  Analogical thinking is a valuable tool in theoretical physics, since it
allows us to take the understanding we have developed in one system and apply
it to another. In this thesis, we study the analogy between two seemingly
unlikely systems: rotating black holes, elusive cosmic entities that push our
theoretical understanding of modern physics to its limits, and bathtub
vortices, an occurrence so common that they can be observed on a day-to-day
basis in almost any household. Despite the clear difference between these two
systems, we argue that lessons from each can be used to learn something about
the other.
  We investigate the equivalence between surface wave propagation in shallow
water and the propagation of a massless scalar field on an effective spacetime,
focussing in particular on the rotating black hole geometry sourced by a
rotating draining vortex flow. Using this analogy, we verify for the first time
that three effects predicted to occur around rotating black holes also occur in
a laboratory experiment. These are superradiance, an energy enhancement process
whereby waves extract rotational energy from the system, quasi-normal ringing,
describing the relaxation of the system toward equilibrium, and the
backreaction, which mediates the exchange of energy between fluctuations and
the background they experience.
