Harnessing Geometric Frustration to Form Band Gaps in Acoustic Channel Lattices

We demonstrate both numerically and experimentally that geometric frustration in two-dimensional periodic acoustic networks consisting of arrays of narrow air channels can be harnessed to form band gaps (ranges of frequency in which the waves cannot propagate in any direction through the system). While resonant standing wave modes and interferences are ubiquitous in all the analyzed network geometries, we show that they give rise to band gaps only in the geometrically frustrated ones (i.e., those comprising of triangles and pentagons). Our results not only reveal a new mechanism based on geometric frustration to suppress the propagation of pressure waves in specific frequency ranges but also open avenues for the design of a new generation of smart systems that control and manipulate sound and vibrations.

Figure 1

Geometric frustration in acoustic networks: (a) A propagating mode with a wavelength twice that of a single channel length (i.e., $\lambda =2L$) can be supported by a rhombic network with (b) $\theta =\pi /2$ and (c) $\theta =\pi /3$ but cannot be supported by (d) the triangular network, causing the system to become frustrated.

Figure 2

Dispersion relations of acoustic networks comprising a periodic array of air channels: (a) square lattice and (b) triangular lattice. Continuous lines and circular markers correspond to analytical and numerical (finite element) results, respectively. The shaded regions in (b) highlight the full band gaps induced by geometric frustration. Lattice configurations, unit cells (highlighted in red), and irreducible Brillouin zones are shown on the left.

Figure 3

Dynamic response of a rhombic lattice with an additional channel of width $d$ along the short diagonal. Dispersion curves along the GM direction are plotted for different values of channel width ratios $d/D$. Mode shapes at the $M$ point are shown for three unit cells characterized by $d/D=0$ (rhombic lattice), $d/D=0.2$, and $d/D=1$ (triangular lattice). Note that for visualization purposes the channel width $D$ is increased to $L/D=20$ (while in the calculations we used $L/D=100$).

Figure 4

Pressure field distribution in finite-sized acoustic networks comprising $6\times 6$ unit cells: (a) square lattice at $\mathrm{\Omega }=0.45$, (b) square lattice at $\mathrm{\Omega }=0.95$, (c) triangular lattice at $\mathrm{\Omega }=0.45$, and (d) triangular lattice at $\mathrm{\Omega }=0.95$. The color indicates the pressure amplitude normalized by the input signal amplitude ($p^{\mathrm{in}}$).

Figure 5

Transmittance of finite-sized networks: fabricated samples with the (a) square and (b) triangular networks. The input chamber is connected to the left edge of the samples, while the microphone to measure the amplitude of the transmitted sound waves is attached to the right edge. The frequency-dependent transmittances for the samples are shown in (c) and (d) for the square and triangular network, respectively. Both experimental (continuous red line) and numerical (dashed blue line) results are shown. The gray regions in (d) highlight the full band gap as predicted for the corresponding infinite structure [see Fig. 2].