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

Abstract

The effect of non-uniform, oscillating bottom profiles on a two-layer stable density stratification model has been examined using the method of weakly non-linear analysis. The study of bottom profiles in the context of two-layered stratified fluids has focused on three specific types: (a) profiles that exhibit a monotonically decreasing pattern, (b) profiles that decay exponentially, and (c) profiles that display Gaussian oscillations. The analysis of the second-order reflection and transmission coefficients for the nonlinear boundary value problem was conducted using a combination of the regular perturbation method and the Fourier transforms technique. The numerical findings pertaining to various physical parameters have been presented, demonstrating the impact of the Class I Bragg resonance in all three profiles and the elevation of the tails in the monotonically decreasing oscillatory profile. Specifically, the presence of high reflections due to the tail-lifting phenomenon is observed in a profile that exhibits a monotonically decreasing pattern, in contrast to the other profiles. The findings of the study indicate that interface modes demonstrate pronounced reflections when the density ratios are low, but divergent results are observed when the density ratios are high. As the density ratio R increases, there is a greater migration of wave energy from the interface mode to the surface mode, resulting in increased levels of reflection. As the value of R approaches 1, it is observed that lower frequencies exhibit significantly more pronounced internal mode reflections compared to surface mode. Several contrasting aspects can be observed in three oscillatory profiles when compared to periodic profiles. These aspects include the disappearance of zero reflection, also known as complete transmission, as well as the absence of oscillations. The findings of this study demonstrate that a monotonically oscillating decreasing profile can be considered as an efficacious Bragg breakwater. Furthermore, this study investigates the energy transfer that occurs during the movement of surface and internal waves across non-periodic oscillatory profiles. As a result, an energy balance relationship is derived, which specifically applies to surface and interface modes. This work has hydrodynamical relevance to wave propagation in coastal regions and to the hydrodynamics of tsunamis in the open ocean, both of which are affected by changes in the bathymetry of the fluid region.