Waves propagating near the event horizon exhibit interesting features such as log-phase singularities and caustics. We consider acoustic horizons in flowing Bose-Einstein condensates whose elementary excitations obey the Bogoliubov dispersion relation. In Hamiltonian ray theory, the solution undergoes a broken pitchfork bifurcation near the horizon, so one might expect the associated wave structure to be given by the Piercy function. But the wavefunction is actually an Airy-type function supplemented by a logarithmic phase term, a new type of wave catastrophe. Similar wavefunctions occur in aeroacoustic streams from jet engines and gravity horizons if they contain dispersions that violate Lorentzian symmetry in the UV. The approach we take differs from previous authors in that we use exponential coordinates to analyze the behavior of the integral representation of the wavefunction. This allows a different treatment of the branches leading to a purely saddle point expansion based analysis that resolves multiple real complex waves interacting with horizons and their attendant caustics. The horizon line is the physical representation of the Stokes surface and indicates where waves are born, and it can be seen that the horizon line and the caustic line generally do not coincide. The finite spatial area between them describes a broad horizontal line.