Quasinormal ringing of acoustic black holes in Laval nozzles: Numerical simulations

Quasinormal ringing of acoustic black holes in Laval nozzles is discussed. The equation for sounds in a transonic flow is written into a Schrödinger-type equation with a potential barrier, and the quasinormal frequencies are calculated semianalytically. From the results of numerical simulations, it is shown that the quasinormal modes are actually excited when the transonic flow is formed or slightly perturbed, as well as in the real black hole case. In an actual experiment, however, the purely-outgoing boundary condition will not be satisfied at late times due to the wave reflection at the end of the apparatus, and a late-time ringing will be expressed as a superposition of boxed quasinormal modes. It is shown that the late-time ringing damps more slowly than the ordinary quasinormal ringing, while its central frequency is not greatly different from that of the ordinary one. Using this fact, an efficient way for experimentally detecting the quasinormal ringing of an acoustic black hole is discussed.

Figure 1

Forms of different Laval nozzles $\pm r(x)$ (thin curves) and the potentials $V(x^{*})$ (thick curves) for stationary transonic flow, as functions of $x$. The fluid is assumed to flow in the negative $x$-direction, and thus $x<0$ is the supersonic region. Also indicated are the maximum point and the height of each $V(x^{*})$ by the vertical and the horizontal dotted line, respectively.

Figure 2

Spacetime distribution of $v$ (left) and $\log |\dot{v}|$ (right) obtained from a numerical simulation of type I. The nozzle parameters are set to $(\alpha ,\beta )=(2,0.01)$. The dashed curve and the dotted curve represent the position of the sonic point and the steady shock, respectively.

Figure 3

Numerical waveform $v_{\mathrm{obs}}(t)=v(t,2.0L)$ (solid curve) obtained from the same simulation as that shown in Fig. 2, compared to the least-damped quasinormal mode (dotted curve) with the frequency ${\omega }_0=(3.60-1.28i)c_{s0}/L$.

Figure 4

Spacetime distribution of $v$ (left) and $\log |\dot{v}|$ (right) obtained from a numerical simulation of type II, with nozzle parameters $(\alpha ,\beta )=(2,0.01)$.

Figure 5

Numerical waveform $v_{\mathrm{obs}}(t)=v(t,2.0L)$ (solid curve) obtained from the same simulation as that shown in Fig. 4, compared to the least-damped quasinormal mode (dotted curve).

Figure 6

Boxed quasinormal frequencies for $(\alpha ,\beta )=(2,0.01)$ and $\mathcal{R}=0.089$, as functions of $x_c$. The curves indicate eight of the numerical solutions to Eq. (34), ${\omega }_{\mathrm{b},m}(m=0,\cdots ,7)$. For comparison, the least-damped ordinary quasinormal frequency ${\omega }_0$ is indicated by the thick horizontal line. The symbols “$\circ$,” “$\times$,” “$▵$,” “$\diamond$,” and “$\star$” represent the “best fitting” frequencies ${\mathit{\omega }}_{\min }$ for the waveforms obtained through type I’ simulations (see Sec. 3b).

Figure 7

Numerical waveform $v_{\mathrm{obs}}(t)=v(t,2.0L)$ (solid) for an early time (upper panel) and a late time (lower panel), obtained from the type I’ simulation with $(\alpha ,\beta ,x_c,\mathcal{R})=(2,0.01,4.0L,0.089)$. For comparison, the least-damped quasinormal mode (dotted line), a superposition of three boxed quasinormal modes (dashed line), and the “best fitting” function (dot-dashed line) are also plotted.

Figure 8

Numerical waveform $v_{\mathrm{obs}}(t)=v(t,2.0L)$ (solid line) for an early time (upper panel) and a late time (lower panel), obtained from the type I’ simulation with $(\alpha ,\beta ,\mathcal{R},x_c)=(2,0.01,1.0,4.0L)$.