This letter reports a tunable acoustic prism featuring continuous beam steering for transverse waves at a single frequency. The proposed prism is comprised of arrayed piezoelectric unit-cells with individually connected inductive shunt circuits. Taking advantage of wave velocity shifting in the vicinity of the local resonant frequency of unit-cells, we can steer the harmonic transverse wave by adjusting the inductive loads, i.e., tuning the inductances. This beam steering mechanism is facilitated by add-on piezoelectric circuitry through electro-mechanical coupling, whereas the host medium is not altered. Our analysis shows that the propagation direction of an acoustic wave has a tuning range of more than 30°. This tunable prism can be used as an acoustic metamaterial for various devices operating at broadband frequencies.

The so-called metamaterials, which are artificial structures that exhibit physical properties not available in natural materials, have shown promising aspects in wave-attenuation and wave-guiding.^{1–5} Acoustic metamaterials, consisting of a periodic host medium and local unit-cells, are capable of manipulating elastic wave propagation. Typically, acoustic metamaterials utilize either Bragg scattering or local resonance.^{6–9} Bragg bandgaps are generated at wavelengths comparable to the spatial scale of the periodicities.^{10,11} The local resonance mechanism utilizes the sub-wavelength resonant behavior at the unit-cell level.^{1,12} The local resonance can modify the frequency-dependent effective mass densities and/or the bulk moduli of the continuum media. In most cases, the internal resonators have highly contrasting elastic properties.^{13–16}

Recently, metamaterial-based acoustic wave-guiding has received significant attention. Various types of unit-cell microstructures are proposed. For example, a two-dimensional gradient index (GRIN) lens, comprised of cylinders with different diameters or unit-cells with cross-shaped apertures, is developed.^{17} Acoustic wave-focusing in phononic crystal plates is demonstrated numerically^{18} and experimentally.^{19} Wave-guiding systems have seen applications in fault detection and energy harvesting. By utilizing local resonance, a two-dimensional planar array can focus acoustic waves for enhanced fault detection sensitivity.^{20} Meanwhile, the Bragg scattering based wave-focusing is exploited for enhanced acoustic energy harvesting.^{21,22} Notably, piezoelectric metamaterials have emerged as a new option in acoustic wave guiding. For instance, piezoelectric transducers are assembled into a rectangular array to generate strong, frequency-dependent directional beaming, which allows beam steering in several directions through excitations at corresponding frequencies.^{23} A spiral-shaped transducer is proposed for frequency-based beam steering and applied to enhanced directional sensing.^{24} By taking advantage of multiple embedded acoustic metamaterial lenses, directional acoustic wave guiding is achieved at corresponding frequencies.^{25} Although promising, these systems share one common limitation that they cannot arbitrarily manipulate acoustic waves under a given frequency since the layout/geometry to facilitate wave-guiding is fixed.

Piezoelectric transducers possess two-way electro-mechanical coupling, and circuitry elements in the electrical domain can easily realize online tunability (e.g., a tunable synthetic inductor is shown in Figure 1(a)). In this research, we demonstrate that the piezoelectric metamaterial can be designed to have features of *adaptive* wave guiding at a single frequency. A unit-cell of the piezoelectric metamaterial is shown in Figure 1(a). It consists of two piezoelectric discs bonded to a host medium. An inductor is connected to the two transducers in parallel. The inductor can be realized by utilizing an op-amp circuit.^{26} This type of the synthetic inductor is adaptive and tunable online. Note that the piezoelectric transducer acts electrically as a capacitor. The combination of the piezoelectric capacitance and the inductive shunt circuit creates a local LC resonant unit. Indeed, there have been recent studies utilizing piezoelectric metamaterials integrated with shunt circuits for wave attenuation and localization.^{16,27–31} It is well-known in optics that natural light can be dispersed through a prism as wave components with different speeds (under different frequencies) in the light yield different refraction angles when passing through the prism. This gives rise to our fundamental idea of tunable beam steering, i.e., to utilize the acoustic wave velocity shifting in the vicinity of the local resonant frequency of the LC shunt in piezoelectric metamaterials, where the tunability is realized by inductance tuning. Our hypothesis is that such shifting of the velocity of acoustic waves, combined with the prism-like triangle-arrangement of unit-cells, will lead to a beam steering effect in the host medium. This metamaterial synthesis is achieved by attaching/bonding arrayed piezoelectric transducers, which are connected with tunable inductors individually, to a structural medium, whereas the structure remains unchanged. For illustration simplicity and without loss of generality, here we adopt an isosceles right-triangle shaped piezoelectric metamaterial prism integrated to a homogenous aluminum plate (Figure 1(b)).

We focus on harmonic transverse waves. Consider a transverse wave propagating in the $\Gamma X$ direction through the area integrated with the prism. Here, the $\Gamma X$ direction indicates the *x*-direction, shown in Figure 1(b). Then, the refraction angles of the acoustic wave are analyzed. According to the well-known Snell law, the ratio of the refraction angle to the incidence angle is proportional to the ratio of phase velocities. As the wave propagates predominantly along the $\Gamma X$ direction, the phase velocity of harmonic transverse waves can be approximated by the phase velocity along this direction. The unit-cell displacement of the *n*th transverse wave along the $\Gamma X$ direction is $w(x,t)=x0ei(nk\Gamma Xx\u2212\omega t)$, where $k\Gamma X$ is the wavenumber in the $\Gamma X$ direction and $x0$ is the amplitude of the acoustic wave. To elucidate the key characteristic of wave propagation in the vicinity of the bandgap, the governing equations of the piezoelectric unit-cell are derived. Extended Hamilton's principle is adopted, and the Bloch-Floquet periodic boundary conditions are considered.^{16,32} The piezoelectric transducers generate electric displacement due to the mechanical strain and are considered as voltage sources in the electrical domain. For the unit-cell shown in Figure 1(a), we have the following lumped-parameter model,

where $Q1$, $Q2$, and *x* are the charge on the top transducer, charge on the bottom transducer, and lumped displacement of the unit-cell; *M*, *K*, *c*, *R*, $k1$, *C*, and *L* represent the mass, stiffness, mechanical damping coefficient, resistance in the shunt circuit, the electro-mechanical constant of the piezoelectric transducer, the capacitance of the piezoelectric transducer, and the tunable inductance, respectively. The piezoelectric transducers on the top and bottom of the host medium are identical. The instantaneous charges on the two transducers are thus equal to each other, i.e., $Q1(t)=Q2(t)$. As shown by Xu *et al.* (2016), the mass, stiffness, and electro-mechanical coupling constant are wavenumber-dependent,

where *r*, $lb$, $wb$, $tp$, $tb$, $h31$, $\rho p$, and $\rho b$ represent the radius of the piezoelectric transducer, the length of the unit-cell substrate, the width of the substrate, the thickness of the piezoelectric transducer, the thickness of the substrate, the piezoelectric constant of the transducer, the mass density of the transducer, and the mass density of the substrate, respectively. The capacitance of one piezoelectric transducer is $C=\pi r2\beta 33tp$, where $\beta 33$ is the dielectric constant. Equations (1a) and (1b) illustrate the dynamic interaction between the mechanical and electrical domains. The LC resonance (Equation (1b)) affects the substrate dynamics (Equation (1a)) owing to the electro-mechanical coupling, thereby influencing the wave propagation characteristics through changing the wave velocity and creating a bandgap.

In this research, we consider a unit-cell consisting of an aluminum substrate (unit-cell size, 25.38 × 25.38 × 3.13 mm^{3}) and two piezoelectric discs (radius, 10.5 mm and thickness, 1 mm) bonded onto the top and bottom surfaces of the substrate. The acoustic prism consists of $13\xd713$ unit-cells that are arranged as an isosceles right triangle (total 91 unit-cells) where 6–10 layers of unit-cells are effectively involved in beam steering. We aim at taking advantages of the wave velocity shifting in the vicinity of the local resonance. The phase velocity of an acoustic wave is proportional to its frequency and inversely proportional to its wavenumber.^{33} Assuming harmonic responses of unit-cell mechanical displacement and charges (on the transducers), the wavenumber with respect to the wave frequency can be solved from the dispersion equation. The dispersion equation of the unit-cell (under un-damped condition) can be obtained from Equations (1a) and (1b) as

It is important to note that, as shown in Equations (2a)–(2c), *M*, *K*, and $k1$ are all wavenumber-dependent. The wavenumber can thus be directly solved from Equation (3). As *L* is included in this fourth-order equation, the wavenumber $k\Gamma X$ is an explicit function of the external inductance *L*. The inductance can be tuned online. Therefore, the bandgap features here have the advantage of being tunable and adaptive. Our basic idea here is that the acoustic wave at one frequency can be steered by adjusting the inductance in the piezoelectric circuit. We rewrite the phase velocity $vp$ as a function of inductance *L*,

We first demonstrate the inductance-dependent features of the dispersion curve and the phase velocity of the acoustic wave. In the following analytical and numerical analyses, the material parameters are chosen as follows: piezoelectric mass density, $\rho p=7500\u2009kg/m3$; piezoelectric Young's modulus, $Ep=106\u2009GPa$; piezoelectric constant, $h31=\u22121.37\xd7109N/C$; dielectric constant, $\beta 33=2.92\xd7108\u2009Vm/C$ host medium mass density, $\rho b=2700\u2009kg/m3$; and host medium Young's modulus, $Eb=62\u2009GPa$. The piezoelectric material constants are from the PZT-5H transducer. Without loss of generality, we choose the inductance value $L0$ to create a local resonance at 19.46 kHz. The corresponding inductance value is $L0=18.3\u2009mH$. Figures 2(a) and 2(b) show the dispersion curves and phase velocity curves, respectively. In Figure 2(a), $k=k\Gamma Xlb/\pi $ is the normalized wavenumber. It can be observed in Figure 2(a) that the LC circuit induces a bandgap around the LC resonant frequency. The frequency of the bandgap decreases with the increase in *L*. As shown in Figure 2(b), the phase velocity of the acoustic wave shifts significantly at a single frequency as bandgaps are adjusted through tuning *L*. For example, the phase velocity of the acoustic wave at 19.46 kHz is increased with increasing *L*. Note that different phase velocities of the acoustic wave yield different refraction indexes. The shifting of phase velocity results in a change in the refraction angle, i.e., yielding the beam steering effect.

Indeed, the relationship of the incidence angle and the refraction angle follows Snell's Law,

where $v0$ is the reference phase velocity of the harmonic transverse wave in the homogenous aluminum medium and $\theta 1$ and $\theta 2$ are the angles of incidence and refraction, respectively. In our analysis, the phase velocity $v0$ in the plate and the incidence angle $\theta 1$ are given constants. On the other hand, the phase velocity in the prism, $vp$, can be adjusted according to the adaptiveness of local resonance. In such a case, the acoustic wave travelling through the prism will have different refraction angles by inductance tuning (Equation (5)). Recall Equation (4). As the inductance in the shunt circuit is continuously adjusted, the phase velocity of the acoustic wave can be continuously tuned. This yields continuous beam steering since the local resonance of the LC shunt circuit depends on the inductance value. One can easily change the frequency of the LC resonance without altering the mechanical part of the prism.

Then, we demonstrate the continuous wave steering through simulations. The finite element method is employed, as it can match well with respect to experimental studies in piezoelectric metamaterial investigations.^{27–29,34} We apply ANSYS 14.5 to simulate the beam steering effects under the harmonic analysis using the sparse solver. The Solid95 element is used to model the substrate, and the Solid226 element is used to model the piezoelectric transducers. Displacement contours with different inductance values are shown in Figures 3(a) and 3(b). Without loss of generality, we choose the frequency of the acoustic wave as 19.46 kHz to illustrate the effectiveness of the prism. In such a case, the acoustic wave in the unit-cell has a fundamental wavelength of 74.4 mm and a fundamental wavenumber of 0.0134 mm^{−1} at 19.46 kHz.

In Figures 3(a) and 3(b), the incident waves propagate from the bottom of the prism, and the results clearly illustrate the wave steering effect at the selected frequency. A nearly planar wave is formed as the forward propagation waves travel through the prism region. It can be observed that when the inductance is decreased to $0.9L0$, the harmonic acoustic wave propagates to the left-hand side. As the inductance is increased to $1.06L0$, the harmonic acoustic wave propagates toward the right-hand side. This indicates that the phase velocity of the acoustic wave is shifted dramatically. Continuous tunable beam steering can be achieved at the selected frequency by simply adjusting the shunt circuit inductance. It is worth emphasizing that the system parameters, e.g., the dimension of the unit-cell and the frequency of the acoustic wave, generally remain unchanged. This is the distinctive feature as compared with previous beam steering investigations.

The relationship of the external inductive load and the steering angle is plotted in Figure 4. The results are obtained under the steady-state condition. The incidence angle is 45°. The inductance is tuned from $0.1L0$ to $1.6L0$ with the step increment of $0.0075L0$. The refraction angles $\theta 2$ are calculated and presented in Figure 4. The theoretical result predicted by Equation (5) is also plotted in Figure 4, which shows good agreement with the finite element result. It can be observed that the refraction angle decreases as we increase *L*. The minimum refraction angle here is 35°. As the inductance approaches $L0$, a conventional bandgap of the piezoelectric metamaterial is induced. Within the bandgap, the harmonic acoustic wave cannot propagate through the prism area. The bandgap is unavoidable because only negative equivalent mass can be achieved here. Further increasing the inductance yields another branch of the beam steering effect. That is, increasing the inductance can decrease the refraction angle from 65° to 47°. The reason is that the phase velocity of the harmonic wave, as shown in Figure 2(b), is decreased to the minimum and then jumps to the maximum value as the inductance is increased in the vicinity of the bandgap. It is worth mentioning that due to the adaptiveness of the LC shunt circuit, the beam steering effect is applicable for the acoustic wave at other frequencies.

In summary, our research finds that the proposed metamaterial-based prism offers the capability of steering the acoustic wave due to the local resonance from the LC shunt circuit. We analyze and demonstrate the tunable acoustic beam steering by choosing the at a single frequency point of 19.46 kHz. The angle of the wave can be tuned between 30° and 60° by adjusting the inductance. The resonant frequency of the shunt circuit can be modified by changing the inductance. Moreover, the full integration of the adaptive piezoelectric metamaterials, tunable inductors, computing resources, and power systems can form a hybrid metamaterial system where acoustic wave guiding can be remotely controlled. The concept proposed here can be applied to adaptive GRIN lenses for a variety of applications.

This research was supported by NSF under Grant No. CPS–1544707.