  Waves propagating near an event horizon display interesting features
including logarithmic phase singularities and caustics. We consider an acoustic
horizon in a flowing Bose-Einstein condensate where the elementary excitations
obey the Bogoliubov dispersion relation. In the hamiltonian ray theory the
solutions undergo a broken pitchfork bifurcation near the horizon and one might
therefore expect the associated wave structure to be given by a Pearcey
function, this being the universal wave function that dresses catastrophes with
two control parameters. However, the wave function is in fact an Airy-type
function supplemented by a logarithmic phase term, a novel type of wave
catastrophe. Similar wave functions arise in aeroacoustic flows from jet
engines and also gravitational horizons if dispersion which violates Lorentz
symmetry in the UV is included. The approach we take differs from previous
authors in that we analyze the behaviour of the integral representation of the
wave function using exponential coordinates. This allows for a different
treatment of the branches that gives rise to an analysis based purely on
saddlepoint expansions, which resolve the multiple real and complex waves that
interact at the horizon and its companion caustic. We find that the horizon is
a physical manifestation of a Stokes surface, marking the place where a wave is
born, and that the horizon and the caustic do not in general coincide: the
finite spatial region between them delineates a broadened horizon.
