An acoustic-scattering calculation matched reference solutions in several modeled open resonators while showing how individual resonant states contribute to the result, according to an arXiv preprint. The resonant-state expansion, or RSE, matched an analytical Mie-theory solution for a homogeneous two-dimensional cylinder and agreed with COMSOL Multiphysics calculations for a sectorally perturbed cylinder and an annular metaatom. In the annular example, separate peaks in the scattering spectrum were linked to different channel patterns and radiation patterns. The document is an arXiv version 1 preprint dated 20 Aug 2026.
A calculation built around resonant states
RSE represents the scattered pressure and particle-velocity fields as sums of normalized resonant states: characteristic patterns associated with the resonator. Each state's excitation amplitude is obtained from overlap integrals involving the incident field and the material perturbation, or change in the modeled material properties. The formulation uses acoustic energy flux to derive extinction, scattering and absorption cross-sections. It also resolves extinction into contributions from individual resonant states, giving the calculation both an overall spectrum and a mode-by-mode view of how that spectrum is assembled.
The simple test came first
The first benchmark was a homogeneous two-dimensional cylinder with an analytical Mie-theory reference solution. Under the reported truncation, with kR < 35 and a maximum azimuthal mode of mmax = 30, the numerical cross-section spectra showed perfect correspondence with Mie theory. The calculated spectral difference approached zero. The paper reports this agreement for the homogeneous-cylinder benchmark at those stated truncation settings.
The calculation also exposed something that a single total curve would hide. The extinction spectrum was resolved into the contribution of every individual resonant state, and some of those modal terms were negative even though total extinction remained positive. A negative term therefore describes one component of the decomposition, not a negative value for the complete extinction spectrum. The modal breakdown is presented alongside the total result so the two are not confused.
A more complicated cylinder
The second test used a cylinder with a C4 sectoral perturbation. The modeled change had an aperture of 45 degrees, a density perturbation equal to 0.1 times the reference density and a compressibility perturbation equal to 0.2 times the reference compressibility. The RSE calculation was compared with a COMSOL Multiphysics simulation. The two agreed excellently, with the reported error approaching zero when the decomposition used N = 5,694 modes. In the paper, this case tests the expansion after both the geometry and material parameters have been altered.
Routing sound through an annular model
The paper's annular example used a material-programmed hard-wall metaatom with an inner radius of 35 mm and an outer radius of 135 mm. The model contained 36 air channels, with 10 radial sections per channel. For this calculation, hard-wall partitions were embedded in the reference geometry rather than represented as a material perturbation. That reference choice is part of the formulation used for the annular case.
At an incidence angle of 20 degrees, the reported scattering spectrum contained local maxima across 2–5 kHz. The near-field pressure patterns at the labeled features A–D occupied different subsets of the channels and had different radiation patterns. In practical terms, the calculation associated different parts of the frequency range with different routes through the annular structure, demonstrating frequency-selective routing. The cross-section spectrum and the near-field pressure distribution were independently validated against a matched COMSOL model.
What the preprint shows—and what it leaves open
The authors interpret the comparisons as showing that RSE can serve as an efficient semi-analytical modal tool for studying acoustic scattering in open resonators with complex geometry. In the examples reported, its value is not limited to producing a cross-section curve: the formulation retains resonant-state contributions, while the annular calculation places spectral features beside near-field pressure patterns and radiation patterns.
There is an important boundary to that interpretation. A complete resonant-state representation may require branch-cut contributions in addition to discrete poles, while the main analysis is restricted to non-dispersive media. The reported agreements are also tied to the finite mode truncations used in the modeled cases. They show how the calculation behaves at the stated settings, rather than resolving every possible contribution in every open geometry.
The supplied evidence is computational: the homogeneous case is checked against Mie theory, while the perturbed cylinder and annular metaatom are checked against COMSOL. The paper does not report an experimental test of a fabricated resonator, a broad quantitative comparison showing that RSE is faster or more accurate than alternative scattering methods, or a demonstration of the proposed extensions to lossy, dispersive or non-Hermitian systems. Those questions remain outside the modeled results presented here.
Financial support was acknowledged from the Russian Science Foundation under grant 25-79-31027. The supplied record identifies the document as arXiv version 1, dated 20 Aug 2026.
Paper data and sources
Original title: Resonant state expansion for acoustic resonators. Part II. Scattering problem
Authors: Egor Domoratskii, Vladimir Igoshin, Nikolay Solodovchenko et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text