Potential Velocity of Water Waves Propagation with Small Bottom Undulation

Basically, when a wave meets a different depth, it will scatter into a transmitted wave and a reflected wave. In this paper, we study the relevance of sinusoidal beds as shoreline protection through the Bragg scattering mechanism. As well known, a relatively small amplitude of sinusoidal beds can reduce the amplitude of incident waves effectivelly, due to Bragg resonance. Bragg resonance will occur if the wavelength of the monochromatic wave is twice the wavelength of sinusoidal beds. We apply the multiple scale asymptotic expansion method to the linear Shallow initial value problem for sinusoidal beds. the effect of Bragg resonance pass through sinusoidal beds has been studied for surface displacement of water waves. We found that a larger amplitude disturbance leads to larger reflected wave amplitude. This result explains that the long shore sandbar indeed can reduce the amplitude of incident wave. Actually, wave's propagation not only derives from surface displacement, but also from potential velocity. At the end, we show that the sinusoidal beds can reduce the potential velocity of incident wave.