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Acoustic Analogy of Quantum Baldin Sum Rule for Optimal Causal Scattering

Sichao Qu1,*, Zixiong Yu2, Erqian Dong1, Min Yang3,†, and Nicholas X. Fang1,4,‡

  • *Contact author: qusichao@hku.hk
  • †Contact author: min@metacoust.com
  • ‡Contact author: nicxfang@hku.hk
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Phys. Rev. Lett. 136, 226902 – Published 5 June, 2026

DOI: https://doi.org/10.1103/dbs8-g68w

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.

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Physics Subject Headings (PhySH)

Focus

No Free Lunch for Sound Waves

Published 5 June, 2026

Sound wave scattering can be increased in one frequency range only by reducing scattering in another range, according to experiments—a discovery relevant for acoustic engineering.

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Outline

Information

Phys. Rev. Lett. 136, 226902– Published 5 June, 2026
  • Received 4 January 2026
  • Revised 16 February 2026
  • Accepted 6 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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