On the localization of shallow water waves by a random bottom

Linear shallow water waves have a velocity which varies with the water depth h(x). We expect that a plane wave (wave vector k parallel to x) will be partially reflected by modulations of the bottom. We argue that this should lead, in the presence of a random modulation of the depth extending over a long enough distance, to exponential localization of all proper modes and to the total reflection of an incident wave, a phenomenon analogous to the one of localization discovered and widely studied in solid state physics. As a first step in the experimental study of localization of water waves, we show the effect of a periodic modulation of the bottom (Bragg scattering) and the additional effect created by using a random amplitude.