Superradiance in acoustic black hole

Rotating superradiance in cylindrical geometries has recently been observed experimentally using acoustic waves, shedding light on the superradiant phenomenon in black holes. In this paper, we study superradiance in acoustic black holes made with solid material for the first time, using theoretical analysis and numerical simulations in COMSOL Multiphysics. We find that superradiance can occur in acoustic black holes when the general superradiance condition is met. We also find that the amplification effect is significantly weaker in acoustic black holes than in regular cylinders, due to absorption within the black holes. Furthermore, we have found that different acoustic black hole models exhibit similar superradiance behavior at the same physical scale, which is also consistent with the phenomena in extremal Kerr black holes. We also present that the solid material ABH model has the most degrees of freedom.

💡 Research Summary

The manuscript investigates the possibility of rotational superradiance in acoustic black holes (ABHs) fabricated from solid materials, a topic that has not been explored experimentally before. The authors begin by reviewing the historical development of superradiance, from its original proposal by Dicke and Penrose to its modern interpretations in rotating astrophysical black holes and laboratory analogues. They emphasize the universal superradiant condition ω − mΩ < 0, where ω is the incident wave frequency, m is the azimuthal mode number, and Ω is the angular velocity of the rotating object. This condition serves as the primary criterion for energy extraction from rotation.

The paper then defines the geometry of a two‑dimensional ABH: a central plateau of thickness h₁ (0 ≤ r ≤ r₁), a tapered region (r₁ < r < R) with thickness h(r) = b(r − r₁)ⁿ + h₁, and an outer uniform region (r ≥ R) of thickness h₂. The authors adopt specific parameters (b = 7.34 × 10⁻⁴ m⁻¹, r₁ = 2 × 10⁻² m, h₁ = 6 × 10⁻⁴ m, n = 2) for the numerical studies.

The theoretical framework starts from the linearized Kirchhoff equations for an inviscid, non‑conducting fluid. In the exterior (air) domain the wave equation reduces to a standard Helmholtz form, while inside the ABH a modified wave equation includes an absorption term α. By transforming to a rotating frame, the absorption term becomes α(ω − mΩ), effectively turning loss into gain when the superradiant condition is satisfied. The radial equations are solved analytically: the exterior solution is a linear combination of Bessel functions Jₘ and Yₘ, while the interior solution involves a complex wavenumber κ = √(ω² + iα(ω − mΩ)) and a radially varying effective sound speed c_eff(r) = √(c_s² − v(r)²). Continuity of pressure and radial velocity at r = R yields impedance matching conditions. The external acoustic impedance is Z₊ = ρ₀c_s, and the internal impedance is Z₋ = ρ_f c_eff(R). By eliminating the coefficients, the authors derive an amplification factor ρ (Eq. 16). Positive ρ indicates superradiant amplification.

For the numerical part, COMSOL Multiphysics is employed with the Pressure Acoustics module and the Delany‑Bazley‑Miki (DBM) model to represent the frequency‑dependent acoustic impedance of porous fiber materials. The DBM impedance is expressed as Z = 1 + A₁X − A₂ − iA₃X − A₄, where X = ρ₀ω/(2πϖ) and ϖ is the flow resistance of the material. Table I lists the coefficients for glass fiber, low‑density glass fiber, and low‑density rock fiber. By varying ϖ (2.5 × 10³ – 5 × 10⁴ Pa·s/m²) the authors explore how material losses affect superradiance.

Two simulation scenarios are presented. First, a static (Ω = 0) case shows the characteristic reduction of the local sound speed toward the ABH singularity, confirming the “black‑hole‑like” trapping of acoustic energy. Second, a rotating case with Ω = 200π rad s⁻¹ is examined for two incident frequencies: ω = 200 Hz (which does not satisfy ω − mΩ < 0) and ω = 20 Hz (which does). In the latter, the pressure field exhibits outward propagation in certain radial zones, indicating that the rotational Doppler shift has turned the absorption term into a gain term, and the reflected wave extracts rotational energy—i.e., superradiance.

The results reveal several key insights. (1) In an ideal ABH with an infinitely sharp taper, almost all incident energy is absorbed, leaving negligible reflection; consequently, superradiant amplification would be weak because there is little reflected wave to be amplified. In realistic ABHs, the central plateau and finite taper reduce absorption, increasing the reflected component and allowing measurable superradiance. (2) The amplification factor obtained from the semi‑analytical formula is modest, whereas the COMSOL simulations—especially when realistic DBM impedances are used—show substantially larger apparent gains. This discrepancy is attributed to the fact that the simulated material impedance does not reach the theoretical optimum, and the finite‑element model captures additional scattering mechanisms. (3) Increasing the flow resistance (i.e., using more dissipative fiber materials) raises the impedance Z, which suppresses both absorption and superradiant gain, confirming the trade‑off between material loss and amplification. (4) Across different ABH designs (varying b, r₁, h₁, n) the superradiant behavior collapses onto a universal curve when plotted against the dimensionless parameter ωR/c_s and the dimensionless rotation rate mΩR/c_s, mirroring the universality observed in extremal Kerr black holes.

Finally, the authors compare solid‑material ABHs with the rotating draining‑bathtub analogue, noting that both share the same governing wave equation in a rotating background and exhibit analogous superradiant thresholds. The solid ABH, however, offers more degrees of freedom (geometry, material composition, flow profile) for experimental optimization. The paper concludes that while absorption in solid ABHs limits the raw amplification compared with lossless cylinders, careful engineering of impedance and rotation can produce observable superradiant amplification, opening a pathway toward tabletop analogues of black‑hole energy extraction.