Extraordinary Sound Isolation Using an Ultrasparse Array of Degenerate Anisotropic Scatterers

Subwavelength acoustic scatterers with extraordinary multipole scattering properties have attracted much research interest. We introduce an ultrasparse anisotropic acoustic scatterer array, which has degenerate monopole and dipole resonances, for total absorption of sound based on scattering cancellation of monopole and dipole moments. The dipolar scattering property of the scatterer is extremely anisotropic so that the dipole resonance can be switched off by rotating the scatterer about its own center by ${90}^{\circ }$. This feature leads to the scatterer-orientation dependence of mode degeneracy as well as the sound-absorption capability. Simulation and experimental results demonstrate the orientation-dependent sound-isolation property of the scatterer array. Moreover, the scatterer array is ultrasparse in that the spatial period exceeds the limit bounded by 4 times the absorption cross section of a subwavelength resonator ($d>4{\sigma }_{\mathrm{abs}}=2\lambda /\pi$). This ultrasparse and controllable sound-isolating material may have applications in many areas such as infrastructures, consumer electronics, and the automotive industry.

Figure 1

Ultrasparse anisotropic degenrate scatterer array for controllable sound absorption. (a) shows that the scatterers are arranged in an array along $y$ direction; each scatterer is uniform in the $z$ direction; a plane wave is normally incident along the $x$ direction perpendicular to the scatterer array. (b) shows the details of the scatterer. The upper panel is a photo of a fabricated sample; the lower panel shows the geometry of the scatterer. The scatterer has an outer radius $r_o\ll \lambda$ and uniform wall thickness $h_{\mathrm{wall}}$; the interior of the scatterer consists of two identical Helmholtz resonators with neck width $h_{\mathrm{neck}}$ and neck length $l_{\mathrm{neck}}$; the internal radius of the cavity is denoted by $r_i$. The orientation of the scatterer $\theta$ ranges from $0^{\circ }$ to ${90}^{\circ }$. Total acoustic absorption is achieved when $\theta =0^{\circ }$, while nearly half of the energy is absorbed when $\theta ={90}^{\circ }$.

Figure 2

Two-dimensional free-field scattering properties of the anisotropic scatterer. (a) and (b) are the scattered pressure fields for $ka=0.44$ corresponding to $x$ and $y$ incidence of a plane wave, respectively. (c) shows the TSCS calculated using $\sigma (ka)=|p_0{|}^{-2}\int |p_{\mathrm{s}}{|}^{2}ds$, where $p_0$ is the amplitude of background wave and $s$ is a closed contour enclosing the scatterer. The black curve in (c) corresponds to the TSCS for $x$ incidence with the maxima at 2114 Hz; the red curve corresponds to $y$ incidence with the maxima at 2159 Hz. The simulated normalized scattering coefficients of the monopole ($M$) and dipole moments ($D_x$ and $D_y$) are plotted in (d) using solid black, red, and blue curves; the analytical monopolar and dipolar scattering coefficients for a rigid cylinder of the same radius are calculated using Eq. (A2) and plotted using dashed curves for comparison. (e) illustrates the lumped elements of the scatterer. The air inside the resonator neck is the acoustic mass $m={\rho }_{\mathrm{air}}{l}^{\mathrm{\prime }}/S$, where ${\rho }_{\mathrm{air}}$ is the density of air, $l^{\mathrm{\prime }}$ is the effective length of the narrow channel, and $S$ is the neck-opening area; the acoustic compliance of the lumped-element model is $C=V/B_{\mathrm{air}}$, where $V$ is the volume of the cavity and $B_{\mathrm{air}}$ is the bulk modulus of air. (f) shows that the monopolar response can be excited by the local pressure, while the dipolar response can be excited by the local velocity.

Figure 3

One-dimensional scattering properties. (a) shows the relation between the resonant frequencies and the array period $d$. The monopole resonant frequency increases more rapidly than the dipole resonant frequency as $d$ decreases, and the modes coalesce at 2111 Hz when $d=0.143$ m. The inset at the lower left corner in (a) show the mode shapes of the two types of resonances; the arrows indicate the direction of particle velocity. (b) shows the normalized monopolar and dipolar scattering coefficients for $d=0.143$ m. The real and imaginary parts of the monopole scattering coefficient ${\alpha }^{pp\mathrm{\prime }}$ are plotted using solid black and blue curves, respectively; the real and imaginary parts of the dipole scattering coefficient ${\alpha }^{vv\mathrm{\prime }}$ are plotted using black and blue circles, respectively. The real parts of the two scattering coefficients are zero so that the pure imaginary ${\alpha }^{p{p}^{\prime }}={\alpha }^{v{v}^{\prime }}=i3.04\times {10}^{-8}$.

Figure 4

Simulated sound-absorption coefficient. (a) shows the top view of wave incident on the scatterer array. The orientation of the scatterer $\theta$ with respect to the propagation direction of the incident plane wave. The corresponding absorption coefficients for different values of $\theta$ are shown in (b). The maximum sound-absorption coefficient reduces as $\theta$ increases. The $\theta =0^{\circ }$ configuration achieves near total absorption, while the maximum absorption for the $\theta ={90}^{\circ }$ case is slightly below 0.5. (c),(d) are the effective bulk modulus and the effective mass density, respectively. The real parts of them indicated by black dots are both zero at 2111 Hz, however, their imaginary parts are nonzero. At 2111 Hz, this double zero-index material has effective impedance $Z_{\mathrm{eff}}=\sqrt{{\rho }_{\mathrm{eff}}{B}_{\mathrm{eff}}}=0.986Z_{\mathrm{air}}$ and sound speed $c_{\mathrm{eff}}=\sqrt{B_{\mathrm{eff}}/{\rho }_{\mathrm{eff}}}=i0.157c_{\mathrm{air}}$.

Figure 5

Simulated sound absorption for different values of period $d$. The maximum absorption frequency increases with the decrease of $d$ and the highest absorption occurs near the optimal period $d=14.3$ mm.

Figure 6

Insertion loss measurement setup and results. (a) illustrates the measurement setup for the insertion loss using only one period of the scatterer array. The dimensions of the waveguide are 50 $\times$ 14.3 $\times$ 4 ${\mathrm{cm}}^3$; the scatterer is placed at the center of the tube. Sound pressure is measured and averaged along the red dashed line at the right side (open end) of the tube. (b) shows the insertion losses corresponding to $\theta =0^{\circ }$ and $\theta ={90}^{\circ }$ using black and red lines. The solid lines correspond to the simulation results of the system shown in (a), and the dashed lines are obtained through measurements. The maximum simulated insertion loss for $\theta =0^{\circ }$ is 15.9 dB at 2102 Hz, while the experimental results have a maximum value of 11.5 dB at 2142 Hz; the simulated and measured maximum values for $\theta ={90}^{\circ }$ are 13.1 dB at 2119 Hz and 8.2 dB at 2148 Hz, respectively. The insertion loss peaks shift to higher frequencies mainly due to the fabrication accuracy of the sample, and hence slightly reduces the sound-isolation performance. However, the relationship between the measured $\theta =0^{\circ }$ and $\theta ={90}^{\circ }$ cases follows that of the simulation results.

Figure 7

Sound-isolation performance of the scatterer array. (a) shows the measurement setup for the 2D sound-pressure field in a 4.2-cm-high waveguide; the period of the scatterer array is 14.3 cm as shown in the upper panel; the area in the black dashed box (30 $\times$ 40 ${\mathrm{cm}}^2$) is located 15 cm away from the scatterer array. (b),(e) show the measured and simulated sound fields without the scatterer array at 2168 Hz, respectively. (c),(f) are the measured and simulated sound-pressure fields with the scatterer array for $\theta =0^{\circ }$ at 2168 Hz, respectively. (d),(g) are the measured and simulated sound-pressure fields with the scatterer array for $\theta ={90}^{\circ }$ at 2168 Hz, respectively. The $\theta =0^{\circ }$ case exhibits better sound-isolation performance in both experiments and simulations. Note that the size of the scatterers are exaggerated in (c), (d), (f), and (g).