Active electroacoustic resonators — loudspeakers whose diaphragm acoustic impedance is set in real time by a digital controller — circumvent the thickness limits that make passive treatments ineffective at low frequencies. Guo, Lissek and Fleury (Phys. Rev. Applied 13, 014018, 2020) experimentally demonstrated a hybrid scheme combining linear feedforward control of the front pressure with nonlinear feedback on the diaphragm displacement (estimated from the rear-cavity pressure), synthesizing a cubic (Duffing-type) nonlinearity at low excitation levels and roughly doubling the bandwidth of absorption coefficient above 0.8 relative to linear control in the 50–500 Hz range. Later work has generalized the programmable nonlinearity itself (cubic, piecewise-linear, and logarithmic restoring forces at low amplitudes). What remains open are the two architectural extensions the source proposed: combining the nonlinear control law with active linear multiple-degree-of-freedom electroacoustic resonator targets, and implementing nonlinear active acoustic metamaterials whose unit cells intrinsically present negative effective bulk modulus. Linear active metamaterials with programmable bulk modulus and nonlinear active metamaterials with other functionalities (nonreciprocity, topological confinement) exist separately, but their combination for broadband or multistable low-frequency absorption has not been demonstrated. The problem is to design, realize, and characterize these architectures and to quantify the absorption performance and nonlinear wave functionality they deliver against linear and single-degree-of-freedom nonlinear baselines.

Background

Absorbing airborne sound at frequencies below roughly 300 Hz is a long-standing difficulty: the acoustic wavelength is of order meters, so porous materials would need impractical thickness, and passive resonant treatments (Helmholtz resonators, quarter-wave tubes, membrane absorbers) absorb efficiently only in a narrow band around each resonance. Electroacoustic resonators offer an active alternative: a moving-coil loudspeaker used as a sound absorber whose diaphragm acoustic impedance — the ratio of the sound pressure acting on the diaphragm to its velocity — is set electronically, either by passive shunt circuits or by real-time digital control driven by microphone measurements, so that a subwavelength cell can present any causal target impedance tuned for maximal absorption (the absorption coefficient being the fraction of incident acoustic power dissipated).

Within this field, Guo, Lissek and Fleury (Phys. Rev. Applied 13, 014018, 2020) introduced a hybrid nonlinear active control architecture on a closed-box loudspeaker prototype. A linear feedforward controller drives the loudspeaker current from the pressure measured by a microphone facing the diaphragm, while a nonlinear feedback controller exploits the proportionality, at low frequency, between the pressure inside the sealed back cavity and the diaphragm displacement — measured by a second microphone — to inject a prescribed nonlinear contribution to the restoring force. Because the nonlinearity is synthesized by software rather than by mechanical elements, it can be triggered at arbitrarily low sound pressure levels (around 95 dB SPL in their experiment) and reprogrammed at will, in contrast to passive nonlinear absorbers such as nonlinear membranes or nonlinear energy sinks (nonlinear attachments that irreversibly drain energy from a primary linear oscillator), whose nonlinear response typically requires levels far above human hearing tolerance. Taking a cubic (Duffing-type) stiffness nonlinearity as an example, the authors showed numerically and experimentally that the hybrid control enlarges the frequency range over which the absorption coefficient exceeds 0.8 to more than 80 Hz — roughly doubling the linear-control result — while radiating only about 0.13% of the incident energy into higher harmonics.

The scheme was demonstrated only in a restricted configuration: the control target is a single-degree-of-freedom (SDOF) resonator — one mass on one spring-damper, characterized by moving mass, compliance, and resistance — and the synthesized nonlinearity is cubic. In their concluding perspectives, the authors proposed two architectural extensions. First, since a digitally driven loudspeaker can be assigned any causal target impedance, the target can be a multiple-degree-of-freedom (MDOF) resonator, i.e. a coupled or parallel set of resonances absorbing over several bands at once; passive MDOF electroacoustic absorbers (coupled resonators in a shared enclosure, Rivet et al. 2018) and linear MDOF target-impedance synthesis were known before, but not their combination with the nonlinear feedback law. Second, an acoustic metamaterial — a structured medium whose unit cells are much smaller than the wavelength and which behaves, after homogenization, as an effective medium characterized by a bulk modulus (the stiffness relating pressure to volume change) and a mass density — could be built from unit cells implementing the same two-microphone scheme: such cells could intrinsically present a negative effective bulk modulus, and combining this linear active control with a nonlinear law was proposed to yield a new family of nonlinear active metamaterials with potentially larger bandwidth or multistable functionalities, multistability meaning that the medium admits several stable dynamical states selectable by the excitation amplitude or history.

Both extensions remain essentially unexplored for absorption. Active acoustic metamaterials with electronically programmable effective bulk modulus have been demonstrated only with linear control laws (Kovacevich and Popa 2024), and the nonlinear active metamaterials realized so far target other wave phenomena — nonreciprocity (Popa and Cummer 2014), nonreciprocal harmonic conversion (Guo et al. 2023), amplitude-driven topological confinement in an active Su-Schrieffer-Heeger-like liner (Padlewski et al. 2023), chiral nonlinearity for topological protection (Guo et al. 2025) — rather than negative-modulus absorption. The open problem is therefore whether the nonlinear active electroacoustic impedance-control framework can be scaled beyond the single cubic SDOF resonator, to MDOF active targets and to nonlinear active metamaterials with negative-effective-bulk-modulus unit cells, and what absorption bandwidth and nonlinear functionality this scaling actually delivers.

Problem Statement

Starting from the experimentally demonstrated hybrid architecture of Guo, Lissek and Fleury (2020) — linear feedforward control of the front pressure combined with nonlinear feedback on the diaphragm displacement estimated from the rear-cavity pressure, implemented on a closed-box electroacoustic resonator — extend the nonlinear active electroacoustic impedance-control framework beyond the demonstrated cubic single-degree-of-freedom configuration, along the two architectural directions proposed in the source's conclusions, with the common objective of further improving broadband low-frequency sound absorption and enabling nontrivial nonlinear wave functionality:

  1. Combine the nonlinear feedback control law with active linear multiple-degree-of-freedom electroacoustic resonator targets, i.e. target impedances corresponding to coupled or parallel multi-resonance systems, and determine whether this combination further improves low-frequency sound absorption beyond what linear MDOF control and nonlinear SDOF control achieve separately.

  2. Implement nonlinear active acoustic metamaterials whose unit cells use the two-microphone active control scheme and thereby intrinsically present negative effective bulk modulus, combined with a synthesized nonlinear law, and investigate the anticipated nontrivial wave phenomena — larger absorption bandwidth or multistable functionalities. For this extension to count as resolved, demonstrating and quantifying at least one of these two phenomena suffices, and a quantified negative outcome for the phenomena examined is an equally acceptable resolution; demonstrating both is not required.

Both extensions proposed in the source's conclusions are mandatory deliverables — the nonlinear feedback law combined with MDOF electroacoustic targets, and the negative-bulk-modulus nonlinear active metamaterial — and resolving only one does not resolve the task. For each extension the expected artifact level is analytic/numerical: specify the control laws and target dynamics; model the loudspeaker-cavity-control system (or the metamaterial unit cell) and thereby realize the proposed architecture in it; characterize stability limits and higher-harmonic distortion; retrieve the amplitude-dependent effective properties in the metamaterial case; propose an experimental validation protocol for the extension; and compare absorption spectra against the linear-SDOF, linear-MDOF, and nonlinear-cubic-SDOF baselines so that any claimed improvement is quantified. Building and measuring a physical prototype is not required to resolve the task.

References

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