Acoustic Analogy of Quantum Baldin Sum Rule for Optimal Causal Scattering

Abstract

The mass law is a cornerstone in predicting sound transmission loss, yet it neglects the constraints of causal dispersion. Current causality-based theories, such as the Rozanov limit, are applicable only to one-port reflective absorbers. Here, we derive a universal sum rule governing causal scattering in acoustic systems, establishing a rigorous analogy to the Baldin sum rule in quantum field theory. This relation reveals that the integral of the extinction cross-section is fundamentally locked by the scatterer's static effective mass and stiffness, which is validated numerically using seminal examples of underwater metamaterials. Furthermore, the proposed sum rule predicts an optimal condition for an anomalously broadened transmission loss bandwidth, as experimentally observed through the spectral shaping effect of an acoustic Fano resonator. Our findings open up an unexplored avenue for enhancing the scattering bandwidth of passive metamaterials.

Figures (5)

• Figure 1: From quantum to acoustic Baldin sum rule. (a) Schematic illustration of light–matter scattering. (b) Schematic illustration of sound–structure scattering. (c) Conceptual diagram of $\sigma_\mathrm{ext}$ spectra shaping, showing the conversion between frequency $\omega$ and wavelength $\lambda$.
• Figure 2: The simulation-based verification of acoustic Baldin sum rule by revisiting seminal examples of underwater metamaterial scatterers. (a) The monopole Helmholtz resonator in a duct. (b) The dipole lead-core resonator in a periodic layout. (c) Extinction spectra ($\sigma_\mathrm{ext}$); insets show the extracted effective properties. (d) The cumulative distribution function $\gamma(\omega)$.
• Figure 3: The experimental validation via airborne sound resonators in ducts. (a) Foam liner (monopole type). (b) Helmholtz resonator (monopole type). (c) Fano resonator (coupled monopole-dipole type). The size of the oscillator sphere represents the amount of dissipation, and the length of the line represents the resonant frequency. Dashed lines represent 2D rotational symmetry axes, and solid lines represent hard boundaries. (d) The simulated (solid lines) and measured (circles) transmission spectra. (e) The cumulative distribution function $\gamma(\omega)$. The inset shows the calculated values of $K_0/K_\mathrm{eff}(0)$ and $\rho_\mathrm{eff}(0)/\rho_0$, along with their ratios and the simulation-extracted values. The experimental $\gamma(\omega)$ below 100 Hz is complemented by low-frequency asymptotic forms derived from simulations (see Ref. labelSM Sec. S4 for details).
• Figure 4: The direct verification of the Kramers–Kronig (KK) relations using transmission data from FEM simulations (left) and experiments (right). (a) $2\mathrm{Re}[1 - T(\omega)]$ or $\sigma_\mathrm{ext}(\omega)$ and its KK-generated counterpart (circles). (b) $2\mathrm{Im}[1 - T(\omega)]$ and its KK-generated counterpart (circles).
• Figure 5: The Figure of Merit [see Eq. (\ref{['eq:fom']})] vs the frequency coverage (beyond 10 dB transmission loss). The data are from our experiments (the sample thickness was $L=2.5\,\mathrm{cm}$ for all three cases) and other reported works wang2023metanguyen2020broadbandjia2025acousticlu2025ultramei2025reconfigurabledong2021ultrabroadbandxu2024broadbandfu2025ultrabroadbandjimenez2017rainbow.