Going Beyond the Causal Limit in Acoustic Absorption

Dictated by the causality constraint, the existence of a minimum absorber thickness has set an almost insurmountable obstacle for the remediation of ultralow-frequency noise in the tens of Hz to 150 Hz regime, owing to the very large absorber thickness required. Here, we show that, by manipulating the boundary condition in a calculated manner on the backside of the absorber, the causal constraint can be circumvented, and broadband near-total absorption of ultralow-frequency acoustic waves may be realized with an absorber thickness that is an order of magnitude less than the causal minimum. Our work delineates a design paradigm for the ultralow-frequency acoustic absorbers.

Figure 1

(a) Photograph of the absorption sample (left panel) and an enlarged microscope image of the metallic mesh (right panel). Sample comprises layers of metallic meshes with an average pore size of $7.2\times {10}^{-3}\mathrm{m}{\mathrm{m}}^2$. (b) Inset, for the open boundary with no sample, we compare the real part of the reflection coefficient, Re[r], as a function of frequency obtained from experiments (red open circles), comsol simulation (green dashed line), and the prediction using the ASBC given by Eq. (2) (blue solid line). Excellent agreement is seen, especially below 300 Hz. Main figure shows the absorption and extinction for a 0.5-cm absorption sample placed in front of the open boundary, where red symbols and red solid lines represent the measured and simulated extinction, respectively, while the green solid line represents the simulated absorption. It is seen that the absorption and extinction results merge and become indistinguishable below 200 Hz, implying a negligible amount of scattered transmission in the low-frequency regime. Measured and simulated absorption results with the hard reflecting boundary are shown as green open circles and dashed line, respectively. Almost no absorption can be seen, as expected from the displacement-velocity nodal condition at the hard boundary.

Figure 2

Simulated absorption performance backed by ASBC (solid line) and AHBC (dashed line) at 100 Hz (red), 150 Hz (blue), 200 Hz (green), and 300 Hz (black), plotted as a function of sample thickness on the logarithmic scale. Two insets schematically illustrate the simulated displacement-velocity-amplitude distribution for a sound wave traveling inside an extremely thick (upper) absorber and a thin (lower) absorber, both backed by ASBC. Red lines indicate the wave amplitude inside the sample.

Figure 3

(a) Placing a AHBC 1.5 cm away from the tube opening creates a “gap,” as illustrated in the inset, which will shift the ASBC 90° from the incident wave direction. Re[r] obtained from experiment (red circles) and simulation (green dashed line) are shown. Blue line represents Re[r] obtained from the best-fitted gap impedance, $Z_{\mathrm{gap}}/Z_0=0.4(kr{)}^2-ik(0.9)$. (b) Simulated Re[r], using the configuration shown in the inset with S = 10, 20, and 30 cm, is plotted as a function of frequency, where S is the distance of the sample’s backside to the opening. This illustrates the effect of the geometric phase (GP) on the ASBC. Dashed lines represent the calculated Re[r] with the GP given by $\cos [k(2S)+\pi ]$. (c) Extinction results obtained from experiment (circles) and simulation (solid lines) plotted as a function of frequency for various gap sizes, D, as indicated. Corresponding absorption results obtained from simulation are plotted as dashed lines.

Figure 4

(a) Red and blue solid lines show the required real and imaginary parts of the decorated-membrane impedance to achieve impedance matching, when it is backed by a cavity with a partial ASBC with three openings on the cavity’s sidewall. Red and blue circles, on the other hand, represent the respective real and imaginary parts of the impedance for the decorated membrane alone. Intersection point indicates the frequency at which the impedance-matching condition is satisfied, as designated by the red arrow. Black solid and dashed arrows point to the first resonance and antiresonance frequencies, respectively. (b) Measured absorption spectrum of a decorated-membrane resonator backed by a cavity with three openings. Extinction data obtained from experiments (red circles) and simulation results (red solid line) are plotted as a function of frequency. Absorption from simulations (green squares) are also shown. Coincidence of the extinction and absorption curves implies that the transmission component is very small. Extremely high and broad absorption peak is seen to be centered at 81 Hz. Inset shows a schematic illustration of the hybrid structure, together with a photograph of the sample. (c) Absorption spectrum of four resonators integrated parallel to each other, as shown in the insets. Effective impedance can be expressed as $Z^{-1}=\varphi (0.25Z_1^{-1}+0.25Z_2^{-1}+0.25Z_3^{-1}+0.25Z_4^{-1})$, with the absorption $A=1-|(Z/Z_0-1)/(Z/Z_0+1){|}^2$. Extinction obtained from experiments (red circles) and absorption obtained from simulations (green solid lines) are shown. Average absorption is 94% in the target range of 80–110 Hz.

Figure 5

Schematic diagram of the potential application configuration, where a combination of metallic mesh and openings is placed at the edges of the enclosed backspace to reduce reflection to nearly zero, as described in previous sections.

Figure 6

Schematic diagram of the model used in comsol to simulate an open-impedance-tube situation, to capture transmitted sound energy through the opening. At the relevant low-frequency regime, the transmitted sound energy is invariably found to be very small.

Figure 7

Airflow velocity and membrane-displacement field at the absorption-peak frequency are plotted in grayscale and colored scale, respectively. Flow velocity is normalized to the source-flow velocity, which is defined as $v_0=P_0/Z_0$, where $P_0$ is the wave’s amplitude. Membrane displacement, on the other hand, is normalized to the maximum displacement when the membrane is backed by the perfect matching layer boundary condition.