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.

使用弱非线性分析方法检验了不均匀、振荡底部轮廓对两层稳定密度分层模型的影响。两层分层流体背景下底部剖面的研究集中在三种特定类型：（a）表现出单调递减模式的剖面，（b）呈指数衰减的剖面，以及（c）显示高斯振荡的剖面。结合正则摄动法和傅里叶变换技术，对非线性边值问题的二阶反射和透射系数进行了分析。已经提出了与各种物理参数相关的数值结果，证明了 I 类布拉格共振在所有三个剖面中的影响以及单调递减振荡剖面中尾部的抬升。具体而言，与其他轮廓相比，在表现出单调递减图案的轮廓中观察到由于尾部提升现象而存在高反射。研究结果表明，当密度比较低时，界面模式表现出明显的反射，但当密度比较高时，会观察到不同的结果。随着密度比 R 的增加，波能从界面模式到表面模式的迁移量更大，导致反射水平增加。当 R 值接近 1 时，可以观察到，与表面模式相比，较低频率表现出明显更明显的内模式反射。与周期性曲线相比，可以在三个振荡曲线中观察到几个对比方面。这些方面包括零反射（也称为完全透射）的消失，以及振荡的消失。本研究的结果表明，单调振荡递减剖面可被视为有效的布拉格防波堤。此外，这项研究还研究了表面波和内波在非周期性振荡剖面上运动期间发生的能量传递。由此得出能量平衡关系，该关系特别适用于表面和界面模式。这项工作与沿海地区的波浪传播和公海海啸的流体动力学具有流体动力学相关性，两者都受到流体区域测深变化的影响。