The preprint presents a computational framework for calculating how material perturbations change the complex eigenfrequencies—the calculated resonance values—and eigenmodes, the corresponding pressure-field patterns, of an open acoustic resonator. It tests the resulting resonant-state expansion against exact analytical solutions and finite-element simulations, reporting excellent quantitative agreement. The supplied analysis does not give numerical error values or uncertainty intervals.
The approach is called a resonant-state expansion, or RSE. It derives a Green function, a mathematical way to represent the resonator’s response, and a generalized acoustic normalization from pressure and velocity. Those ingredients are used to build a matrix eigenvalue problem for the perturbed modes.
The demonstration is theoretical and computational. It uses a homogeneous circular-cylinder reference model and examines material perturbations with homogeneous, radial and sectoral profiles. The analysis concerns model configurations and resonant modes rather than a human or animal participant sample.
A deliberately controlled test case
The reference cylinder has a radius of 10 centimetres. Its density is set at 10 times the background density, while its sound speed is one-half of the background value; the surrounding medium is air. The model begins from these values before evaluating the listed material perturbations.
The framework examines homogeneous, radial and sectoral variations in density and compressibility. In plain terms, those cases represent an overall material change, a change that varies with distance from the centre, and a pattern divided into angular sectors. The paper uses them to examine tuning, changes in the radial shape of modes and coupling associated with broken symmetry.
For this two-dimensional open problem, the Green-function spectrum includes a branch-cut contribution as well as discrete resonant poles. The formulation therefore includes more than a list of isolated resonances. For the reference resonator, the transcendental equation defining the eigenfrequencies was solved with Cauchy’s argument principle together with Newton’s method.
What the different perturbations changed
With a homogeneous perturbation, the reported coupling was confined to resonant states with the same azimuthal number. In other words, modes carrying different angular labels did not mix in that case. This is the symmetry selection rule reported for the homogeneous model.
In the radial case, azimuthal order was preserved while the radial modal profile changed, so the calculated mode had a different structure from the centre outward. The RSE results were reported to be close to numerical simulations and to require substantially less computational time. The supplied analysis gives neither timing values nor numerical error values.
Sectoral perturbations coupled different azimuthal harmonics, meaning angular components that were separate in the symmetric setup appeared together in the calculation. For this case, the reported RSE basis size was 5,694. Low-quality-factor sectoral modes did not require cut-mode contributions, while cut poles were significant for homogeneous and radial perturbations.
Symmetry breaking also reshaped the resonances
In the sectoral tests, the paper reported significant eigenfrequency shifts and pronounced quality-factor reductions for whispering-gallery modes. More sectors were associated with greater loss, even though no additional absorptive material loss was included.
Together, the calculations represent several distinct responses within one framework: homogeneous perturbations preserve azimuthal separation, radial perturbations reshape the radial profile while preserving azimuthal order, and sectoral perturbations mix harmonics. The paper reports these as different outcomes for the modeled resonator.
A useful framework with a narrow demonstrated scope
The validation is limited to a two-dimensional cylindrical open resonator, selected density and compressibility perturbations, analytical reference results and finite-element simulations. It does not establish accuracy for real experimental devices, and no experimental validation is included in the supplied evidence.
The reported RSE calculations are restricted to nondispersive media and frequency-independent perturbations. The supplied analysis does not demonstrate performance in dispersive media, three-dimensional systems or geometrically complex resonators; those settings remain proposed extensions rather than tested cases.
The supplied account reports qualitative agreement but contains no numerical error metric, uncertainty interval, convergence result or timing value. It also gives no code-availability or dataset-availability statement. As a result, the computational advantage described for radial perturbations is not quantified.
Open questions include whether the formulation can be extended to scattering observables and experimental resonator behavior, and how basis size and convergence govern branch-cut contributions for high-quality-factor or high-azimuthal-order modes. The analysis also identifies dispersive and three-dimensional extensions as tests still to be done.
What the preprint reports
The front matter identifies the work as arXiv:2608.19979v2, dated 21 August 2026. The authors acknowledge financial support from the Russian Science Foundation under grant 25-79-31027.
Paper data and sources
Original title: Resonant state expansion for acoustic resonators. Part I. Eigenvalue 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