Resonant state expansion for acoustic resonators.
Part I. Eigenvalue problem

In general, the linear acoustic equations for the media with density $\rho_{\omega}(\mathbf{x})$ and $\beta_{\omega}(\mathbf{x})$ compressibility have the following form:

where $p(\mathbf{x})$ – pressure, $\mathbf{v}(\mathbf{x})$ – velocity, and $\mathbf{f}(\mathbf{x})$ is the force volume density 22; 51. Time-dependent parts of pressure and velocity are given by $\exp(-i \omega t)$ with the complex frequency $\omega$. The material parameters are connected with each other by the sound velocity $c_{\omega}(\mathbf{x})=[\rho_{\omega}(\mathbf{x})\beta_{\omega}(\mathbf{x})]^{-1/2}$. Here and further, the subscript $\omega$ denotes frequency dependence.

Let us write Eqs. (1) in the following inhomogeneous matrix form

Here, we introduce the matrix operator $\hat{\mathbb{D}}(\mathbf{x})$ and the material parameters matrix $\hat{\mathbb{P}}_{\omega}(\mathbf{x})$ as

and acoustic field vector $\vec{\mathbb{F}}(\mathbf{x})$ and force source vector $\vec{\mathbb{J}}(\mathbf{x})$ as

The solution of Eq. (2) is given by the dyadic GF as follows:

Here, the integral is taken over the whole system volume $V$ with all inhomogeneities and singularities of the material parameters matrix $\hat{\mathbb{P}}_{\omega}(\mathbf{x})$. Additionally, the GF satisfies the equation

If the GF is meromorphic, then it can be expanded in terms of its resonant poles as

where $\vec{\mathbb{F}}_{n}(\mathbf{x})$ and $\omega_{n}$ are the RSs and resonant frequencies satisfying Eq. (2) without external sources:

Obviously, the GF (5) has poles when the perturbation frequency matches the acoustic resonator eigenfrequencies $\omega=\omega_{n}$ in the complex $\omega$-plane. In general, however, the analytic structure of the Green’s function may contain additional non-pole contributions. In particular, for the two-dimensional open acoustic system considered below, the Green’s function possesses a branch cut in the complex-frequency plane. Therefore, the discrete set of RS poles alone does not constitute a complete spectral representation, and the branch-cut contribution must be included explicitly. The corresponding Green-function expansion, containing both the discrete resonant poles and the branch-cut contribution, is introduced in Eq. (23).

It is important to emphasize that RSs $\vec{\mathbb{F}}_{n}(\mathbf{x})$ of an open acoustic resonator are not normal modes of the Hermitian systems. They satisfy purely outgoing-wave boundary conditions and have complex eigenfrequencies $\omega_{n}$ with $\mathrm{Im}(\omega_{n})\leq 0$ for the time dependence chosen as $\exp(-i\omega t )$. Since eigenfrequencies imaginary part $\mathrm{Im}(\omega_{n})<0$, this outgoing solution grows as $\exp(-\mathrm{Im}(\omega_n) r / c)$ at large distances when $r\rightarrow+\infty$. In fact, this exponential divergence is not a physical growth of the acoustic pressure $p$, but rather a consequence of the analytic continuation of the scattering problem to the complex-frequency plane. Physically, an RS describes the field remaining in the resonator in the absence of an external excitation. The temporal dependence of the RSs decays as $\exp\left[\mathrm{Im}(\omega_{n})t\right]$, while the spatially increasing outgoing tail at a fixed time can be interpreted in terms of radiation emitted at earlier retarded times $t-r/c$, when the amplitude of the decaying resonance was larger. Thus, an individual RS is a generalized eigenmode rather than a square-integrable physical field. A measurable acoustic field excited at a real frequency is obtained from a causal superposition of RSs and non-resonant background contributions, for which such divergences cancel. This is why the normalization of RSs requires a generalized bilinear form with surface terms, rather than the usual energy integral over the whole space 21; 3.

The RSs of Eq. (6) form an orthonormal and complete set of functions. We consider the normalization relation for the RSs is given as follows (see Sec. I of the Supplemental material 46 for details):

The first integral in Eq. (7) is taken over the entire system volume $V$ with all inhomogeneities, and the second one is taken over the closed surface $S$ on the boundaries of volume $V$. The surface contribution in Eq. (7) is essential for open systems. Since acoustic RSs obey outgoing boundary conditions at complex eigenfrequencies $\omega_{n}$, they generally diverge exponentially outside the resonator and cannot be normalized by the standard energy integral over all space, as noted above.

Although the acoustic equations and the normalization relation above are written in a general form that allows for frequency-dependent material parameters, in the following we restrict the RSE to nondispersive media and frequency-independent perturbations. Accordingly, we set $\rho_{\omega}(\mathbf{x})\equiv\rho(\mathbf{x})$, $\beta_{\omega}(\mathbf{x})\equiv\beta(\mathbf{x})$, and $\hat{P}_{\omega}(\mathbf{x})\equiv\hat{P}(\mathbf{x})$, with $\Delta\hat{P}_{\omega}(\mathbf{x})\equiv\Delta\hat{P}(\mathbf{x})$. Under this assumption, the perturbed eigenvalue problem remains linear in the complex eigenfrequency $\Omega$, and the perturbation matrix elements are independent of $\Omega$. The extension of the present formalism to dispersive acoustic media, for which the material operators must be evaluated at the perturbed eigenfrequency and the resulting eigenvalue problem becomes nonlinear, is beyond the scope of this work.

We now formulate how a perturbation of the material parameters modifies the RS eigenvalue problem. When the acoustic system is perturbed, i.e., $\hat{\mathbb{P}}^{\prime}=\hat{\mathbb{P}}+\Delta\hat{\mathbb{P}}$, as shown in Fig. 1, the eigenvalue problem (6) transforms into

where $\vec{\mathbb{U}}(\mathbf{x})$ and $\Omega$ are the eigenvectors and complex eigenvalues of the perturbed system, respectively. In the most general linear constitutive description, the perturbation operator $\Delta\hat{\mathbb{P}}$ need not be block diagonal. Its off-diagonal components couple the pressure and particle-velocity fields and describe acoustic bianisotropy, commonly referred to as Willis coupling 35; 48; 52. In the following, however, we restrict our analysis to block-diagonal perturbations associated with spatial variations of the mass density and compressibility.

We obtain the solution of Eq. (8) through the expansion of the perturbed system RSs $\vec{\mathbb{U}}(\mathbf{x})$ into the unperturbed system RS $\vec{\mathbb{F}}(\mathbf{x})$, using Eq. (5) in Eq. (3) and considering $\vec{\mathbb{J}}(\mathbf{x})=-\Omega\Delta\hat{\mathbb{P}}(\mathbf{x})\vec{\mathbb{U}}(\mathbf{x})$:

Let us expand the perturbed field $\vec{\mathbb{U}}(\mathbf{x})$ inside the system into the unperturbed RS $\vec{\mathbb{F}}_{n}(\mathbf{x})$ as

With the previous expansion, Eq. (9) transforms to the following linear matrix form eigenvalue problem

where the matrix perturbation element $V_{nn^{\prime}}$ is

The structure of Eq. (11) corresponds to the Brillouin-Wigner form of perturbation theory in both quantum mechanics 2; 27 and electrodynamics 38, rather than to the conventional Rayleigh-Schrödinger perturbation expansion 23. It has a direct physical meaning, because it measures the overlap between the spatial profile of the perturbation $\Delta\hat{\mathbb{P}}(\mathbf{x}^{\prime})$ and two RSs $\vec{\mathbb{F}}_{n}(\mathbf{x})$ of the unperturbed system. In brief, $V_{nn^{\prime}}$ plays the same role as a perturbation matrix element in quantum mechanics. Diagonal elements of Eq. (12) with $n=n^{\prime}$ describe the self-action of a mode and determine, to first order, the shift of its complex eigenfrequency $\omega_{n}$, whereas off-diagonal elements of Eq. (12) with $n\neq n^{\prime}$ describe the perturbation-induced coupling between different RSs. In a Hermitian quantum-mechanical problem, such off-diagonal matrix elements are directly related to transition amplitudes, and transition probabilities are proportional to $|V_{nn^{\prime}}|^{2}$. Instead, in a non-Hermitian acoustic problem, the matrix element $V_{nn^{\prime}}$ is a complex coupling amplitude between leaky RSs, responsible for modal hybridization, avoided crossings, resonance splitting, and changes of the radiative linewidth. In the RSE framework, the RS $\vec{\mathbb{U}}_{n}(\mathbf{x})$ of the perturbed system is instead expanded over the whole basis of unperturbed RS $\vec{\mathbb{F}}_{n}(\mathbf{x})$ with Eq. (10), and the resulting matrix problem given by Eq. (11) is solved directly.

One can reformulate Eq. (11) as a matrix eigenvalue problem of the form $\hat{W}\mathbf{c}=\Omega\mathbf{c}$, where the matrix elements are given by $W_{nn^{\prime}}=\omega_{n}(\delta_{nn^{\prime}}+V_{nn^{\prime}})^{-1}$, and the RSE coefficients $c_{n}$ form its eigenvector $\mathbf{c}$. Diagonalization of the matrix $\hat{W}$ yields the eigenfrequencies $\Omega$ as well as the corresponding expansion coefficients $c_{n}$ for the perturbed RS $\vec{\mathbb{U}}_{n}(\mathbf{x})$.

Since the RSE framework allows us to expand the perturbed field into an unperturbed series, each pressure field RS will contribute to the perturbed state. Consequently, the perturbed RS field needs to be renormalized numerically with Eq. (7). Let us introduce functional $N[\vec{\mathbb{F}}_{n}]\sim\vec{\mathbb{F}}_{n}^{2}$ formed by the both integrals from Eq. (7). Substituting Eq. (10), one can obtain the following form of the functional $N[\vec{\mathbb{F}}_{n}]$

Here, $\hat{\mathbb{M}}_{nn^{\prime}}(\Omega)$ is the normalization matrix element, defined by both volume and surface contributions from Eq. (7). If the eigenvector $\mathbf{c}$ of $\hat{W}\mathbf{c}=\Omega\mathbf{c}$ is found, we can define the value of the normalization functional Eq. (13) of the RS, written in the basis of unperturbed eigenstates as $\mathcal{N}=\mathbf{c}^{\mathsf{T}}\hat{M}\mathbf{c}$. Perturbed system acoustic field vector is defined with the normalization as

We first consider the source-free acoustic problem, $\mathbf{f}(\mathbf{x})=0$, for which the RS are defined as the eigenmodes of Eqs. (1). Eliminating the velocity field from these equations by substituting the second equation into the first one, one obtains the Helmholtz equation $\nabla\cdot[\rho^{-1}(\mathbf{x})\nabla p(\mathbf{x})]+\omega^{2}\beta(\mathbf{x})p(\mathbf{x})=0$ for the pressure field $p(\mathbf{x})$, with sound speed $c(\mathbf{x})=[\beta(\mathbf{x})\rho(\mathbf{x})]^{-1/2}$, which in general depends on coordinate $\mathbf{x}$. Once the pressure eigenmodes are found, the corresponding velocity field $\mathbf{v}(\mathbf{x})$ is obtained from the second of Eqs. (1). These pressure and velocity fields together form the RSs of the acoustic system.

Figure 1 illustrates the main concept of the RSE framework. Various inclusions, such as porous materials or metamaterial absorbers, may be present within the system and influence the eigenfrequencies and eigenfunctions. This significantly increases the complexity of the problem and often renders it analytically intractable. The solution to this challenge is provided by the RSE, which allows one to express the solution of a complex system with $\{\Omega_{n},\vec{\mathbb{U}}_{n}(\mathbf{x})\}$ in terms of an analytically solvable reference system, characterized by $\{\omega_{n},\vec{\mathbb{F}}_{n}(\mathbf{x})\}$, with an added perturbation $\Delta\hat{\mathbb{P}}$. In this work, three types of perturbations are considered for both analytical and numerical investigation in order to develop an acoustic RSE framework and to validate its accuracy against other simulation methods. Firstly, in the case of homogeneous perturbation, the initial unperturbed resonator is replaced with another material, whose parameters are constant throughout the entire internal space of the resonator. This represents the simplest type of perturbation and is used to demonstrate the accuracy of the RSE method. Next, the radial perturbation of the material parameters is considered. In this case, the perturbation is angularly independent and changes with respect to radius. That perturbation modifies the radial distribution of the resonator eigenmodes, whereas axial symmetry remains unchanged. Finally, we consider sectoral perturbation, which breaks the axial symmetry $C_{\infty}$ of the initial cylindrical resonator. In this case, the resonator is separated into different sectors with their own material parameters. The analysis of these types of perturbations within the RSE framework enables the investigation of wave scattering by complex structures with arbitrary spatial variations of material parameters.

As a starting point, we choose a two-dimensional analytically solvable reference system. Specifically, a homogeneous circular cylinder of radius $R=10~\mathrm{cm}$, embedded in a homogeneous background medium (air with sound velocity $c_{b}=343~\mathrm{m/s}$ and density $\rho_{b}=1.2~\mathrm{kg/m}^{3}$). The cylinder material parameters are normalized relative to the background air parameters, $\rho_{0}=10\rho_{b}$ and $c_{0}=c_{b}/2$. Here, we neglect the frequency dependence of material parameters and choose these specific values, which are corroborated by numerical analyses of experimentally acquired data for analogous resonant systems, specifically, labyrinth-like and coil structures, operating at comparable frequencies 25; 33; 58. Hence, compressibility for both materials is $\beta=1/\rho c^{2}$.

This system is used as the unperturbed basis of the RSE. More general two-dimensional acoustic systems, such as cylinders with radial or azimuthal inhomogeneity, are then treated as perturbations of this reference configuration. Since such inhomogeneous systems generally do not admit closed-form analytical solutions, the RSE provides an efficient and systematic way to determine their resonances.

For this cylindrically symmetric system, the Helmholtz equation takes the form

where all material parameters including mass density $\rho$, compressibility $\beta$ and sound velocity $c$ are azimuthally homogeneous and depend only on the radial coordinate:

Due to the cylindrical symmetry, the azimuthal index $m$ is a good quantum number that takes integer values, giving the number of field oscillations around the cylinder. The RSs of the unperturbed system can therefore be written as:

Here $\chi_{m}(\varphi)$ representing angular part of RS (16) is defined by

and satisfies the orthonormality relation

The radial part of Eq. (16) has the form

where $J_{m}(k_{n}r)$ and $H_{m}(k_{n}^{ex}r)$ are cylindrical $m$-order Bessel functions and Hankel functions of the first kind, respectively, $k_{n}=\omega_{n}/c_{0}$ and $k_{n}^{ex}=\omega_{n}/c_{b}$. It follows from Eqs. (1) and (16) that the velocity components corresponding to the pressure eigenmode can be written in cylindrical coordinates as

The normalization constant for RSs in cylindrical coordinates for the non-dispersive materials, for which $\partial\hat{\mathbb{P}}/\partial\omega=0$, in accordance with the orthonormality relation (7), takes the form:

We find that the eigenfunctions are normalized according to (20) with the normalization constant

The general form of the constant normalization for a homogeneous cylinder in a medium is simplified in two specific cases. First, for azimuthally homogeneous modes $m=0$, the last term is eliminated. Second, if the resonator density $\rho_{0}$ and the background medium density $\rho_{b}$ are equal, regardless of the azimuthal moment, the normalization constant is simplified to $A_{n}^{m}=\left(R\sqrt{\beta_{b}-\beta_{0}}\right)^{-1}$. Once $\beta_{b}=\beta_{0}$, physically, a resonator does not exist, as well as a complete and orthogonal set of RSs; meanwhile, the pressure field can not be normalized in open space.

The boundary condition at the surface of the cylindrical resonator, specifically the continuity of the pressure derivative normalized to the corresponding density, yields the following secular equation for the RS frequencies

where $\gamma=\rho_{b}c_{b}/\rho_{0}c_{0}$. We find the solution of Eq. (22) numerically for the eigenmodes with azimuthal number $m=0~\mathrm{to}~60$. Eigenstates characterized by a real part approaching zero and a large imaginary component correspond to evanescent waves outgoing from the resonator area (shown in Fig. 2 by shading with white colour). In contrast, eigenfrequencies with a small imaginary part are associated with high-order RSs, ”whispering gallery”-like modes, and generally localized inside the considered resonator. These modes exhibit extremely high quality factors $Q$ (shaded in red colour in Fig. 2). Leaky modes of the resonator, characterized by a pressure field distributed throughout the entire considered area, exhibit significantly higher energy losses to the surrounding environment when compared to the "whispering gallery"-like modes, which are illustrated in Figure 2 and shaded in blue. To accelerate the simulation and improve the accuracy of eigenvalue calculations, we employ Cauchy’s argument principle 56; 40 paired with the Newton method to solve the transcendental equation (22).

In addition to the discrete resonant poles, the Green’s function may contain non-pole singularities that must be included in its spectral representation. In the two-dimensional acoustic problem considered here, the Green’s function possesses a branch cut along the imaginary-frequency axis. Its spectral representation therefore consists of a sum over the discrete RS poles supplemented by an integral along the branch cut 9,

where $\sigma_{m}(\omega)$ is the branch-cut weight function, which determines the contribution of the continuum along the cut and is given by

Considered two-dimensional GF (23) corresponds to the dyadic GF (5), satisfying the Eq. (4) with the factor $2\omega$. We discuss the details of the two-dimensional acoustic problem GF in Sec. II and III of the Supplemental material 46. The above-noted weight function $\sigma_{m}(\omega)$ introduces cut-pole contributions to the eigenfrequencies’ positions in the perturbed system spectra. For the low azimuthal numbers, its contribution is insubstantial, as shown in 46, unlike the high azimuthal numbers, where one needs to take into account a large number of cut poles, leading to long-time simulations.

Before proceeding to particular perturbation profiles, we introduce the perturbation matrix element in its most general pressure-only form. Using Eq. (12) and eliminating the velocity field via Eq. (1), the perturbation matrix elements can be expressed solely in terms of the pressure eigenfunctions of the unperturbed system as follows:

with the integration restricted to the region where the material parameters are perturbed. This form clearly separates the effects of compressibility and density variations and provides a unified starting point for treating homogeneous, radial, and sectoral perturbations considered below (see details in Sec. IV of the Supplemental material 46).

First and foremost, we begin with the simplest perturbation of the reference system: a homogeneous modification of the material parameters inside the cylinder (inset figure in panel (c) in Fig. 3). Such a perturbation is independent of the polar angle and can be written as $\Delta\rho(r)=\Delta\rho\ \theta(R-r)$ and $\Delta\beta(r)=\Delta\beta\ \theta(R-r)$. Since this perturbation preserves the cylindrical symmetry of the unperturbed resonator, the azimuthal index $m$ does not couple. According to the orthonormality of the angular functions $\chi_{m}(\varphi)$, given by Eq. (18), the perturbation matrix is diagonal in the azimuthal basis, i.e. $V_{mm^{\prime}}^{\ell\ell^{\prime}}=V_{m}^{\ell\ell^{\prime}}\delta_{mm^{\prime}}$, and only RSs with the same azimuthal number $m$ are coupled. The perturbation matrix element (12) has a block-diagonal structure $V=\mathrm{diag}\left\{V_{m=0},V_{m=1},\ldots,V_{m}\right\}$, where each matrix block $\hat{V}^{(m)}$ may still contain off-diagonal elements with respect to the radial indices $\ell$ and $\ell^{\prime}$ define field distribution along the cylindrical resonator radius, but different azimuthal numbers $m$ and $m^{\prime}$ do not couple.

We choose the perturbations in the material parameters to be related to those of the unperturbed resonator, with $\Delta\rho=0.1\rho_{0}$ and $\Delta\beta=0.1\beta_{0}$. Within the RSE framework, matrix perturbation elements are determined analytically for the non-diagonal case $n\neq n^{\prime}$ and the diagonal case $n=n^{\prime}$ (see Supplemental Material 46, Sec. IV A). Importantly, since the homogeneous perturbation does not change the circular symmetry of the unperturbed problem, the secular equation (22) with parameter $\gamma$ changed, still gives us the RSs of the perturbed system. To illustrate the accuracy of the RSE, we compare numerically acquired results for the perturbed system, obtained via Eq. (22), with the RSE calculations discussed above, as shown in Fig. 3. In addition, for reference, we also include the unperturbed spectra for specific azimuthal numbers $m=3$ and $m=10$.

Figure 3(a) demonstrate the spatial distributions of the pressure field modulus for specific resonator eigenmodes with $m=3$ and $m=10$. In both cases, the field distribution structure exhibits symmetry with respect to a specific value of the azimuthal number $m$, and displays twice the number of local field maxima compared to the $m$ value. The degree of radial degeneration depends on the radial quantum number $\ell$, which determines the number of local maxima along the cylinder radius. For example, the field A with $m=3$, $\ell$ equals 4 shown in Fig. 3(a). In contrast, the field B in Fig. 3(a) with $m=10$ has the radial index $\ell$ equals 2.

The homogeneous perturbation is especially useful because it provides an analytically verifiable test case for the acoustic RSE. In this situation, the perturbed resonant frequencies can be determined independently from the exact secular equation of a homogeneous cylinder with modified material parameters. Therefore, the RSE results can be benchmarked directly against the exact solution, without the need for additional numerical reference calculations.

Next, we consider a radial perturbation of the cylinder material parameters, preserving cylindrical symmetry. In this scenario, the perturbation depends solely on the radial coordinate, altering the material uniformly in all directions from the axis. This resembles an isotropic perturbation, as shown in Fig. 4. The orthogonality of the angular functions $\chi_{m}(\varphi)$ ensures that each azimuthal index $m$ remains unaffected by the others and stays separated.

We introduce this angularly isotropic radial perturbation as $\Delta\rho(r)=\Delta\rho f_{\rho}(r)\theta(R-r)$ and $\Delta\beta(r)=\Delta\beta f_{\beta}(r)\theta(R-r)$. Here, $f_{\rho}(r)$ and $f_{\beta}(r)$ are dimensionless functions. They describe the dependence of the material parameters on the distance from the resonator center. We discuss the details of RSE for radial perturbation of various forms in Supplemental Material 46, Sec. IV B. In general, the integral (25) is taken for arbitrary dimensionless functions $f_{\mathcal{\rho}}(r)$ and $f_{\mathcal{\beta}}(r)$, dividing into two parts. However, in the absence of radial dependence, i.e., $f_{\mathcal{\rho}}(r)=f_{\mathcal{\beta}}(r)=\mathrm{const.}$, the integrals simplify to the form of the homogeneous perturbation. Finally, as noted, they are taken for diagonal $n=n^{\prime}$ and non-diagonal $n\neq n^{\prime}$ cases.

To be specific, we define radial characteristic functions as $f_{\rho}(r)=f_{\beta}(r)=r/R$ for numerical simulations. This leads to a smooth linear variation of material parameters along the cylinder radius, transitioning from $\{\rho_{0},\beta_{0}\}$ in the center of the structure to $\{\rho_{0}+\Delta\rho,\beta_{0}+\Delta\beta\}$ at the cylinder boundary. Likewise to the homogeneous perturbation, for the numerical analysis, we choose a perturbation of material parameters related to the material parameters of the initial resonator as $\Delta\rho=0.2\rho_{0}$ and $\Delta\beta=0.4\beta_{0}$.

In order to validate the accuracy of RSE for the considered perturbation form, we numerically simulate an open cylindrical resonator with radial dependence of the material parameters. Numerical computations were performed in COMSOL Multiphysics using the finite-element method with a user-controlled mesh and an extra-fine element size. Initially, to accelerate calculations, the one-dimensional axisymmetric model was used. The cylindrical wave radiation condition was used in order to simulate open boundary conditions. The Pressure acoustics module was used in eigenfrequency calculation for Fig. 4.

As shown in Fig. 4, the RSE results are in close agreement with the numerical simulations, while requiring substantially less computational time by expanding the perturbed states in the basis of the unperturbed resonator. The contribution of cut poles to the eigenfrequency spectra remains significant due to the high-$Q$ factor modes analysis. Considered radial perturbation changes material parameter values close to the cylinder border and changes the radial modal profile while preserving its azimuthal order, as shown in panel (a) in Fig. 4, in general.

Finally, we introduce a radially independent perturbation with a nontrivial angular dependence, $\Delta\rho(r,\varphi)=\Delta\rho S_{\rho}(\varphi)\theta(R-r)$ and $\Delta\beta(r,\varphi)=\Delta\beta S_{\beta}(\varphi)\theta(R-r)$. Unlike the homogeneous and radial perturbations considered above, this perturbation explicitly reduces the rotational symmetry of the unperturbed cylindrical resonator from $C_{\infty}$ to a discrete symmetry $C_{N}$. Here, the dimensionless characteristic functions $S_{\rho}(\varphi)$ and $S_{\beta}(\varphi)$ describe the angular distribution of the material perturbation and are equal to 1 inside the corresponding sectoral regions (for details see Supplemental Material 46, Sec. IV C). In the following, we consider identical even profiles, $S_{\rho}(\varphi)=S_{\rho}(-\varphi)$ and $S_{\beta}(\varphi)=S_{\beta}(-\varphi)$, characterized by a rotational symmetry $C_{N}$, where $N$ is the order of the rotational axis and the corresponding rotation angle is $\varphi_{N}=2\pi/N$. Such discrete rotational symmetries are common in sectoral and space-coiling acoustic meta-atoms.

We consider two representative perturbation profiles corresponding to $N=1$ and $N=4$, i.e., perturbations with $C_{1}$ and $C_{4}$ symmetry, respectively, as shown in Fig. 5. Analytical evaluation of the angular part of the perturbation matrix element (25) shows that the rotational symmetry imposes selection rules on the coupling between different azimuthal modes. As a result, the perturbation matrix acquires a block structure in which certain coupling terms vanish depending on the symmetry of the angular profiles $S_{\rho}(\varphi)$ and $S_{\beta}(\varphi)$. The corresponding derivation and explicit expressions for these symmetry-dependent coupling terms are given in Supplemental Material 46, Sec. IV C.

For numerical analysis, we set the material parameter perturbations as $\Delta\rho=0.2\rho_{0}$ and $\Delta\beta=0.4\beta_{0}$. The half-cylinder perturbation occupies an interval of $\varphi=180^{\circ}$, whereas the $C_{4}$-symmetry profile consists of alternating sectors of width $\varphi=45^{\circ}$. Likewise to the radial perturbation, we validate the RSE solution with the numerical one, obtained by COMSOL Multiphysics, as shown in Fig. 5. Similarly to the previous cases, we present the reference solution for the unperturbed system.

In sectoral perturbation, different azimuthal harmonics are coupled. This coupling prevents a frequency shift or changes in the pressure field distribution for any single mode, unlike previous perturbation types. Consequently, the pressure field exhibits a complex form for both sectoral perturbation symmetries, as shown in Fig. 5. One may observe that the asymmetric field distribution of the chosen mode is due to the difference between material parameters in the half-cylinder perturbation, which leads to a higher degree of field localization and the appearance of another field maximum for the half-cylinder profile in Fig. 5(a). Similarly, the $C_{4}$-symmetric mode chosen in Fig. 5(c), is generally localized within the perturbed sector areas, but in contrast has a symmetric spatial distribution.

More importantly, the sectoral perturbation produces a significant shift in the eigenfrequency and a pronounced reduction in the quality factor $Q$ of the “whispering-gallery” modes. Increasing the number of perturbation sectors leads to a greater loss increase, as shown in the comparison in Figs. 5(b) and 5(d). In scattering terms, the reduction of the Q factor originates from enhanced radiative leakage induced by the sectoral perturbation, which breaks the original symmetry and couples the resonant state to additional radiative channels. Since the perturbations $\Delta\rho$ and $\Delta\beta$ are real, no additional absorptive material losses are introduced.

Here, we use a number of basis sizes equal to $N=5694$ for the RSE calculation. One may observe RSE’s high accuracy in comparison with numerical simulations obtained by COMSOL Multiphysics. Previously, we discussed the significance of the presence of cut poles for the RSE in the case of homogeneous and radial perturbations. However, low-$Q$ modes in sectoral perturbations do not require cut-mode contributions due to the high magnitude of the imaginary part of the eigenfrequency, as shown in panel (d) in Fig. 5. That fact leads to a substantial reduction in numerical simulation time. Notably, the RSE system also influences the number of found eigenfrequencies in a finite time, unlike numerical simulation solutions, which give fewer numbers. This makes it possible to accurately calculate natural frequencies for resonators of arbitrary shape without solving the problem of eigenvalues for them.