Resonances of a rotating black hole analogue

Abstract

Under certain conditions, sound waves in a fluid may be governed by a Klein-Gordon equation on an ``effective spacetime'' determined by the background flow properties. Here we consider the draining bathtub: a circulating, draining flow whose effective spacetime shares key features with the rotating black hole (Kerr) spacetime. We present a complete investigation of the role of quasinormal (QN) mode and Regge pole (RP) resonances of this system. First, we simulate a perturbation in the time domain by applying a finite-difference method, to demonstrate the ubiquity of ``QN ringing.'' Next, we solve the wave equation in the frequency domain with the continued-fraction method, to compute QN and RP spectra numerically. We then explore the geometric link between (prograde and retrograde) null geodesic orbits on the spacetime, and the properties of the QN/RP spectra. We develop a ``geodesic-expansion'' method which leads to asymptotic expressions (in inverse powers of mode number $m$) for the spectra, the radial functions, and the residues. Next, the role of the Regge poles in scattering and absorption processes is revealed through the application of the complex angular momentum method. We elucidate the link between the Regge poles and oscillations in the absorption cross section. Finally, we show that Regge poles provide a neat explanation for ``orbiting'' oscillations seen in the scattering cross section.