In the present manuscript, the time-dependent capillary gravity wave motion in the presence of a current and an undulated permeable bottom is analyzed. The spectral method is used to simulate the time-dependent surface elevation. Also, the Laplace–Fourier transform method is used to obtain the integral form of the surface elevation, and the asymptotic form of the associated highly oscillatory integral is derived using the method of stationary phase. The reflection and transmission coefficients due to the small bottom undulation are obtained using the perturbation method and the Fourier transform method and also alternatively using Green’s function technique and Green’s identity. The nature of wave energy propagation obtained from plane capillary gravity wave motion is verified through time domain simulation and using the spectral method. It is found that, in the case of co-propagating waves, the wave energy propagates faster and also the surface profiles in terms of wave packets move faster for larger values of the Froude number. Also, the maximum value of the reflection and transmission coefficients decreases due to increasing values of the Froude number. For the sinusoidal bottom topography, the Bragg resonance occurs if the ratio of the wave numbers of the wave and the rippled bed is one by two.

中文翻译：

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波状底部随时间变化的波浪运动

在本手稿中，分析了在存在电流和起伏的可渗透底部的情况下与时间相关的毛细管重力波运动。谱法用于模拟随时间变化的表面高程。此外，使用拉普拉斯-傅立叶变换方法获得表面高程的积分形式，并使用固定相方法推导出相关的高振荡积分的渐近形式。使用微扰法和傅立叶变换方法以及交替使用格林函数技术和格林恒等式来获得由于小底部起伏引起的反射和透射系数。通过时域模拟并使用谱方法验证了从平面毛细管重力波运动获得的波能传播的性质。发现，在共同传播的波的情况下，对于较大的弗劳德数值，波能传播得更快，并且波包方面的表面轮廓移动得更快。此外，反射系数和透射系数的最大值由于弗劳德数的增加而减小。对于正弦底部地形，如果波的波数与波纹床的波数比为一比二，就会发生布拉格共振。