Acoustic resonance coupling for directional wave control: from angle-dependent absorption to asymmetric transmission

We investigate acoustic wave coupling between two resonators of very different intrinsic losses, e.g. lossy and lossless resonators as an extreme case. We find that the resonator pair placed on a reflector exhibits angle-dependent absorption, showing an order of magnitude difference for opposite angles of incidence. The results obtained from a harmonic oscillator model show excellent agreement with numerical results. Moreover, our analytical model explicitly describes the contribution of radiation leakage and coupling to the angle-dependent absorption, enabling one to design a simple resonant structure demonstrating asymmetrical transmission.

Interaction between resonators with a large loss contrast has drawn attention, because it is often avoidable and, more importantly, functions as an essential building block of non-Hermitian metamaterials, such as a lossy acoustic metasurface [16,17], a lossy acoustic topological insulator [18] and a passive PT symmetry structure [19]. To study coupled resonance in optics, couple-mode theory has proven useful for understanding the interaction between resonance modes [20,21]. Similarly, in acoustics, formalisms based on a Lorentzian model have been used to capture physical characteristics as well as spectral behaviors of coupled resonators [7,22]. However, such models were considered for simple scenarios, e.g. wave propagation in a waveguide [22]. In addition, these models often implemented a lumped loss parameter accounting for both internal loss and resonance coupling [7], partly because of complexity involved in characterizing radiation leakage and coupling between resonators [23].
Here, we investigate the interaction between a pair of lossy and lossless resonators placed on an acoustic reflector using an analytical formula based on a harmonic oscillator model. As an isolated system, we study the absorption cross section of the resonator pair for acoustic waves incident on the same side of the reflector. Our formalism includes closed-form expressions of radiation leakages from the individual resonators and radiation coupling between the two resonators, thereby enabling to characterize the radiatively-coupled resonators from an interference perspective. We observe angle-sensitive absorption in the resonator pair, and our simulations show an order of magnitude difference in absorption between two oblique incidence angles (tilted to the right and left, at opposite angles). The analytical results of the harmonic oscillators show excellent agreement with numerical results obtained using a finite-element-method simulator. Such angle-dependent absorption comes from the interference of the radiated waves from the resonators, which is explicitly explained with the analytical model. In particular, we find that high absorption in the lossy resonator occurs when the effective radiation leakage of the lossy resonator is reduced by the lossless resonator. Moreover, based on the coupling between resonators, we show a transmissive resonant structure enabling asymmetrical transmission.

Theory
To study coupled resonances, we consider an isolated pair of lossy and lossless harmonic oscillators on an acoustic reflector in a two-dimensional (2D) domain, as illustrated in figure 1(a). These two resonators operate at the same resonance frequency (f 0 ), but only the left oscillator has an intrinsic loss (otherwise identical with the same mass,m and spring constant, k). The harmonic oscillators, whose widths (s) are identical and subwavelength (s c f , / c the speed of sound), isotropically radiate acoustic waves to the half-space as like a point scatterer. The resonators placed with a spacing of d are coupled each other via the radiation. Note that these harmonic oscillators interacting with incident acoustic waves represent acoustic resonators such as Helmholtz resonators, quarter-wavelength resonators, and membrane-type resonators, enabling us to focus on coupling phenomena applicable to all these resonator types instead of limiting to a specific type of acoustic resonators. In addition, such a simple model system as the isolated resonator pair has relevance to practical systems consisting of acoustic sources (speaker) or detectors (microphone) in combination of adjacent lossless resonators, and in this system, the lossless resonators radiatively coupled with the acoustic sources and detectors can control the characteristics (e.g. directivity and sensitivity) of the sources and detectors.
The coupling between the resonators is characterized by radiation coupling rate ( c h ), while the interaction between the resonators and free space is characterized as radiation leakage rate ( r h ). Using the radiation leakage rate ( r h ) and coupling rate ( c h ), the coupled equation of motion is expressed by [24,25] where the subscripts 1 and 2 denote the individual resonators, respectively, y is the vibration amplitude, l h is the internal loss rate having the same unit (kg s -1 ) as the leakage and coupling rates, and Fis the force acting on the resonators due to incident acoustic waves. The radiation leakage rate serves as the damping term of the coupled equations (equation (1)), indicating that a larger radiation leakage leads to a decrease in the vibration amplitude. Assuming a plane wave excitation under normal incidence 0 , q = ( ) we have F t F t .
Z is the acoustic impedance, λ is the wavelength, and 2 o abs s l p = / (i.e. the upper limit of a single-resonance subwavelength resonator in 2D) [27].
Our formulism is based on explicit descriptions of radiation leakage ( r h ) and radiation coupling ( c h ) rather than using a lumped loss ( l r h h + ) as a fitting parameter or considering c h as a real value (i.e. lack of phase information). To find closed-form expressions of r h and , c h we consider the radiated pressure field induced by a harmonically-vibrating resonator. The radiation leakage rate ( r h ) and radiation coupling rate ( c h ) per unit depth (kg ms -1 ), both derived from the 2D dipole Rayleigh integral, are expressed, respectively, by where Z is the acoustic impedance, s is the width of the resonator, g is the Euler's constant of 0.5332, g = and H 0 2 ( ) is the zero-order Hankel function of second kind. The detailed derivation can be found in appendix A.
As plotted in figure 2, the leakage r h ( ) and coupling c h ( ) rates have complex values. The real part of each complex value corresponds to leakage or coupling resistance, whereas its imaginary part is related to acoustic reactance, which does not contribute to acoustic power transfer [28]. The radiation leakage rate of equation (3a) corresponds to acoustic radiation impedance derived in [29]. Note that the radiation leakage rate is proportional to the square of the resonator width s ( ) relative to the wavelength (l), i.e. s . figure 2(a), as decreasing the width s , ( ) the leakage rate (its magnitude) is reduced and its reactance becomes more dominant than its resistance, i.e. Im Re .
Similarly, the coupling rate c h ( ) is gradually decreased with the distance d, whereas its phase angle is drastically varied by d (see figure 2(b)). From equation (2a), the peak absorption occurs at , Equation (4) indicates that the peak absorption frequency ( p w ) depends on the imaginary parts of r h and .
c h When both the imaginary parts are zero, p w is equal to the natural frequency (

Angle-dependent absorption and its mechanism
Based on the resonance coupling discussed above, we demonstrate angle-dependent acoustic absorption, applicable to acoustic sensing or acoustic antennas (directive responses are desired).
The numerical results (symbols) show excellent agreement with the analytical results (solid lines) obtained from the full analytical model of equation (2), validating the closed-form expressions of r h and c h (equation (3)). For other angles of incidence, absorption cross section spectra can be found in appendix B. The numerical results are obtained by using 2D finite-difference frequencydomain simulation (COMSOL Multiphysics 5.3). In the numerical calculation, the vibrating masses of the Typical acoustic absorbers on a reflector are known to have a characteristic of quasi-omnidirectional sound absorption (rather insensitive to incident angles) [30], which is typically desirable for noise reduction. In contrast, the angle-dependent acoustic absorption, observed in our study, can be useful in applications requiring directional acoustic sensing. In such an example, the lossy harmonic oscillator may be replaced by an acoustic detector (transducer), and a lossless acoustic resonator (e.g., Helmholtz resonator) can be placed nearby so as to enable high-sensitivity detection of acoustic waves from a specific direction while preventing sensing of acoustic waves from the opposite angle.
To find conditions for high absorption, equation ( where n is the integer number. The strongest coupling occurs by the smallest d when n 0, In figure 3(b), we confirm that the positive angle of 40 q = +  meets the condition of equation (6), leading to (constructive interference). For this constructive interference, the effective radiation leakage is increased, and thus the vibration of the lossy resonator is suppressed.
The angle-dependent absorption in this paper is based on the ideal system consisting of the lossy and lossless resonator. In a realistic system, losses may be unintentionally introduced. We observe the angle-dependent absorption even in a realistic system consisting of two Helmholtz resonators (see appendix D). The lossless Helmholtz resonator is realized by using a short neck length, and thus the loss in the neck is negligible. For losses comparable to that of the lossy resonator, the transmission contrast is compromised while the angle-dependent absorption is preserved.

Acoustic focusing
In the high absorption condition, the resonance coupling allows the lossless resonator to effectively transfer acoustic energy into the lossy resonator. In other word, acoustic energy is focused without using a focusing lens. From equation (6), we can notice that the resonance coupling is also dependent on the distance (d) of the two resonators. Thus, at a given angle of incidence, the absorption cross section as well as the resonance coupling can be controlled by d. Thus, effective acoustic focusing requires a proper selection of the distance (d). For the normal incidence ( 0 q = ), with no excitation phase difference   figure 3(c). The discrepancy arises because the direct interaction between the lossless resonators negatively affects overall absorption by dampening each other's vibration due to the distance between the lossless resonators (1.2 0 l ) corresponding to 0 f » (i.e. increase of the effective radiation leakage).

Asymmetric acoustic transmission
Based on the resonance coupling, we propose a structure for unidirectional acoustic transmission, as illustrated in figure 5(a). Such a unidirectional acoustic transmission is desirable for practical application scenarios. For example, acoustic waves can be separated and moved from their noise sources such that formation of standing waves is inhibited. In other applications, acoustic energy harvesters enclosed by such panels enabling asymmetric transmission can collect acoustic energy without losing them once acoustic waves pass through the panels.
The  Note that the transmission contrast of the forward to backward waves reaches approximately 15 at 795 Hz. In addition, we find that there is the distinct difference in the total pressure fields between the propagation directions, as observed in figures 5(c), (d).
Similarly, [32] demonstrated a structure consisting of a channel ( 2l width) for asymmetric acoustic transmission, which is based on wave-vector direction conversions. Although the peak transmission of our device is relatively low compared to that of [32], our proposed design shows the large transmission contrast and is enabled by the simpler structure using the resonance coupling. The forward transmission in our design can be further increased by engineering the middle resonator (e.g. replacing it by a monopole resonator such as a Helmholtz resonator). The substitution of the monopole resonator is feasible as long as the monopolar resonator interacts with both top and bottom acoustic domains (see appendix E).

Discussion
We have developed an analytical model that describes the interaction between two acoustic resonators on an acoustic reflector. Our model captures the physical characteristics of the radiatively-coupled resonators. Based on the understanding of the resonant coupling, we have shown angle-dependent absorption as well as asymmetric acoustic wave transmission. Although our formulism is based on the isolated resonator pair, it offers physical insights into periodic structures [33][34][35]. Our results, based on only one lossy resonator, is different from the previous studies on resonant coupling [5,36,37], where two lossy (lossless) resonators had a spectral overlap for a higher absorption (transmission). Moreover, our approach enabling the angle-sensitive absorption can be useful for applications, e.g. acoustic antennas capable of absorbing the sound wave in a specific angle while isolating unwanted acoustic waves (noise) from other angles [38].
Appendix A. Derivation of the radiation leakage rate ( r h ) and radiation coupling rate ( c h ) We start by considering the radiated pressure field induced by one resonator harmonically vibrating with a velocity of u , 0 which is given by the 2D dipole Rayleigh integral as The difference between the leakage and coupling is that the radiation coupling c21 h is determined by radiation from the neighboring resonator (S S = = ) and reciprocity. 2D cylindrical waves emitted from a point source are described with the 2D Green's function (modified for an infinite reflector) given by [29] g x y x y i H kR , , where H 0 2 ( ) is the zero-order Hankel function of second kind, and R x x y y .

Appendix B. Absorption cross section spectra for other angles of incidence
The large contrast in abs s between q  at resonance, seen in figure 2(a) h ( ) as shown in figure C.1(c). The harmonic oscillator pair, considered in the main text, can be implemented with a realistic model composed of lossy and lossless Helmholtz resonators (HRs), as illustrated in figure D.1(a). The two HRs have the same neck width (w n ), but different neck lengths (h n ). Although our design is based on the lossless HR in combination of the lossy HR, the lossless HR inevitably have thermal-viscous losses. To minimize the loss in the neck, the lossless HR has a smaller neck length than the lossy HR. The thermal and viscous losses of the HRs occurring in the necks are considered by using the Thermoviscous Acoustic module in COMSOL Multiphysics. The angle-dependent absorption of the HR pair is shown in figure D.1(b). In addition, for normal incidence ( 0 q = ), the distancedependent absorption is observed, as shown in figure D.1(c). These results indicate that our design approach is valid for the realistic system, where the lossless resonator has losses.
When considering non-negligible losses in the lossless resonator, the coupled resonator exhibits a decrease in the absorption contrast while it preserves the angle dependence. Figure D