#TITLE_ALTERNATIVE# - Noviantri (NIM 10103015), Viska
ABSTRACT: Long shore sandbars are often found along gentle beaches. The number of bars can range from a few to dozens and the spacing from tens to hundreds of meters. The bar amplitudes can be as high as meter. We are interested in studying how this natural sinusoidal sandbars affect the amplitude of incoming waves. In this final project we show that sinusoidal sandbars may lead to Bragg resonance. Basically, when a wave meets a different depth, it will scatter into a transmitted wave and a redirected wave. Let us imagine an incident wave running above a flat bottom with a finite patch of sinusoidal disturbance. During its evolution there occurs many scattering processes. Bragg resonance happens when the wavelength of incident wave is twice of the wavelength of the periodic bottom disturbance. Applying multiple scale expansion to get the results. We also find that a larger amplitude disturbance leads to a larger reflected wave amplitude. This result explains that the long shore sandbar indeed can reduce the amplitude of incident wave and protect a beach from a large amplitude incident wave. It turns out that the same phenomena occurs for water waves in a long channel with corrugated boundaries. Starting from the full governing equation for flat bottom channel, if the channel width is small compared with the characteristic wavelength, we obtain a one dimensional wave equation, in which its coeficient depend explicitly on the function of the left and right boundaries. Bragg resonance is then occurs under analogous condition. But here we observe that a large amplitude disturbance does not directly lead to a larger reflected wave amplitude.