  Understanding the ability of particles to maneuver through disordered
environments is a central problem in innumerable settings, from active matter
and biology to electronics. Macroscopic particles ultimately exhibit diffusive
motion when their energy exceeds the characteristic potential barrier of the
random landscape. In stark contrast, wave-particle duality causes electrons in
sufficiently disordered media to come to rest even when the potential is weak
-- a remarkable phenomenon known as Anderson localization. Here, we present a
hydrodynamic active system with wave-particle features, a millimetric droplet
self-guided by its own wave field over a submerged random topography, whose
dynamics exhibits localized statistics analogous to those of electronic
systems. Consideration of an ensemble of particle trajectories reveals a
suppression of diffusion when the guiding wave field extends over the
disordered topography. We rationalize mechanistically the emergent statistics
by virtue of the wave-mediated resonant coupling between the droplet and
topography, which produces an attractive wave potential about the localization
region. This hydrodynamic analog, which demonstrates how a classical particle
may localize like a wave, suggests new directions for future research in
various areas, including active matter, wave localization, many-body
localization, and topological matter.
