Spatial Decomposition of a Broadband Pulse Caused by Strong Frequency Dispersion of Sound in Acoustic Metamaterial Superlattice

materials

Article Spatial Decomposition of a Broadband Pulse Caused by Strong Frequency Dispersion of Sound in Acoustic Metamaterial Superlattice

Yuqi Jin 1,2 , Yurii Zubov 1, Teng Yang 1,3, Tae-Youl Choi 2, Arkadii Krokhin 1,4 and Arup Neogi 1,4,*

1 Department of Physics, University of North Texas, P.O. Box 311427, Denton, TX 76203, USA; [email protected] (Y.J.); [email protected] (Y.Z.); [email protected] (T.Y.); [email protected] (A.K.) 2 Department of Mechanical Engineering, University of North Texas, 3940 North Elm Suite, Denton, TX 76207, USA; [email protected] 3 Department of Materials Science and Engineering, University of North Texas, 3940 North Elm Suite, Denton, TX 76207, USA 4 Center for Agile and Adaptive Additive Manufacturing, 3940 North Elm Suite, Denton, TX 76207, USA * Correspondence: [email protected]

Abstract: An acoustic metamaterial superlattice is used for the spatial and spectral deconvolution of a broadband acoustic pulse into narrowband signals with different central frequencies. The operating frequency range is located on the second transmission band of the superlattice. The decomposition of the broadband pulse was achieved by the frequency-dependent refraction angle in the superlattice. The refracted angle within the acoustic superlattice was larger at higher operating frequency and verified by numerical calculated and experimental mapped sound fields between the layers. The spatial dispersion and the spectral decomposition of a broadband pulse were studied using lateral position-dependent frequency spectra experimentally with and without the superlattice structure along the direction of the propagating acoustic wave. In the absence of the superlattice, the acoustic Citation: Jin, Y.; Zubov, Y.; Yang, T.; propagation was influenced by the usual divergence of the beam, and the frequency spectrum was Choi, T.-Y.; Krokhin, A.; Neogi, A. unaffected. The decomposition of the broadband wave in the superlattice’s presence was measured Spatial Decomposition of a by two-dimensional spatial mapping of the acoustic spectra along the superlattice’s in-plane direction Broadband Pulse Caused by Strong to characterize the propagation of the beam through the crystal. About 80% of the frequency range of Frequency Dispersion of Sound in the second transmission band showed exceptional performance on decomposition. Acoustic Metamaterial Superlattice. Materials 2021, 14, 125. Keywords: acoustic metamaterial; ultrasound pulse; pulse decomposition; acoustic superlattice https://doi.org/10.3390/ma14010125

Received: 30 November 2020 Accepted: 28 December 2020 Published: 30 December 2020 1. Introduction Artificial phononic periodic structures have been studied for decades and are gener- Publisher’s Note: MDPI stays neu- ally termed phononic crystals [1]. A phononic crystal can manipulate the propagation of a tral with regard to jurisdictional clai- wave based on the structure of the crystal. It depends on the frequency and direction of the ms in published maps and institutio- incident wave with respect to the crystal plane, which can be very different from natural nal affiliations. crystals [2]. In the long-wavelength limit, the phononic crystals’ physical properties can be artificially designed according to the effective medium theory [3–5]. Once the operating wavelength approaches the periodicity or smaller, the eigenmodes’ dispersion relation

Copyright: © 2020 by the authors. Li- becomes highly nonlinear and may exhibit anomalous group velocity [6]. In this region, the censee MDPI, Basel, Switzerland. phononic crystals behave as metamaterials [7–10]. The potential wave steering function- This article is an open access article alities of elastic [11], acoustic [12], and thermal waves [13] using 1D, 2D, or 3D phononic distributed under the terms and con- periodic structures have been demonstrated along with the fundamental principles in the ditions of the Creative Commons At- existing studies. In those transmission bands, the abnormal behavior includes negative tribution (CC BY) license (https:// refraction [14] and ﬂat regions of the equifrequency surface [9,15]. These unique properties creativecommons.org/licenses/by/ provide opportunities to realize long-distance acoustic collimator [8] and super-resolution 4.0/).

Materials 2021, 14, 125. https://doi.org/10.3390/ma14010125 https://www.mdpi.com/journal/materials Materials 2021, 14, x 2 of 8 Materials 2021, 14, 125 2 of 8

properties provide opportunities to realize long-distance acoustic collimator [8] and su- monostaticper-resolution [16,17 monostatic] and bistatic [16,17] [18– 20 and] lenses, bistatic which [18–20 showed] lenses, great which improvement showed great in acoustic im- detectionprovement [21 ] in and acoustic ultrasonic detection imaging [21] [ and22– 24 ultrasonic] and elastography imaging [22 [25–24–]27 and]. elastography [25The–27]. acoustic superlattice structure considered in this work is a 1D layered and periodic structure,The whichacoustic possesses superlattice most structure of the propertiesconsidered ofin metamaterialsthis work is a 1D [28 layered]. These and structures peri- areodic commonly structure, designed which possesses to modify most the of propagation the properties of thermalof metamaterials phonons [28]. in thermo-electrical These struc- devicestures are [29 ].commonly A recent studydesigned shows to modify that the the introduction propagation of of randomness thermal phonons in the periodicityin thermo- of a superlatticeelectrical devices structure [29]. leadsA recent to Andersonstudy shows localization that the introduction of thermal phononsof randomness [30], leading in the to higherperiodicity efﬁciency of a superlattice than conventional structure designs. leads to At Anderson the macroscope localization scale, of solid-solidthermal phonons periodic superlattices[30], leading are to used higher to efficiencysteer a propagating than conventional Rayleigh designs. waveand At therealize macroscope ﬁltering scale,[31] and waveguidingsolid-solid periodic [32]. superlattices are used to steer a propagating Rayleigh wave and realize Thisfiltering study [31] proposes and waveguiding a numerical [32]. and experimental realization of the spectral and spatial deconvolutionThis study of anproposes acoustic a broadbandnumerical and pulse experimental into frequency realization components of the using spectral an acoustic and metamaterialspatial deconvolution superlattice. of an The acoustic superlattice broadband design pulse was introducedinto frequency in ourcomponents previous using work on non-spreadingan acoustic metamaterial propagation superlattice. of acoustic beam The superlattice [8]. Here, we design focus was on theintroduced decomposition in our pre- behav- iorvious of the work superlattice. on non-spreading The decomposition propagation of the of signalacoustic into beam multi-spectral [8]. Here, we components focus on occursthe duedecomposition to the frequency-dependent behavior of the index superlattice. of refraction The decomposition and frequency-dependent of the signal refraction into multi angle.- Thisspectral rainbow component effect originatess occurs due from to frequency the frequency dispersion-dependent and index is well-known of refraction in optics. and fre- Here quency-dependent refraction angle. This rainbow effect originates from frequency disper- we demonstrate that it can be strongly enhanced for sound waves due to strong frequency sion and is well-known in optics. Here we demonstrate that it can be strongly enhanced dispersion of the eigenmodes at the second transmission band. With a broadband pulse for sound waves due to strong frequency dispersion of the eigenmodes at the second source incident on the superlattice’s facet, the monochromatic components with different transmission band. With a broadband pulse source incident on the superlattice’s facet, the wavelengths propagate through the superlattice with self-sorted refraction paths. The fre- monochromatic components with different wavelengths propagate through the superlat- quencytice with components self-sorted from refraction short paths. to long The wavelength frequency arecomponents transmitted from with short different to long refractionwave- angleslength from are largetransmitted to small. with different refraction angles from large to small. 2. Materials and Methods 2. Materials and Methods 2.1. Design of the Superlattice Structure 2.1. Design of the Superlattice Structure The periodic lattice structure was assembled from 10 pieces of 178 mm wide square The periodic lattice structure was assembled from 10 pieces of 178 mm wide square shape stainless steel 409 plates with a thickness of 0.891 mm. The plates are arranged shape stainless steel 409 plates with a thickness of 0.891 mm. The plates are arranged pe- periodicallyriodically with with 10 10 mm mm period. period. The The plates plates are are held held by by stereolithography stereolithography type type Formlabs Formlabs (Somerville,(Somerville, MA, MA, USA) USA) Form Form 22 printedprinted sound transparent transparent resin resin corner corner holders, holders, as shown as shown inin Figure Figure1A. 1A. The The superlattice superlattice is issubmerged submerged in water. The The scanned, scanned, transmitted transmitted area area was was 4040 mm mm apart apart from from thethe lastlast stainless steel steel plate. plate. The The scanned scanned line line was was 90 mm 90 mmlong longwith with1 1 mmmm interval.

Figure 1. Design of the 10 layers superlattice structure. (A) Photograph of the real fabricated superlattice structure. The corner holders were made by water-property-like resin. (B) The band structure calculated for 10◦ angle of incidence. (C) Schematic diagram of the experimental setup. In the ﬁgure, β refers to the single element planar transducer with 100 kHz fundamental frequency, µ indicates the 10-layer superlattice structure, and γ is the needle type underwater hydrophone. Materials 2021, 14, 125 3 of 8

Figure1B shows the superlattice’s band structure calculated for the angle of incidence of 10◦ measured from axis x. The geometrical parameters of the unit cell (inset) are thickness of water (steel) layer a = 9.109 mm (b = 0.891 mm) and period d = 10 mm. The frequency bandwidth of the source pulse in this study was from 50 to 150 kHz. A pulse excites the eigenmodes in the second transmission band, which extends from 79 to 108 kHz. The bandgaps cut the harmonics with frequencies below 79 kHz and above 108 kHz.

2.2. Numerical Simulation The numerical simulations were performed using ﬁnite element analysis (FEA) COM- SOL Multiphysics 5.5 (Burlington, MA, USA) software. The experimental setup was modeled by placing the superlattice in the center of the 250 × 450 mm2 water tank. The parameters of DI water are: the speed of sound is 1480 m/s, and the density is 1000 kg/m3. The Young’s modulus, speed of sound, and density of the stainless steel plates are 200 GPa, 5750 m/s, and 7800 kg/m3, respectively. The transducer has a diameter of 1 inch (25.4 mm, the same size as the real transducer in the experiment). It is located 25 mm under the water level at the center of the left horizontal boundary. The distribution of pressure was calculated in the frequency domain for the angle of incidence of 10◦.

2.3. Experimental Setup Figure1C illustrates a monostatic setup of the experiment. The superlattice is emerged in a large (550 × 550 × 550 mm3) DI water tank. Unfocused Ultran (State College, PA, USA) 25.4 mm diameter planar transducer NCG100-D25 generates 100 kHz fundamental frequency broadband pulse (from 50 to 200 kHz) in every 10 milliseconds, and the signal is measured by Müller-Platte (Oberursel, Germany) 1 mm diameter tip needle hydrophone. Pulse signal to transducer was generated by Imaginant (Pittsford, NY, USA) JSR 300 pulser/receiver. The measured time-domain signal from needle hydrophone was forwarded to Tektronix MDO 3024 (Beaverton, OR, USA) oscilloscope for further acquir- ing. The transducer was held and rotated in X-Y plane by a Thorlabs (Newton, NJ, USA) CR1-Z6 angular translation stage for variable incident angles. The needle hydrophone was held and moved in X-Y plane by a two-dimensional 200 by 200 mm translation range Newmark (Zimmerman, MN, USA) LC 200 linear translation stage, which was controlled by Newmark (Zimmerman, MN, USA) NSG-G2-X2 stepper motor controller. Before each experiment, the angular stage turned the transducer to 10◦ incident angle. The automatic raster scan was completed by a pre-prepared MathWorks (Natick, MA, USA) MATLAB R2020b code, which moved the needle hydrophone to each location and paused for 20 s for data acquirement. After recording each time window, the code moved the hydrophone to the next position.

3. Results The nature of the physical decomposition of the acoustic pulse by superlattice is the same as splitting white light by a spectrum of different colors when passing through a prism, namely the refraction coefﬁcient’s frequency dispersion. Here we explore the frequency dispersion of sound at the second transmission band of the designed superlattice. Figure2 demonstrates different refraction angles for monochromatic sound beams at three different frequencies that were incident on the superlattice at the same angle of incidence. On passing through 10 periods of the superlattice, the beams emerge out of the crystal at different spatial positions. The angle of refraction gradually increases with frequency. For frequencies 95, 105, and 115 kHz, we respectively obtain ng = 0.33, 0.24, and 0.21. MaterialsMaterials 20212021,, 1414,, x 125 44 of of 8 8

Materials 2021, 14, x 4 of 8

◦ FigureFigureFigure 2. 2. NumericalNumerical 2. Numerical simulation simulation ofof of refractionrefraction refraction ofof of monochromatic monochromatic monochromatic beams beamsbeams incident incidentincident to toto the thethe 10 1010 layers layers layers superlattice superlattice superlattice structure structure structure at at10 at 10°angle. 10°angle. Differentangle. Different Different refraction refraction refraction angles angles areangles observed are are observed observed for (A) for 95for kHz,( A(A) )95 (95B )kHz, kHz, 105 kHz, ((BB)) 105105 and kHz, (C) 115 andand kHz. ((CC)) 115 The115 kHz. acoustickHz. The The pressureacoustic acoustic distributionpressure pressure distribution maps showed in normalized linear scale. 휃휃 refers to refraction angle. Subindex s was indicated simulation mapsdistribution showed in maps normalized showed linear in normalized scale. θr referslinear toscale. refraction 푟푟 refers angle. to refraction Subindex angle.s was indicatedSubindex simulations was indicated results. simulation results.results. Figure3 shows the experimental results for monochromatic beam refraction for the sameFigure frequenciesFigure 3 3shows shows used the the in experimental experimental numerical simulation resultsresults for in monochromaticmonochromatic Figure2. The pressure beam beam refraction refraction map for for each for the the fre- samequencysame freq freq isuencies obtaineduencies used used by Fourierin in numerical numerical transformation simulationsimulation of experimentallyinin FigureFigure 2.2. The The recordedpressure pressure time-dependentmap map for for each each frequencywavefrequency envelopes. is isobtained obtained The by time-domain by Fourier Fourier transformation transformation data were measured ofof experimentallyexperimentally at each middle recorded recorded point time time between-depend--depend- the entlayers.ent wave wave The envelopes. envelopes. one-dimensional The The time time raster--domaindomain scan datadata is formed were measuredmeasured from nine at at lineseach each formiddle middle constructing point point between between the con- thetourthe layers plotslayers. inThe. The Figure one one-3dimensional-.dimensional The incident raster raster beam scanscan hits is the formed ﬁrst layer fromfrom nearnine nine lines the lines point for for constructing withconstructing coordinates the the contour(0,contour 25) on plots X-Yplots in plane. in Figure Figure The 3. 3. centers The The incident incident of the outcoming beambeam hits the refractedthe firstfirst layer layer beams near near with the the frequenciespoint point with with coor- 95 coor- kHz, dinates (0, 25) on X-Y plane. The centers of the outcoming refracted beams with frequen- dinates105 kHz, (0, and 25) 115on X kHz-Y plane. were detected,The centers respectively, of the outcoming around therefracted points beams (90, 70), with (90, frequ 90), anden- cies 95 kHz, 105 kHz, and 115 kHz were detected, respectively, around the points (90, 70), cies(90, 95 110). kHz, Experimentally 105 kHz, and 115 observed kHz were angles detected of refraction, respectively in Figure, around3 are in the good points agreement (90, 70), (90, 90), and (90, 110). Experimentally observed angles of refraction in Figure 3 are in good (90,with 90), the and numerical (90, 110). results Experimentally in Figure2 .observed angles of refraction in Figure 3 are in good agreement with the numerical results in Figure 2. agreement with the numerical results in Figure 2.

FigureFigure 3. Experimental 3. Experimental results results of of broadband broadband pulse pulse sourced sourced soundsound intensity maps maps between between the the superlattice superlattice layers layers with with 10° 10 ◦ incidenceincidence to theto the structure. structure. The The subﬁgures subfigures illustrate illustrate the the singlesingle frequency dependent dependent refraction refraction angles angles inside inside the the superlattic superlatticee Figureobtained 3. Experimental from Fouri resultser transformation of broadband of the pulse recorded sourced time sound domain intensity data. mapsThe acoustic between intensity the superlattice distribution layers maps with are 10° incidenceobtained to from the Fourierstructure. transformation The subfigures of illustrate the recorded the single time frequency domain data. dependent The acoustic refraction intensity angles inside distribution the superlattic maps aree obtainedshown in from normalized Fourier linear transformation scale. (A) 95of kHz.the recorded (B) 105 kHz. time ( Cdomain) 115 kHz. data.θr refersThe acoustic to refraction intensity angle. distribution Subindex emapsindicates are experimental results.

Materials 2021, 14, x 5 of 8

Materials 2021, 14, 125 5 of 8 shown in normalized linear scale. (A) 95 kHz. (B) 105 kHz. (C) 115 kHz. 휃푟 refers to refraction angle. Subindex e indicates experimental results.

ExperimentalExperimental results obtained results for the obtained broadband for thepulse broadband are presented pulse in areFigure presented 4. The in Figure4. transmission Thespectra transmission were measured spectra at were two symmetric measured atpoints two symmetricbehind the pointssuperlattice. behind For the superlattice. reference, theFor same reference, measurements the same were measurements performed werewithout performed the superlattice, without theand superlattice, the re- and the sultant pressureresultant maps pressureare shown maps in Figure are shown 4A. The in Figure acoustic4A. beam The acoustic spreads beam symmetrically spreads symmetrically with respect towith the respect horizontal to the axis, horizontal giving rise axis, to giving two practically rise to two equal practically transmission equal transmission spec- spectra tra in Figure 4B.in FigureUnlike4 B.this, Unlike the spectra this, the obtained spectra at obtained two symmetrical at two symmetrical points 40 mm points above 40 mm above and below theand superlattice below the axis superlattice (see Figure axis 4C) (see are Figure quite4 different.C) are quite They different. are shown They by are red shown by red and blue curvesand in blue Figure curves 4D. inBecause Figure of4D. frequency Because ofdispersion frequency inside dispersion the superlattice, inside the superlattice,the the broadband pulsebroadband harmonics pulse arrive harmonics at different arrive vertical at different positions. vertical Therefo positions.re, the Therefore, spectral the spectral compositionscompositions of the signals ofmeasured the signals above measured and below above the superlattice and below theaxis superlatticeare quite dif- axis are quite ferent, as showndifferent, in Figure as shown 4D. The in measured Figure4D. spectra The measured exhibit narrow spectra peaks exhibit at narrow92 and peaks112 at 92 and kHz. Geometrical112 kHz. positions Geometrical of these positionspeaks coincide of these well peaks with coincide the positions well withcalculated the positions from calculated Snell’s Law usingfrom the Snell’s definition Law using of the the group deﬁnition index ofof therefraction group index[33]. of refraction [33]. s 2 2 푘 푐 /휔 kyc0/ω ∂kx ( 푦 )0= − 휕푘푥 + 푛푔(휔) = −ng ω √1 + ( ) 1 (1) (1) (휕푘 /휕푘 ) ∂kx/∂ky휕푘푦 ∂ky ω 푥 푦 휔 ω 휔

where 푘푥 is thewhere Blochkx isvector the Bloch of propagating vector of propagating wave, 푘푦 is wave, the parallelky is the to parallel the plates to thecompo- plates component nent of the waveof the vector, wave which vector, is which conserved is conserved at refraction, at refraction, and 푐 andis thec0 speedis the of speed sound of soundin in water. 0 water. The derivativeThe derivative (휕푘 /휕푘∂kx)/∂ isky calcisulated calculated from fromthe dispersion the dispersion of the ofsecond the second trans- transmission 푥 푦 휔 ω mission bandband at a given at a given frequency frequency 휔. ω.

Figure 4. Experimental results of pulse decomposition measured at two symmetrical vertical Figure 4. Experimental results of pulse decomposition measured at two symmetrical vertical points. (A) and (B) show the points. (A) and (B) show the symmetric pressure map and almost equal transmission spectra ob- symmetric pressuretained in map free and water almost without equal superlattice. transmission (C) spectra and (D obtained) show different in free waterspectral without compositions superlattice. of the (C) and (D) show differentsignals spectral measured compositions 40 mm ofabove the signals and below measured superlattice 40 mm axis. above The and two below spectra superlattice were measured axis. The at the two spectra were measureddistance at the 6 distance mm behind 6 mm the behind last layer the last of the layer superlattice. of the superlattice.

Materials 2021, 14, x 6 of 8

Continuous spatial distribution of pulse harmonics is given by color map in Figure 5. The transmitted intensity of sound was measured along the vertical 90 mm long vertical line (from −45 to +45 mm) parallel to the last plate at a distance 2 mm behind it. White dashed lines mark the region’s boundaries where the essential part of acoustic energy is transmitted. Inclinations of these lines are a direct manifestation of pulse decomposition in a medium with frequency dispersion. Materials 2021, 14, 125 The harmonics with frequencies from 82.25 to 95 kHz arrive within the spatial6 of 8 interval from −45 to 30 mm. The frequency harmonics from 100 to 110 kHz are refracted within the interval from 30 to 45 mm. The decomposition effect turns out to be relatively poor, since a broad transmission band from 88 to 108 kHz is observed between 21 to 36 mm. dashed lines markThe decomposition the region’s boundaries performance where is much the better essential for the part lower of frequency acoustic range energy from is 82.25 transmitted. Inclinationsto 94.5 kHz. of The these decomposition lines are a direct performance manifestation at higher of frequencies, pulse decomposition between 102 in to 110 a medium withkHz, frequency is also rather dispersion. effective. These harmonics are located from 37 to 45 mm on the vertical axis.

Figure 5. ExperimentalFigure 5. Experimental result of superlattice result decomposition of superlattice performance decomposition of the entire performance second transmission of the entire band secondalong the scanned line at 2 mm parallel to the last layer of the superlattice. At different locations along the vertical axis, transmission transmission band along the scanned line at 2 mm parallel to the last layer of the superlattice. At peaks centered at different frequencies are observed. The 10-layer superlattice decomposes the pulse into narrow bands propagateddifferent along different locations refraction along thepaths vertical that finally axis, arrive transmission at different peaks locations centered on the at scanned different line. frequencies are observed. The 10-layer superlattice decomposes the pulse into narrow bands propagated along different refraction4. Discussion paths that ﬁnally arrive at different locations on the scanned line. Fundamental studies of periodic structures of acoustic metamaterials have been of The harmonicshigh interest with frequencies for decades. from However, 82.25 to most 95 kHz of the arrive practical within applications the spatial of interval acoustic met- from −45 to 30amaterials mm. The have frequency been mostly harmonics used for from filtering, 100 to waveguiding, 110 kHz are refractedand lensing. within This study the interval fromdemonstrates 30 to 45 mm. that an The ac decompositionoustic metamaterial effect superlattice turns out can to spatially be relatively disperse poor, a broad- since a broad transmissionband acoustic band pulse from into 88narrowband to 108 kHz signals is observed with different between central 21 to frequencies. 36 mm. The Due to decompositionrefraction, performance the various is much components better for ofthe the lowerbroadband frequency wave travel range with from different 82.25 velocities to 94.5 kHz. The decompositionwithin the crystal performance and arrive at at different higher frequencies,locations at the between end of the 102 superlattice. to 110 kHz, To the is also rather effective.best of our These knowledge, harmonics a mechanical are located device from has 37 yet to to 45 be mm used on for the the vertical spectral axis. decomposition of a broadband pulse. Signal decomposition is conventionally done applying signal 4. Discussion Fundamental studies of periodic structures of acoustic metamaterials have been of high interest for decades. However, most of the practical applications of acoustic metamaterials have been mostly used for ﬁltering, waveguiding, and lensing. This study demonstrates that an acoustic metamaterial superlattice can spatially disperse a broadband acoustic pulse into narrowband signals with different central frequencies. Due to refraction, the various components of the broadband wave travel with different velocities within the crystal and arrive at different locations at the end of the superlattice. To the best of our knowledge, a mechanical device has yet to be used for the spectral decomposition of a broadband pulse. Signal decomposition is conventionally done applying signal processing software and noise cancelation methods. Here we propose an alternative pure mechanical method, which is simple, robust, free from time delay, and does not require sophisticated numerical technique. Materials 2021, 14, 125 7 of 8

The decomposition effect is observed for the frequencies where the corresponding transmission band exhibits strongly nonlinear frequency dispersion. The proposed acoustic device’s performance can be increased if the sound wave propagates a long distance inside the superlattice and selects the interval of frequencies where the dispersion of the transmitting bands is stronger. The intensity of the input acoustic beam is strongly reduced passing through 10 layers of the superlattice. The principal source of the weakness of the output signal is reﬂection at the water–superlattice interface. By estimating the maximum intensity of the beam obtained in numerical simulations, we conclude that average over the spectrum loss is about 11.8 dB. Experimentally, the loss averaged over all the frequency components is 22.9 dB. Higher experimental losses are due to viscous dissipation, which is ignored in numerical simulation and radiative losses to water environment generated by vibrating plate holders.

Author Contributions: Conceptualization: Y.J. and A.N.; methodology: Y.J. and T.Y.; software: Y.Z.; validation: A.K., T.-Y.C. and A.N.; formal analysis: Y.J. and Y.Z.; investigation: A.N.; resources: A.N. and T.-Y.C.; data curation: Y.J.; writing—original draft preparation: Y.J., Y.Z. and T.Y.; writing—review and editing: A.K. and A.N.; visualization: A.K.; supervision: A.N.; project administration: T.-Y.C., A.K. and A.N.; funding acquisition: A.N. All authors have read and agreed to the published version of the manuscript. Funding: This work is supported by an Emerging Frontiers in Research and Innovation (EFRI) grant from the National Science Foundation (NSF) Grant No. 1741677, and the support from the infrastructure and support of Center for Agile & Adaptive and Additive Manufacturing (CAAAM) funded through State of Texas Appropriation #190405-105-805008-220 is also greatly acknowledged. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data available from corresponding author. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References 1. Kushwaha, M.S.; Halevi, P.; Dobrzynski, L.; Djafari-Rouhan, B. Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 1993, 71, 2022. [CrossRef][PubMed] 2. Dowling, J.P. Sonic band structure in ﬂuids with periodic density variations. J. Acoust. Soc. Am. 1992, 91, 2539–2543. [CrossRef] 3. Krokhin, A.A.; Arriaga, J.; Gumen, L.N. Speed of sound in periodic elastic composites. Phys. Rev. Lett. 2003, 91, 264302. [CrossRef] [PubMed] 4. Hou, Z.; Wu, F.; Fu, X.; Liu, Y. Effective elastic parameters of the two-dimensional phononic crystal. Phys. Rev. E 2005, 71, 037604. [CrossRef][PubMed] 5. Sánchez-Dehesa, J.; Krokhin, A. Introduction to acoustics of phononic crystals. Homogenization at low frequencies. In Phononic Crystals; Springer: New York, NY, USA, 2016; pp. 1–21. 6. Lu, M.-H.; Feng, L.; Chen, Y.-F. Phononic crystals and acoustic metamaterials. Mater. Today 2009, 12, 34–42. [CrossRef] 7. Shen, C.; Xie, Y.; Sui, N.; Wang, W.; Cummer, S.; Jing, Y. Broadband Acoustic Hyperbolic Metamaterial. Phys. Rev. Lett. 2015, 115, 254301. [CrossRef] 8. Zubov, Y.; Djafari-Rouhani, B.; Jin, Y.; Soﬁeld, M.; Walker, E.; Neogi, A.; Krokhin, A. Long-range nonspreading propagation of sound beam through periodic layered structure. Commun. Phys. 2020, 3, 1–8. [CrossRef] 9. García-Chocano, V.M.; Christensen, J.; Sánchez-Dehesa, J. Negative refraction and energy funneling by hyperbolic materials: An experimental demonstration in acoustics. Phys. Rev. Lett. 2014, 112, 144301. [CrossRef] 10. Page, J.H.; Yang, S.; Liu, Z.; Cowan, M.L.; Chan, C.T.; Sheng, P. Tunneling and dispersion in 3D phononic crystals. Z. für Krist.-Cryst. Mater. 2005, 220, 859–870. [CrossRef] 11. Mousavi, S.H.; Khanikaev, A.B.; Wang, Z. Topologically protected elastic waves in phononic metamaterials. Nat. Commun. 2015, 6, 1–7. [CrossRef] 12. Ma, G.; Sheng, P. Acoustic metamaterials: From local resonances to broad horizons. Sci. Adv. 2016, 2, e1501595. [CrossRef] [PubMed] 13. Guo, Y.; Cortes, C.L.; Molesky, S.; Jacob, Z. Broadband super-Planckian thermal emission from hyperbolic metamaterials. Appl. Phys. Lett. 2012, 101, 131106. [CrossRef] Materials 2021, 14, 125 8 of 8

14. Li, J.; Liu, Z.; Qiu, C. Negative refraction imaging of acoustic waves by a two-dimensional three-component phononic crystal. Phys. Rev. B 2006, 73, 054302. [CrossRef] 15. Kaya, O.A.; Cicek, A.; Ulug, B. Self-collimated slow sound in sonic crystals. J. Phys. D Appl. Phys. 2012, 45, 365101. [CrossRef] 16. Walker, E.L.; Jin, Y.; Reyes, D.; Neogi, A. Sub-wavelength lateral detection of tissue-approximating masses using an ultrasonic metamaterial lens. Nat. Commun. 2020, 11, 1–13. [CrossRef][PubMed] 17. Yang, T.; Jin, Y.; Choi, T.-Y.; Dahotre, N.; Neogi, A. Mechanically tunable ultrasonic metamaterial lens with a subwavelength resolution at long working distances. Smart Mater. Struct. 2020, 30, 015022. [CrossRef] 18. Zhang, S.; Yin, L.; Fang, N. Focusing ultrasound with an acoustic metamaterial network. Phys. Rev. Lett. 2009, 102, 194301. [CrossRef] 19. Sun, F.; Guo, S.; Li, B.; Liu, Y.; He, S. An acoustic metamaterial lens for acoustic point-to-point communication in air. Acoust. Phys. 2019, 65, 1–6. [CrossRef] 20. Ma, C.; Kim, S.; Fang, N.X. Far-field acoustic subwavelength imaging and edge detection based on spatial filtering and wave vector conversion. Nat. Commun. 2019, 10, 1–10. [CrossRef] 21. Walker, E.L.; Reyes-Contreras, D.; Jin, Y.; Neogi, A. Tunable Hybrid Phononic Crystal Lens Using Thermo-Acoustic Polymers. ACS Omega 2019, 4, 16585–16590. [CrossRef] 22. Wells, P.N. Ultrasound imaging. Phys. Med. Biol. 2006, 51, R83. [CrossRef][PubMed] 23. Chan, V.; Perlas, A. Basics of ultrasound imaging. In Atlas of Ultrasound-Guided Procedures in Interventional Pain Management; Springer: New York, NY, USA, 2011; pp. 13–19. 24. Fenster, A.; Downey, D.B.; Cardinal, H.N. Three-dimensional ultrasound imaging. Phys. Med. Biol. 2001, 46, R67. [CrossRef] [PubMed] 25. Jin, Y.; Walker, E.; Krokhin, A.; Heo, H.; Choi, T.-Y.; Neogi, A. Enhanced instantaneous elastography in tissues and hard materials using bulk modulus and density determined without externally applied material deformation. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 2019, 67, 624–634. [CrossRef][PubMed] 26. Jin, Y.; Walker, E.; Heo, H.; Krokhin, A.; Choi, T.-Y.; Neogi, A. Nondestructive ultrasonic evaluation of fused deposition modeling based additively manufactured 3D-printed structures. Smart Mater. Struct. 2020, 29, 045020. [CrossRef] 27. Jin, Y.; Yang, T.; Heo, H.; Krokhin, A.; Shi, S.Q.; Dahotre, N.; Choi, T.-Y.; Neogi, A. Novel 2D Dynamic Elasticity Maps for Inspection of Anisotropic Properties in Fused Deposition Modeling Objects. Polymers 2020, 12, 1966. [CrossRef] 28. Ostrovskii, I.V.; Nadtochiy, A.B.; Klymko, V.A. Velocity dispersion of plate acoustic waves in a multidomain phononic superlattice. Phys. Rev. B 2010, 82, 014302. [CrossRef] 29. Chowdhury, I.; Prasher, R.; Lofgreen, K.; Chrysler, G.; Narasimhan, S.; Mahajan, R.; Koester, D.; Alley, R.; Venkatasubramanian, R. On-chip cooling by superlattice-based thin-film thermoelectrics. Nat. Nanotechnol. 2009, 4, 235–238. [CrossRef] 30. Arregui, G.; Lanzillotti-Kimura, N.D.; Sotomayor-Torres, C.M.; García, P.D. Anderson photon-phonon colocalization in certain random superlattices. Phys. Rev. Lett. 2019, 122, 043903. [CrossRef] 31. Yudistira, D.; Boes, A.; Janner, D.; Pruneri, V.; Friend, J.; Mitchell, A. Polariton-based band gap and generation of surface acoustic waves in acoustic superlattice lithium niobate. J. Appl. Phys. 2013, 114, 054904. [CrossRef] 32. Yang, G.Y.; Du, J.K.; Huang, B.; Jin, Y.A.; Xu, M.H. Surface acoustic waves in acoustic superlattice lithium niobate coated with a waveguide layer. AIP Adv. 2017, 7, 045206. [CrossRef] 33. Christensen, J.; Garcia de Abajo, F.J. Negative refraction and backward waves in layered acoustic metamaterials. Phys. Rev. B 2012, 86, 02431. [CrossRef]