broadening the understanding of wave-bottom interactions in coastal and ocean environments.\N\NTo address this problem, the authors employ weakly nonlinear analysis combined with regular perturbation methods and Fourier transforms. The weakly nonlinear approach allows for capturing the second-order effects of wave interaction with bottom topographies, while the perturbation technique simplifies the solution of the nonlinear boundary value problems that arise in the system. Numerical simulations are conducted to compute the reflection and transmission coefficients for both surface and interface modes, which are key physical parameters indicating the extent to which wave energy is either reflected back or transmitted through the fluid layers. The authors validate their model by comparing it with previously established results for sinusoidal bottom profiles.\N\NThe key findings of the study reveal that the nature of the bottom profile plays a significant role in determining the wave behaviour. For instance, in the case of a monotonically decreasing oscillatory profile, the authors observe a tail-lifting phenomenon, where high levels of wave reflection occur due to the lifting of wave energy in the profile's tail. This behaviour contrasts with that of periodic profiles, where complete transmission, or zero reflection, is often observed. Moreover, the study highlights that the class I Bragg resonance -- a phenomenon where waves are amplified due to constructive interference with the bottom undulations -- occurs across all three types of profiles, but with distinct characteristics depending on the profile shape. Another notable observation is that as the density ratio between the fluid layers increases, there is a greater transfer of energy from the interface modes to the surface modes, leading to enhanced reflection at higher densities.\N\NThe study's findings have significant implications for both coastal engineering and tsunami hydrodynamics. The insights gained from the analysis of non-periodic oscillatory profiles could inform the design of more effective coastal defences, such as Bragg breakwaters, which rely on the principle of wave reflection to protect shorelines. Additionally, understanding how energy is distributed between surface and internal waves in response to changes in bathymetry is crucial for predicting the behaviour of tsunamis as they propagate from the deep ocean towards coastal areas.\N\NIn conclusion, this manuscript provides valuable contributions to the field of wave dynamics in stratified fluids, particularly in the context of realistic seabed profiles. The combination of nonlinear analysis with detailed numerical simulations offers new insights into the behaviour of surface and internal waves over non-periodic bottom topographies, with direct applications to coastal engineering and natural disaster mitigation strategies. The research presented is of high relevance to the study of wave propagation in marine environments, where irregular bathymetry plays a pivotal role in influencing wave behaviour.