Arrays of discrete, lossy Helmholtz resonators loaded on a waveguide can absorb low-frequency sound almost perfectly when each resonance is critically coupled — the balance of a resonator's energy leakage into the waveguide against its intrinsic viscothermal loss, represented spectrally as a zero of the reflection coefficient lying exactly on the real frequency axis. Designs optimized for band-averaged absorption, including the source system of Herrero-Durá et al. (Appl. Sci. 10, 1690, 2020), leave the zeros slightly off the real axis: five coupled resonators deliver absorption above 0.995 across 400–560 Hz, yet not every resonance is exactly critically coupled, and the same tension recurs throughout the foundational critical-coupling literature (Sci. Rep. 6, 19519 (2016); Sci. Rep. 7, 13595 (2017)). The open question is whether a rigid-backed one-dimensional array of N coupled Helmholtz resonators admits a geometry placing all N reflection zeros exactly on the real frequency axis within the target band while the band-averaged absorption is retained at the level of the best near-critical designs — or whether a fundamental tradeoff couples per-resonance coupling exactness to band-averaged performance.

Background

Absorbing low-frequency airborne sound with compact structures is a classic difficulty of acoustics: porous and fibrous materials must be comparable to a wavelength in thickness to work well below a few hundred hertz, which motivates resonant absorbers whose building blocks are deep-subwavelength. A Helmholtz resonator (HR) is the canonical such element: a rigid-walled cavity opening to the surrounding medium through a narrow neck, behaving at low frequency as a single-degree-of-freedom oscillator whose resonance frequency is set by the neck and cavity dimensions. Mounted along a waveguide (or on a rigid wall), each HR is an open, lossy resonator: it leaks energy back into the propagating acoustic field while dissipating energy through viscothermal (viscous and thermal) losses concentrated in the narrow neck. Because the acoustic wavelength is much larger than both the resonators and the array period (the point-resonator and plane-wave conditions), the whole array can be modeled one-dimensionally with the transfer matrix method (TMM), which chains waveguide-propagation matrices with point-scattering matrices for each resonator; absorption is measured experimentally in an impedance tube, where the plane-wave reflection coefficient is extracted with the two-microphone transfer-function method.

For a single lossy resonance in a reflection problem (a one-port, rigid-backed system), absorption is maximal when the resonator's energy leakage rate equals its intrinsic loss rate. This is the critical coupling condition, equivalent to the resonator's acoustic impedance matching that of the waveguide, and it has a compact spectral representation: in the complex frequency plane, the reflection coefficient R(f)R(f) of the lossless system has zeros mirrored above the real axis, and adding loss shifts them downward; a zero lying exactly on the real frequency axis means zero reflection, hence perfect absorption (α=1\alpha = 1), at that frequency (Romero-García et al. 2016). With several resonators the array supports several hybridized resonances and several zeros, and two design objectives compete. Herrero-Durá et al. (2020) optimized the geometry of a five-resonator cascade for the highest global absorption over the band 400–560 Hz, reaching α>0.995\alpha > 0.995 across the band, and observed that the resulting zeros of ∣R∣|R| lie 'very close, but not exactly at the real frequency axis, i.e., not all the resonances are critically coupled', whereas — in their wording — exact placement of all zeros on the real axis 'is the case if the system is optimized to obtain perfect absorption at N discrete frequencies'. The same tension recurs across the foundational literature: the multi-resonant scatterer of Romero-García et al. (2016) attains broadband absorption above 93% with one resonance exactly critically coupled and its neighbor off-axis (α≈0.96\alpha \approx 0.96), and the rainbow-trapping absorber of Jiménez et al. (2017) states explicitly that 'using a wide-bandwidth cost function does not ensure that all the resonances remain critically coupled', while in the manufactured system 'not all the zeros are located exactly on the real axis'. Whether exact critical coupling at every resonance and high band-averaged absorption are simultaneously achievable in a coupled resonator array — and, if not, what quantitative tradeoff governs the design space — is the unresolved question posed here.

Problem Statement

For the reflection (one-port, rigid-backed) problem of a one-dimensional array of NN discrete, mutually coupled, lossy Helmholtz resonators loaded on a waveguide, operating in the point-resonator and plane-wave regime (acoustic wavelengths much larger than the resonator and array-period dimensions) with viscothermal losses determined by the resonator geometry, determine whether there exists a choice of geometry — the neck and cavity dimensions that set each resonator's frequency, energy leakage, and intrinsic loss, together with the inter-resonator spacings — such that all NN hybridized resonances are simultaneously critically coupled, i.e., all NN zeros of the reflection coefficient R(f)R(f) lie exactly on the real frequency axis within the target band, while the band-averaged absorption coefficient is retained at the level of the best near-critical-coupling designs. The question is posed for the reference instantiation of Herrero-Durá et al. 2020: N=5N = 5 resonators, target band 400–560 Hz, with the band-averaged absorption over 400–560 Hz retained at the level α>0.995\alpha > 0.995 achieved by the band-optimized source design. If such geometries exist, provide a constructive design or optimization method that finds them and validate it with transfer-matrix or finite-element modeling within the standard viscothermal-loss model class — the equivalent-fluid model of the source system, in which the narrow neck and cavity are described by frequency-dependent complex effective density and bulk modulus (Section 2.1, Eqs. (1)–(3) of Herrero-Durá et al. 2020, following the classic effective-property model for narrow slits) — with impedance-tube experimentation as the strongest confirmation. If they do not exist within this model class, prove this and characterize the fundamental tradeoff between per-resonance coupling exactness and band-averaged absorption — specifically, a bound relating the number of exactly critically coupled resonances, the band-averaged absorption, and the available geometric degrees of freedom.

References

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