Accurate quasinormal modes of the analogue black holes

Abstract

We study the quasinormal modes of the spherically-symmetric $(2+1)$-dimensional analogue black hole, modeled by the ``draining bathtub'' fluid flow, and the $(3+1)$-dimensional canonical acoustic black hole. In the both cases the emphasis is on the accuracy. Formally, the radial equation describing perturbations of the $(2+1)$-dimensional black hole is a special case of the general master equation of the 5-dimensional Tangherlini black hole. Similarly, the $(3+1)$-dimensional equation can be obtained from the master equation of the 7-dimensional Tangherlini black hole. For the $(2+1)$-dimensional analogue black hole we used three major techniques: the higher-order WKB method with the Pad\'e summation, the Hill-determinant method and the continued fraction method, the latter two with the convergence acceleration. In the $(3+1)$-dimensional case, we propose the simpler recurrence relations and explicitly demonstrate that both recurrences, i.e., the eight-term and the six-term recurrences yield identical results. Since the application of the continued-fraction method require five (or three) consecutive Gauss eliminations, we decided not to use this technique in the $(3+1)$-dimensional case. Instead, we used the Hill-determinant method in the two incarnations and the higher-order WKB. We accept the results of our calculations if at least two (algorithmically) independent methods give the same answer to some prescribed accuracy. Our results correct and extend the results existing in the literature and we believe that we approached assumed accuracy of 9 decimal places. In most cases, there is perfect agreement between all the methods; however, in a few cases, the performance of the higher-order WKB method is slightly worse.