Acoustic metamaterials with spinning components

We show that an acoustic metamaterial consisting of an array of spinning cylindrical inclusions can possess many unusual properties, including folded bulk bands and interface-state bands. The folding of bands inside the first Brillouin zone is made possible by a rotation-induced antiresonance of compressibility and the rotational Doppler effect. Both bulk and interface-state band dispersions exhibit remarkable filling-fraction-dependent features such as the emergence of a cutoff frequency when the filling fraction exceeds a critical value. Robust one-way transport properties are supported by nondegenerate interface states, but within the same band, interface states at different frequencies can have different propagation directions.

Figure 1

(a) A two-dimensional square array of rotating cylinders in a fluidic background. Each cylinder rotates about its symmetric axis at a constant angular velocity Ω. (b) The band structures calculated by MST and EMT at a filling ratio $f=0.15$ showing two rotation-induced gaps (shaded in gray) sandwiching a “folded” band in between. The bands intercept the Brillouin zone center and boundaries at ${\omega }_1,{\omega }_2,{\omega }_3$, and ${\omega }_4$. (c) Effective-mass densities and compressibility vs frequencies at $f=0.15$.

Figure 2

(a) Frequencies of four band-edge states vs filling fraction $f$. (b)–(f) Evolution of bulk bands with increasing $f$. When $f>1/3$, ${\omega }_3$ becomes a cutoff frequency below which acoustic waves cannot propogate.

Figure 3

(a) Interface state dispersion relations calculated by MST and EMT at $f=0.15$. ${\omega }_1$ is the frequency of the flat band. ${\omega }_1^{\prime },{\omega }_2^{\prime },{\omega }_3^{\prime }$, and ${\omega }_4^{\prime }$ denote the frequencies at the Brillouin zone center and boundaries. The green shaded regions represent the projected bulk bands along the $x$ direction. (b) Effective-mass densities and compressibility. Parts (c) and (d) are the absolute values of the pressure field distribution of two eigenstates in the first and second interface-state bands, respectively. Parts (e) and (f) are the intensities of the average field inside all cylinders on the $y>0$ side for the two states in (c) and (d), respectively. The MST calculated field intensities can be well fitted using exponential functions decaying away from the boundaries at $y=0$. For (e) and (f), the fitted exponential functions are $f\left(y\right)=0.0741exp\left(-3.026y\right)-0.00027$ and $f\left(y\right)=0.1611exp\left(-1.232y\right)-0.0044$, respectively.

Figure 4

The real and imaginary parts of an interface state that belongs to (a) a flat band and (b) a folded band, respectively.

Figure 5

(a) Frequencies of band edges as a function of the filling fraction. (b)–(f) The evolution of interface states (solid red lines) as the increase of the filling fraction increases. The green shaded regions represent the projected bulk bands along the $x$ direction. When $f>1/3$, the dispersion relation is similar to that of (f), in which a band gap appears below a cutoff frequency, showing that the rotating inclusions stop the propagation of low-frequency longitudinal waves in the bulk of the composite material.

Figure 6

(a) The projected bulk band (green shaded region) and interface states (red solid line) at a filling fraction $f=0.2111$. (b) The wave with frequency $\omega a/2\pi c_0=0.0173$ and that with $\omega a/2\pi c_0=0.0181$ propagating along a convex-shaped interface in a finite system. The inset is the configuration of the structure. The blue line denotes the interface. The red arrows indicate the directions in which the cylinders rotate. The red stars mark the positions of point sources.