Low acoustic transmittance through a holey structure

The “acoustic double fishnet” is a structure with holes running from its front to back faces, yet at a characteristic frequency it transmits very little sound. The transmittance of this structure, which is comprised of a pair of closely spaced, periodically perforated plates, is determined experimentally and analytically. The surprising acoustic properties are due to hybridization between a two-dimensional resonance within the gap between the plates, and pipe modes within the holes. At the center of the stop band the input impedance is imaginary, interpreted as a negative product of effective bulk modulus and density.

Figure 1

Schematic of nine unit cells of the acoustic double fishnet (ADF) structure. This is comprised of a pair of identical plates of thickness $d_{\mathrm{p}}$ drilled with a square array of holes of radius $a$ and pitch $\Lambda$ (in experiments $a$ $=$ 1.2 mm and $\Lambda$ $=$ 8 mm), separated by a gap $d_{\mathrm{g}}$ which is small compared to the wavelength of the normally incident planar sound waves.

Figure 2

(Top) Resonance frequencies of the ADF structure given by Eqs. (6) and (7). Odd-order modes are solid lines and even-order modes are dashed. (Bottom) Amplitude transmission coefficient as a function of frequency for varying plate thickness $d_{\mathrm{p}}$ obtained using Eqs. (1)–(5) with $a$ $=$ 1.2 mm, $\Lambda$ $=$ 8 mm, and $d_{\mathrm{g}}$ $=$ 0.47 mm. The black regions correspond to transmission of less than 1$\%$, and the two shades of dark gray correspond to transmission of less than 5$\%$ and 10$\%$. The dashed white lines indicate the three plate thicknesses for which experimental data were obtained, and the points labeled A to I correspond to the pressure profiles in Fig. 3. The arrow on the horizontal axis of the bottom panel indicates the frequency that corresponds to $f$/$f_{\mathrm{gap}}$ $=$ 1; the arrow on the vertical axis of the bottom panel indicates the plate thickness that corresponds to $L$/$\Lambda$ $=$ 1.

Figure 3

Pressure profiles of the resonant peaks labeled in Fig. 2. The shape of the even-order mode remains essentially unchanged as the plate thickness varies; however, the odd-order mode changes due to leakage into the gap region.

Figure 4

Experimentally determined transmittance (blue) and modal-matching theory (red) for ADFs with the following plate thickness $d_{\mathrm{p}}$ and gap $d_{\mathrm{g}}$: (a) 3 mm and 0.47 mm, (b) 3 mm and 0.94 mm, (c) 5.9 mm and 0.47 mm, (d) 5.9 mm and 0.94 mm, (e) 12 mm and 0.47 mm, (f) 12 mm and 0.94 mm.

Figure 5

Experimentally determined transmittance of the ADF with $d_{\mathrm{p}}$ $=$ 3 mm (blue) and a single 5.9 mm perforated plate (red).

Figure 6

(Left) Experimentally determined dispersion properties of the ADF with $d_{\mathrm{p}}$ $=$ 12 mm separated by a gap $d_{\mathrm{g}}$ $=$ 0.47 mm. The graph is incomplete as data could only be obtained between normal incidence and an azimuth angle of 40°. (Right) Theoretical dispersion properties of the ADF with the same configuration as the experiment.