Time localization of energy in disordered time-modulated systems

Wave localization induced by spatial disorder is ubiquitous in physics. Here, we study the temporal analog of such phenomenon on water waves. Our time disordered media consists in a collection of temporal interfaces achieved through electrostriction between water surface and an electrode. The wave field observed is the result of the interferences between reflected and refracted waves on the interfaces. Although no eigenmode can be associated to the wave field, several common features between space and time emerge. The waves grow exponentially depending on the noise level in agreement with a 2D matrix evolution model such as in the spatial case. The relative position of the momentum-gap appearing in the time modulated systems plays a central role in the wave field evolution. When tuning the excitation to compensate for the damping, transient waves, localized in time, appear on the liquid surface. They result from a particular history of the multiples interferences produced by a specific sequence of time boundaries.