An arXiv preprint presents a Green-function-based resonant-state expansion (RSE) for open acoustic resonators and reports excellent agreement for complex eigenfrequencies—resonance values that also reflect modeled loss—and eigenmodes, the associated resonant patterns. The checks used exact analytical solutions and finite-element simulations in two-dimensional cylindrical models.
The method is designed for material perturbations: changes in mass density and compressibility that can be uniform, vary with radius or follow sectoral patterns.
A basis built from a solvable cylinder
The calculation begins with a homogeneous circular cylinder 10 centimetres in radius. Its analytically solvable resonant states serve as the reference basis, from which the authors derive explicit perturbation matrices for uniform, radial and sectoral changes in density and compressibility.
At the centre is a Green function, a mathematical representation of the system’s response. For this two-dimensional open problem, its spectral representation contains discrete resonant poles as well as a branch-cut integral—the continuum contribution alongside the individual resonances.
To build the reference spectrum, the study found resonances for azimuthal numbers from 0 to 60, using Cauchy’s argument principle together with the Newton method. These numbers label the angular families of resonances used in the perturbation calculations.
Checks against independent calculations
One test changes the material uniformly. The modified cylinder still has an exact homogeneous-cylinder secular equation—an equation that determines its resonant frequencies—so the frequencies can be obtained independently and compared directly with the RSE result.
Another test uses a radial profile, fρ(r) = fβ(r) = r/R, with Δρ = 0.2ρ0 and Δβ = 0.4β0. For that profile, the paper reports close agreement between RSE results and numerical simulations, along with substantially less computational time.
The reported validation covered both complex eigenfrequencies and eigenmodes, with excellent quantitative agreement against analytical and finite-element results.
When the circle’s symmetry is broken
A separate set of models uses sectoral perturbations. These reduce the cylinder’s continuous rotational symmetry, written C∞, to a discrete CN symmetry and impose selection rules—restrictions on which azimuthal modes can couple.
In the modeled whispering-gallery modes, the paper reports a significant eigenfrequency shift and a pronounced reduction in quality factor, the quantity it uses to track resonant loss. Cases with more sectors corresponded to greater loss. The perturbations were real and introduced no additional absorptive material loss; the reported loss was radiative.
The sectoral calculation used an RSE basis of 5,694 states and was reported to have high accuracy against COMSOL Multiphysics simulations.
The need for cut-mode contributions depends on the regime: in the setting examined, low-quality-factor sectoral modes were reported not to require them.
The result has a clear boundary
The framework is illustrated in a two-dimensional open cylindrical resonator, with tests covering homogeneous, radial and sectoral perturbations. Its accuracy for more complex geometries and three-dimensional systems is not established by these calculations.
One explicit boundary is dispersive acoustic media, where the eigenvalue problem becomes nonlinear. The paper places that extension beyond the scope of this work.
The reported matches are described as close or excellent, but the analysis supplies no numerical error bounds, confidence intervals, convergence tables or runtime values. That leaves the quantitative trade-offs of changing the basis size and cut-pole terms open.
The document is an arXiv version 1 preprint dated 20 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