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This paper conducts numerical studies on superradiance and Hawking radiation of a specific rotating acoustic black hole model characterized by parameters A and B, within the framework of analogue gravity. The standard radial wave equation for scalar perturbations in the effective metric of this model is solved numerically by using an adaptive Runge-Kutta method with tortoise coordinates; this approach necessitates careful numerical inversion of the coordinate transformation near the horizon via a root-finding algorithm. By imposing appropriate boundary conditions, we extract the reflection coefficient $\mathcal{R}$ and transmission coefficient $\mathcal{T}$ in a range of frequencies ω. Our results clearly demonstrate superradiance, with the reflectivity $|\mathcal{R}|^2$ exceeding unity for $\omega < m\varOmega_{\rm{H}} = 1$ (where $m=-1$ and $\varOmega_{\rm{H}}=-1$), which confirms energy extraction from the rotating background. The high accuracy of our method is validated by the flux conservation relation, $|\mathcal{R}|^2 + [(\omega - m\varOmega_{\rm{H}})/\omega]|\mathcal{T}|^2 = 1$, which typically has a numerical precision of $ 10^{-8}$. Furthermore, using the derived Hawking temperature and the rotation modified Bose-Einstein distribution, we calculate the Hawking radiation power spectrum $P_\omega$, and use the numerically obtained transmission coefficient $|\mathcal{T}|^2$ as the greybody factor of the model. A prominent feature of $P_\omega$ is its sharp enhancement (or divergence) as ω approaches the threshold $m\varOmega_{\rm{H}}$ from above, which is a characteristic directly related to the denominator of the Bose-Einstein factor. This research also reveals that superradiant amplification and Hawking spectrum characteristics are significantly dependent on the specific values of flow parameters A and B, even when the superradiant threshold $m\Omega_H$ is kept unchanged. This detailed numerical study provides quantitative results for the scattering and radiation properties of this model, and also for strong support for the analogue gravity framework.
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Keywords:
- acoustic black hole /
- superradiance /
- Hawking radiation /
- numerical calculation
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图 1 反射系数模$ |{\cal{R}}| $(实线)与透射系数模$ |{\cal{T}}| $(虚线)随入射声波频率ω的变化关系
Figure 1. Modulus of reflection coefficient $ |{\cal{R}}| $ (solid line) and transmission coefficient $ |{\cal{T}}| $ (dashed line) as functions of incident wave frequency ω.
图 2 放大系数$ |{\cal{R}}|^{2}-1 $(黑色实线)随入射声波频率ω的变化. 该系数大于零的区域$ (0<\omega \lesssim 1.0) $代表超辐射存在的区域. 此图确定了超辐射发生的频率范围, 其上限对应理论阈值$ m \varOmega_{{\rm{H}}}=1 $
Figure 2. Amplification factor $ |{\cal{R}}|^{2}-1 $ (black solid line) as a function of incident wave frequency ω. The region where this factor is greater than zero $ (0<\omega \lesssim 1.0) $ represents the existence of superradiance. This figure determines the frequency range for superradiance, with the upper limit corresponding to the theoretical threshold $ m \varOmega_{{\rm{H}}}=1 $.
图 3 霍金辐射功率谱$ P_{\omega} $(黑色实线)随频率ω的变化, 特别展示了超辐射临界频率($ m \varOmega_{{\rm{H}}}=1 $)附近的行为. 图中$ P_\omega $为采用自然单位制($ \hbar=k_{\rm{B}}=c=1 $)及特征长度$ r_0=1 $计算得到的无量纲功率谱密度, 代表以$ \hbar c / r_0 $为单位的能量
Figure 3. Hawking radiation power spectrum $ P_{\omega} $ (black solid line) as a function of frequency ω, specifically showing the behavior near the superradiance critical frequency ($ m \varOmega_{{\rm{H}}}=1 $). The plotted $ P_\omega $ is the dimensionless power spectrum density calculated using natural units ($ \hbar= $$ k_{\rm{B}}=c=1 $) and characteristic length $ r_0=1 $, representing energy in units of $ \hbar c / r_0 $.
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