We start by investigating the eigenmode properties of Helmholtz-like resonator system shown in Fig. 2a. Such an open system can be treated as a closed rectangular acoustic cavity suject to deformation. A small air gap is opened to introduce the interaction between the cavity and the exterior environment. The role of the acoustic waveguide connecting to the neck is to guide the incoming acoustics waves, enabling measurement in experiments, as will be described in the later section. Because of its non-Hermitian nature, this open system turns the closed cavity modes into leaky modes (also denoted as quasinormal modes). The leaky modes have complex eigenfrequencies ω= ω0-iγ, where ω0 and γ are the resonant frequency and radiative decay rate, respectively. The radiative Q-factor can be derived from Q = ω0/(2γ). Here, each leaky mode's complex eigenfrequency is calculated by COMSOL Multiphysics. Following the definition in Ref [20], the leaky modes are labelled as Mpq, where p and q are the number of maxima in the pressure field along the x- and y-axes, respectively.

We demonstrate that the reflection or transmission spectrum of such Helmholtz resonators can be perfectly reproduced with a complex eigenfrequency of leaky modes based on coupled-mode theory (CMT)25 (See Sect. 1 and Fig.S1 in supporting material (SM)). Thus, the goal in searching BICs is to find leaky modes with infinitely large Q-factor. Since a BIC is also accompanied by vanishing linewidth of the acoustic resonance in the reflection or transmission spectrum, we calculate the reflection coefficient as a function of frequency and size ratios in Fig. S2 while assuming the system is lossless. Many BICs can be found by checking the vanishing linewidth of resonant peaks. These BICs can be categorized into three types (See Fig. 1): symmetry protected BICs (Fig. 1a-b), Friedrich-Wintgen BICs induced by two mode interference (Fig. 1c-d) and mirror-effect induced BICs (Fig. 1e-f). Except where explicitly mentioned, only the lossless case is studied for calculating the leaky modes. The realistic system's losses will deteriorate the total quality factor by 1/Q = 1/Qabs+1/Qrad and will be discussed in the experiment section.

**2.1 Symmetry-protected quasi-BIC**

Referring to Fig. 2a, we consider a rectangular cavity of dimensions Lx by Ly. If only one open port is introduced on the left of the acoustic resonator, symmetry is broken along the y axis but still maintained along the x axis. Thus, it will support several symmetry protected BICs as long as the centre of neck and resonator are symmetric with respect to the x-axis. The left waveguide's existence will further lead to broken symmetry along the x-axis, and thus turn an ideal BIC into a quasi BIC. Without loss of generality, we use mode M12 in the square cavity as an example to describe the effect of neck width w and centre shift yc on the Q-factor of BIC. Interestingly, from Fig. 2b, the Q factor is still larger than 104 for the system with protected symmetry along the y axis even when the neck's width is half of the right cavity width. The Q-factor increases exponentially with decreasing neck width. This can be understood by treating the neck as a perturbation of the square cavity. The smaller neck width, the smaller the perturbation, and thus the larger the Q-factor is. Another interesting finding is that the Q-factor is proportionally to 1/(yc)2 when the neck width is small compared to the width of the right cavity, as shown in Fig. 2c. This phenomenon has been observed and proved for BIC in photonic systems26. Additional symmetry protected quasi BICs can be found in such structures as long as q is even number for mode Mpq(See Fig.S3).

**2.2 Friedrich-Wintgen BIC induced by mode interference.**

Friedrich and Wintgen demonstrated that in quantum mechanics full destructive interferences of two degenerate modes gives rise to the avoided crossing of eigenvalues, accompanied by the formation of a BIC27. This type of BIC can be easily constructed in our system by tuning the size ratio of a rectangular cavity. For a closed rectangular cavity, cavity modes Mpq and Mp+2,q−2 will become degenerate at a certain size ratio. When an air gap is introduced on one side of the rectangular cavity, strong coupling between these two modes results in the giant enhancement of Q-factor for one mode but suppresses it for another. We use paired modes M23 and M41 as an example to illustrate this principle. The pressure distributions for these two modes are shown in the inset of Fig. 2d. It is found that the eigenfrequencies for these two modes cross at R = Lx/Ly = 1.42 in a closed cavity. All the eigenmodes become leaky modes with complex eigenfrequencies when introducing the neck to couple the acoustic waveguides to the cavity. Based on the eigenmodes analysis, we find that the real part of eigenfrequencies exhibits an avoided crossing. Simultaneously, the Q-factor for mode M23 is enhanced to a maximum of 6.87⋅107 but suppressed to minimum 86.76 for mode M41 at R = 1.398, as shown in Fig. 2e-f.

Moreover, the avoided crossing suggests that these two modes interchange with each other after the size ratio passes through the critical size ratio. This interesting phenomenon is confirmed by the mode evolution, as shown in Fig.S4. Indeed, the upper branch mode, for example, evolves from mode M23 into mode M41 when the size ratio increases from 1.3 to 1.5. Following a similar strategy, more BIC induced by mode interference, such as M13-M33 and M33-M51, can be found by merely constructing avoided crossing (See Fig.S5a-d). Besides, we find that BIC can also be found in mode crossing for M24 and M42 (See Fig.S5e-f). Here, we emphasize that not all crossings of two modes of a closed cavity can bring about BICs. The two modes must have the same parity along both x- and y-directions. This is also the reason why we choose paired modes Mpq and Mp+2q−2 here.

**2.3 Mirror-induced BIC**

In addition to the abovementioned two types of BICs that have been intensively studied in the photonics community, we also find a new type of BIC: mirror-symmetry induced BIC (See Fig. 1e-f). Because all the outer boundary conditions are set as a hard wall in simulation, we can view the rightmost boundary as a partial mirror. All the eigenmodes with almost symmetric pressure distribution in the full-size resonator can also be found in a half-size resonator. Thus, many other BICs can also be constructed by simply shrinking the width to half based on this mirror effect. For example, one can easily find a BIC at R = 0.498 for mode M13 (Fig. 2g), which is indeed half of the critical size ratio for BIC M13 in a full resonator (Fig. 2f).

Moreover, we find that the mirror effect also occurs for the x-axis. As shown in Fig. 2i, a BIC can be found at R = 1.986. Following a similar approach, more BICs can be constructed (See Figs.S6-7). The mirror effect indicates that one can achieve extreme pressure confinement even with reduced size in the 2D case, suggesting an effective way to engineer the Purcell factor that is the key to realize enhanced acoustic emission28.

Note that the above three BIC types are not limited to regular rectangular shaped resonators. We can also find them in the elliptical resonator (See Figs.S8). The only difference is that the size ratio is defined as R = a/b, where a and b are semi-major and semi-minor axes, respectively. Besides, the conclusion drawn in the 2D case can be straightforwardly generalized to three-dimensional (3D) open resonators (e.g. cuboid resonators) (See Figs.S9-11). More freedom is provided in the 3D case because three parameters including length, width and height, are involved in the mode calculation.

**2.4 Experimental verification of BIC**

Next, we switch to the experimental demonstration of all three types of BICs. We fabricate two acoustic cuboid resonators shown in Fig. 3a: full resonator and half resonator. In experiments, the left circular tube's diameter is fixed as d = 29mm while the length, width, and height for the neck are set as 40mm, 20mm, 20mm, respectively. Figure 3b shows the measurement set up while Fig. 4a-c depicts the structure's schematic. Figure 3c shows the measured transmission spectra for the full resonator and half resonator, and Fig. 3d-g corresponds to the zoomed-in range in the vicinity of the BICs while Fig. 3h shows the pressure distribution of BICs. Excellent agreement can be found between simulation and experiment over the full spectrum (See Fig.S12). Other leaky modes that are not BICs are given in Fig.S13. In the following, we discuss all three types of BIC observed in experiments.

To study the symmetry protected BIC, we fix Lx = Ly = Lz = 60mm (Lx = 30mm, Ly = Lz = 60mm) for full (half) resonator and only change the centre shift yc and zc for the cuboid resonator with respect to the symmetric axis of the neck. From Fig. 3d-f, it can be found that there are three symmetry protected BICs. For example, when yc increases from 0 to 1mm for the half resonator, a dip shows at around 2920Hz (green curve), which corresponds to mode M121 (See Fig. 3h-BIC 6). However, we did not observe M121 (BIC 1) for the full resonator when the same asymmetry parameter yc = 1mm is introduced. This can be explained by that this mode's Q-factor decreases relatively slowly with respect to yc and makes the resonance vanish in the low-Q mode background spectrum. Further enlarging yc may help to excite this mode.

Interestingly, another symmetry protected BIC M221 (BIC 2) appears for a full resonator with yc = 1(blue curve), evidenced by a shallow dip in the transmission spectrum (See Fig. 3e). For mode M222 (BIC 3), it is not enough to introduce the asymmetry by shifting yc. The mode becomes visible in the transmission spectrum when yc and zc are adjusted to 4mm simultaneously, as shown in Fig. 3f (magenta curve). We also systematically study the role of yc (or yc=zc) on the Q factor for BIC mode M121, M221 and M222. The simulated and measured transmission spectra for the modes M121 and M221 are presented in Fig. 4d-e and Fig. 4g-h, respectively while the cases of M222 can be found in Fig.S14. Good agreement can be found between these two. When yc is reduced to zero, the vanishing line width of resonances indicates the BIC's appearance. The Q-factor can be obtained by fitting the spectrum with the Fano formula29 (See Sect. 2 in SI and Fig.S15). Figure 5a-b shows the measured Q-factor vs yc for the former two modes while the Q-factor of M222 is put in Fig.S16. The maximum Q-factor for these two modes is only 250, lower than the theoretical prediction (see Fig. 5e-f). This is because there are loss in the real system due to the thermo-viscous boundary layers, which degrades the Q-factor. Moreover, the Q-factor reduces with the increasing yc, matching the trend of Q-factor vs yc. Here, it is worth noting that the Q-factor for yc = 0mm should be larger than 250. However, we cannot retrieve the exact value because they are almost indistinguishable from the background, which is exactly the signature of BIC.

For the two-mode interference induced BIC, we fix Lx = Ly = 60mm and yc = zc = 0mm, but only vary Lz. The size ratio Rx and Ry are defined as Rx = Lz/Lx and Ry = Lz/Ly, respectively. Thus, we have R = Rx = Ry for Lx = Ly. For the fixed Lz = 60mm (Rx = Ry = 1), we can find from Fig. 3g that there are two BICs (BICs 4–5) in the range 5750-5950Hz, where the pressure field distributions are shown in Fig. 3h (mode 4 and 5 for the full resonator, and mode 7–8 for the half resonator). Unlike symmetry-protected BICs, these two BICs always exist regardless of the value of yc. Their Q-factors only depend on the size ratio. The simulated and measured transmission spectra for full resonator are presented in Fig. 4f and Fig. 4i where resonator height Lz varies from 59.5mm to 60.5mm. The measurement results of the half resonator with different Lz can be found in Fig.S17. Figure 4b-c shows the Q-factor of two modes as functions of size ratio. For the full resonator case, the maximum Q-factor is 583 at R = 1 for mode B(M113) while it is 485 at R = 1.008 for mode A (M131). Note that the trend of measured Q factor vs R devitates from the theoretical prediction (See Fig. 5e-f) because of inevitable intrinsic loss in the real system. The theoretical calculated Q-factor for mode A is maintained at a high value because the size ratio Rxy=Ly/Lx is always 1, which is precisely the critical size ratio. For mode B, the Q-factor reaches the maximum at critical size ratio Rx = Ry = 1. However, the majority of Q-factors are ranged between 400–500. Such high-Q factors are already good enough for real applications, such as ultra-narrowband acoustic absorbers and enhanced acoustic emission. A similar phenomenon can also be observed for the half resonator. The only difference is that the maximum Q-factor for mode A (BIC 4) and B (BIC 5) are 393 and 327, respectively, which are lower than the full resonator case. The reduction of Q-factor may be attributed to increased thermo-viscous losses arising from the acoustic resonator's smaller volume.