Quasinormal modes and classical wave propagation in analogue black holes

Many properties of black holes can be studied using acoustic analogues in the laboratory through the propagation of sound waves. We investigate in detail sound wave propagation in a rotating acoustic $(2+1)$-dimensional black hole, which corresponds to the “draining bathtub” fluid flow. We compute the quasinormal mode frequencies of this system and discuss late-time power-law tails. Because of the presence of an ergoregion, waves in a rotating acoustic black hole can be superradiantly amplified. We also compute superradiant reflection coefficients and instability time scales for the acoustic black hole bomb, the equivalent of the Press-Teukolsky black hole bomb. Finally we discuss quasinormal modes and late-time tails in a nonrotating canonical acoustic black hole, corresponding to an incompressible, spherically symmetric $(3+1)$-dimensional fluid flow.

Figure 1

In the left panel we show the real part of the fundamental QN frequency ${\omega }_{\mathrm{R}}$ (and in the right panel we show the imaginary part ${\omega }_{\mathrm{I}}$) as a function of the rotation parameter $B$, for selected values of $m$. For $m>0$ ($m<0$) ${\omega }_{\mathrm{R}}$ and $|{\omega }_{\mathrm{I}}|$ increase (decrease) with rotation.

Figure 2

Reflection coefficient $|R_{\omega m}{|}^2$ as a function of $\omega$ for $m=1$ (left panels) and $m=2$ (right panels). Each curve corresponds to a different value of $B$, as indicated. The top panels show that the reflection coefficient decays exponentially at the critical frequency for superradiance, ${\omega }_{\mathrm{S}\mathrm{R}}=mB$. The middle panels show a close-up view in the superradiant regime for $B<1$: at $B=1$ the maximum amplification is 21.2% ($m=1$) and 4.7% ($m=2$). The bottom panels show that superradiant amplification can become much more efficient for values of the rotation parameter $B>1$.

Figure 3

Real part (left) and imaginary part (right) of the fundamental BQN frequency and the first two overtones as a function of mirror location $r_0$. Both panels refer to $m=1$ and $B=1$ (but the $B$-dependence of the real part of the BQN frequency is very weak). With excellent accuracy ${\omega }_I$ crosses zero (and the instability abruptly shuts off) at the critical radius predicted by Eq. (48), as can be verified taking a glance at the first row of Table .

Figure 4

Growth time scales for BQNMs, as a function of mirror location $r_0$, for different values of $B$. The left panel refers to $m=1$, the right panel to $m=2$. Notice the different scales for the ${\omega }_I$ axis.