Waves Generated by a Submerged Topography for the Whitham Equation

In this work, we study numerically the generation of waves due to a current-topography interaction when the current speed is near a critical value using the forced Whitham equation as a model. When the topography is taken as a bump and in the critical value of the current speed, sharp solitary waves are periodically generated upstream and the onset of wave breaking occurs. We show that the time of this onset increases with the width of the obstacle. In particular, it shows that Whitham equation captures features of the full Euler equations, including wave breaking which the forced Korteweg-de Vries equation fails to predict. Besides, when the topography is taken as a hole, waves remain over the cavity gaining energy for a while. Some of them are later emitted upstream as solitary waves while others have their amplitude growing towards a maximum value with sharp gradient. This indicates that a wave breaking may occur above the hole. Furthermore, considering a solitary wave as the initial state of the free surface, we find regimes in which this wave bounces back and forth above the hole remaining trapped for large times.