Time-dependent wave motion with undulated bottom

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.