Acoustic Band Structure Research Articles

Phononic band gap and wave propagation on polyvinylidene fluoride-based acoustic metamaterials

In the present work, the acoustic band structure of a two-dimensional phononic crystal (PC) containing an organic ferroelectric (PVDF-polyvinylidene fluoride) and topological insulator (SnTe) was investigated by the plane-wave-expansion (PWE) method. Two-dimensional PC with square lattices composed of SnTe cylindrical rods embedded in the PVDF matrix is studied to find the allowed and stop bands for the waves of certain energy. Phononic band diagram ω = ω(k) for a 2D PC, in which non-dimensional frequencies ωa/2πc (c-velocity of wave) were plotted vs. the wavevector k along the Г–X–M–Г path in the square Brillouin zone shows five stop bands in the frequency range between 10 and 110 kHz. The ferroelectric properties of PVDF and the unusual properties of SnTe as a topological material give us the ability to control the wave propagation through the PC over a wide frequency range of 103–106 Hz. SnTe is a discrete component that allows conducting electricity on its surface but shows insulator properties through its bulk volume. Tin telluride is considered as an acoustic topological insulator as the extension of topological insulators into the field of “topological phononics”.

Acoustic band structures and time reversal of elastic waves in two- and three-dimensional phononic crystals

The present study aims to investigate experimentally the acoustic band structures of elastic waves in two-dimensional (2D) and three-dimensional (3D) phononic crystals (PCs) consisting of periodic square arrays of cylindrical air holes in a solid matrix made of cubic polyoxymethylene. In the 2D PC, the first band gap appeared at frequencies between 0.50 and 0.77 MHz, and the second pass band at frequencies between 0.77 and 1.17 MHz. On the other hand, the 3D PC exhibited a wider band gap extending from a frequency of 0.44 MHz with noise level transmission at a frequency around 0.93 MHz. The time compressions of the time-reversed waves through the 2D and the 3D PCs were also obtained by using a time reversal process. The 2D PC exhibited better time compression than the 3D PC in the time-reversed waves, consistent with the fact that the time compression achieved by time reversal is sensitive to the number of frequency information grains.

Acoustic band structures in two-dimensional phononic crystals consisting of periodic square arrays of stainless-steel cylinders in water

The present study aims to investigate experimentally and theoretically the acoustic band structures in two-dimensional phononic crystals (PCs) consisting of periodic square arrays of stainlesssteel solid cylinders with diameters of 1.0 mm in water. Two types of PCs with different lattice constants (a), 1.5 and 3.0 mm, were assembled to investigate the influence of structural parameters on their band gaps. The dispersion relations were calculated along the ΓX direction of the first Brillouin zone by using the finite element method. The experimental and the theoretical transmission coefficients were obtained for 2, 4, 6, 8, and 10 layers of cylinders along the ΓX direction. The first band gap of the PC with a = 1.5 mm was found around a frequency of 0.5 MHz while that of the PC with a = 3.0 mm was found around 0.25 MHz. Relatively wide band gaps were observed in the PC with a = 1.5 mm compared to that with a = 3.0 mm. The combination of the two different PCs in tandem with different structural parameters was demonstrated to lead to a superposition of their individual band gaps.

수중에서 정방형 격자를 갖는 2차원 포노닉 크리스탈의 음향 밴드 구조

포노닉 크리스탈이란 기저물질 내에 주기적으로 배열된 산란체로 구성된 복합물질로서 포노닉 크리스탈에 입사된 음파가 특정 주파수 대역에서 차단되는 현상인 밴드 갭이라는 중요한 특성을 갖는다. 본 연구에서는 수중에서 산란체로서 1 mm의 직경을 갖는 원기둥 형태의 스테인리스 스틸 막대가 1.5 mm의 격자상수를 가지며 정방형으로 배열된 2차원 포노닉 크리스탈의 음향 밴드 구조를 이론 및 실험적으로 고찰하였다. 2차원 포노닉 크리스탈의 밴드 구조를 예측하기 위해 유한요소법을 이용하여 첫째 브릴루앙 영역의 <TEX>${\Gamma}X$</TEX> 방향에 대해 주파수와 파동벡터에 대한 분산관계를 계산하였다. 초음파가 입사되는 방향과 수직한 스테인리스 스틸 막대 층의 개수를 1, 3, 5, 7, 9개로 변화시켜가며 투과계수 및 반사계수를 측정하였다. 계산된 분산관계로부터 2 MHz 이하의 주파수 대역에서 5개의 밴드 갭이 존재하는 것으로 예측되었으며, 첫째 밴드 갭은 0.5 MHz를 중심으로 나타났다. 투과계수 및 반사계수로부터 실험적으로 확인된 밴드 갭은 분산관계로부터 예측된 밴드 갭과 잘 일치하는 것으로 나타났다. Phononic crystals are composite materials consisting of a periodic arrangement of scattering inclusions in a host material. One of the most important properties of phononic crystals is the existence of band gaps, i.e., ranges of frequencies at which acoustic waves cannot propagate through the structure. The present study aims to investigate theoretically and experimentally the acoustic band structures in two-dimensional (2D) phononic crystals consisting of periodic square arrays of stainless steel solid cylinders with a diameter of 1 mm and a lattice constant of 1.5 mm in water. The theoretical dispersion relation that depicts the relationship between the frequency and the wave vector was calculated along the <TEX>${\Gamma}X$</TEX> direction of the first Brillouin zone using the finite element method to predict the band structures in the 2D phononic crystals. The transmission and the reflection coefficients were measured in the 2D phononic crystals with 1, 3, 5, 7, and 9 layers of stainless steel cylinders stacked in the perpendicular direction to propagation at normal incidence. The theoretical dispersion relation exhibited five band gaps at frequencies below 2 MHz, the first gap appearing around a frequency of 0.5 MHz. The location and the width of the band gaps experimentally observed in the transmission and the reflection coefficients appeared to coincide well with those determined from the theoretical dispersion relation.

Tunable solid acoustic metamaterial with negative elastic modulus

We report in this letter on a tunable solid acoustic metamaterial with negative elastic modulus by means of piezoelectric composite. The theoretical formulae for one-dimensional layer-stacked metamaterial embedding a piezoelectric material by means of external shunted inductors are presented. The acoustic band structure of the composite is calculated by the transfer matrix method. Results show that a band gap can be opened and tuned by the resonant behavior of the LC circuit. It is found further by the formulae that piezoelectric material with large piezoelectric constant and small elastic modulus will be beneficial for opening a wide band gap. The effective elastic constant of the system is also calculated by the unit-cell-boundary-averaging method. Result shows that the system behaves as an effective medium with a negative elastic modulus. This property is quite different from the typical solid metamaterial achieved by dispersing heavy inclusions coated with a soft layer into a matrix for which only the negative mass density can be obtained.

Symmetry breaking induces band gaps in periodic piezoelectric plates

In this study, we propose a phononic plate model with periodically corrugated surfaces and investigate the propagation behavior of Lamb waves in it. In contrast to the traditional phononic crystals which are composed of isotropic materials, the electroelastic coupling is taken into account in this investigation. Based on the Floquet's theorem, the Lamb wave solution is expanded in the series form of spatially harmonic plane waves. Band structures in the first Brillouin zone are presented graphically. The band gaps can be tuned by adjusting the structural parameters. We find that the structural symmetry of the plate has significant influence on the band gap properties. The symmetry breaking of the plate structure leads to more band gaps appearing in the first Brillouin zone. Moreover, the band gaps are also sensitive to the symmetry of the electric boundary conditions, which provides a feasible way to electrically control the acoustic band structure.

The Brillouin zones and band gaps of a two-dimensional phononic crystal with parallelogram lattice structure

We present a detailed theoretical study on the acoustic band structure of two-dimensional (2D) phononic crystal. The 2D phononic crystal with parallelogram lattice structure is considered to be formed by rigid solid rods embedded in air. For the circular rods, some of the extrema of the acoustic bands appear in the usual high-symmetry points and, in contrast, we find that some of them are located in other specific lines. For the case of elliptic rods, our results indicate that it is necessary to study the whole first Brillouin zone to obtain rightly the band structure and corresponding band gaps. Furthermore, we evaluate the first and second band gaps using the plane wave expansion method and find that these gaps can be tuned by adjusting the side lengths ratio R, inclined angle θ and filling fraction F of the parallelogram lattice with circular rods. The results show that the largest value of the first band gap appears at θ=90° and F=0.7854. In contrast, the largest value of the second band gap is at θ=60° and F=0.9068. Our results indicate that the improvement of matching degree between scatterers and lattice pattern, rather than the reduction of structural symmetry, is mainly responsible for the enhancement of the band gaps in the 2D phononic crystal.

Loop filters as resonant elements for acoustic metamaterials and stop band structures

Two geometries of acoustic loop filters side-loaded onto waveguides are explored using audio-frequency impulse response measurements. Finite element modeling is used to elucidate the interference mechanisms in each loop filter configuration. These simulations also yield transmission amplitude results, which are in good agreement with those determined experimentally. The ability to create broad forbidden transmission regions is demonstrated by the use of a sub-wavelength spaced series of slightly detuned filters. Finally, the phase information from the impulse response measurements is analyzed to show that regions of negative bulk modulus exist at frequencies in the vicinity of the loop interference minima.

Brillouin scattering from porous silicon-based optical Bragg mirrors

Brillouin light scattering experiments were done on porous silicon-based optical Bragg mirrors with modulation wavelengths of ∼100 nm. By using a combination of pseudo-reflection and backscattering geometries, phonon dispersion curves along the superlattice modulation axis were mapped. Excellent agreement is obtained with the bulk acoustic mode band structure calculated using a one-dimensional elastic continuum model. In addition to zone-folding of the bulk longitudinal mode dispersion curve, the samples are marked by a surface-localized acoustic mode at the superlattice-air interface. The frequency of this mode lies near the upper edge of a phononic band gap centered at ∼16 GHz. These results, along with optical reflectance data showing visible-range photonic band gaps, reveal that these samples are one-dimensional hypersonic phononic-photonic crystals.

Matryoshka locally resonant sonic crystal

The results of numerical modeling of sonic crystals with resonant array elements are reported. The investigated resonant elements include plain slotted cylinders as well as their various combinations, in particular, Russian doll or Matryoshka configurations. The acoustic band structure and transmission characteristics of such systems have been computed with the use of finite element methods. The general concept of a locally resonant sonic crystal is proposed that utilizes acoustic resonances to form additional band gaps that are decoupled from Bragg gaps. An existence of a separate attenuation mechanism associated with the resonant elements that increases performance in the lower frequency regime has been identified. The results show a formation of broad band gaps positioned significantly below the first Bragg frequency. For low frequency broadband attenuation, a most optimal configuration is the Matryoshka sonic crystal, where each scattering unit is composed of multiple concentric slotted cylinders. This system forms numerous gaps in the lower frequency regime, below Bragg bands, while maintaining a reduced crystal size viable for noise barrier technology. The finding opens alternative perspectives for the construction of sound barriers in the low frequency range usually inaccessible by traditional means including conventional sonic crystals.

The Study of Characteristics of Sphere-Radial Phononic Crystals

By deducing the spherical elastic wave equation in theory, the concept of sphere-radial phononic crystal is proposed, and then the equations to determine the acoustic band structures is deduced. A numerical example is given for steel/nitrile rubber phononic crystal. The numerical simulation results suggest that the band gaps of sphere-radial phononic crystals do exist, which have better attenuation characteristics and practical application performance than the one-dimensional phononic crystals.

Acoustic Band Structures and Homogenization of Periodic Elastic Media

AbstractDispersion curves and frequency dependent velocity surfaces of periodic heterogeneous elastic media are determined from unit‐cell problems using quasi periodic boundary conditions. Effective material parameters are obtained using a least square optimization of the phase velocities in the heterogeneous and the effective, generally anisotropic, homogeneous medium. For efficiency, a preconditioned iterative eigenvalue solver and a model order reduction scheme are used. This “dynamic” homogenization procedure is validated for frequencies below the first stop band employing FEM calculations on a finite structure consisting of an array of unit‐cells and the effective homogeneous material. (© 2010 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Effect of basis configuration on acoustic band structure in two-dimensional complex phononic crystals

The effect of basis configuration on acoustic band structure in two-dimensional complex phononic crystals （PCs） was investigated. We found that when the basis configuration was changed, the irreducible Brillouin zone （BZ） of PCs might be changed too，and part of band extrema were no longer on the high-symmetry lines. In particular，the PCs with some basis configurations were forced to render the whole first BZ irreducible. Therefore，besides the high symmetrical PCs，only by throughout investigation of the BZ can we obtain the authentic band structures and band gaps for the lower symmetrical PCs with complex crystal lattice.

ACOUSTIC BAND GAPS TUNED BY TRANSLATION GROUP SYMMETRY IN TWO-DIMENSIONAL PERIODIC COMPOSITES

The acoustic band structures of two kinds of acoustic crystals (two-dimensional periodic arrays of rigid solid rods embedded in air with two different configurations) have been studied by the plane-wave expansion (PWE) method based on super cell calculation. The translation group symmetry of the acoustic crystal is changed by changing the area of the cross section of adjacent rods. We found that by changing the translation group symmetry, one can effectively adjust the acoustic band gaps (ABGs). In the case that the cross section of scattering rods is square without any rotation, the decrease of translation group symmetry is advantaged to form ABGs. But when the cross section of scattering rods is square with a rotation of 45°, the decrease of translation group symmetry is disadvantaged to form ABGs.

Surface acoustic waves propagating on microstructured phononic crystals

We investigate the interaction between high frequency surface acoustic waves and periodic microstructured patterns that form phononic crystals. The experimental method combines an optical pump-probe setup with interferometric detection and provides picosecond temporal and micron spatial resolutions. Surface acoustic waves with frequency components up to 1.3 GHz are imaged in real-time propagating over the periodic metamaterial. We used a DRIE (deep reactive ion etching) process to fabricate 2D air-silicon phononic crystals in the form of a square lattice. We present real-time animations of surface acoustic waves scattered by the phononic crystals. In particular we describe the frequency and angular dependence of the surface acoustic wave reflection from a 2D phononic crystal boundary. Fourier analysis allows us to reveal details of the acoustic band structure including gaps. The presence of such phononic band gaps enables us to visualize surface acoustic waves in waveguides, cavities and other phononic circuits at GHz frequencies.

Acoustic wave propagation in one-dimensional phononic crystals containing Helmholtz resonators

One-dimensional acoustic waveguide containing subwavelength-sized Helmholtz resonators is known to exhibit novel physical phenomena. However, no systematic theoretical study on this system has been carried out so far except on a few limited cases. We present a thorough theoretical calculation on the acoustic wave propagation in phononic crystals containing Helmholtz resonators without any geometrical size restrictions. The band structures, transmission spectra, and defect states are studied for diverse geometries using the interface response theory. It is shown that the acoustic band structure of the model is fundamentally different from the conventional acoustic–elastic cases and richer due to the coexistence of the resonant and the Bragg gaps. It is also shown that the presence of a defect resonator in the system can give rise to a localized mode inside the resonance gaps. The results clearly show that the presence of the Helmholtz resonators singly or periodically can play a prominent role in designing any acoustic band gap materials.

Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures

We present a full acoustic band structure calculation for periodic elastomeric solids at different levels of deformation. We demonstrate the ability to use deformation to transform phononic band gaps. Periodic elastomeric structures are subjected to axial compression and are found to undergo a transformation in their patterned structure upon reaching a critical value of applied load. During the initial linear regime of the nominal stress-strain behavior, the band gaps evolve in an affine and marginal manner. Upon reaching the critical load, the pattern transformation is found to strongly affect the in-plane phononic band gaps, resulting in the closure of existing band gaps and in the opening of new ones. The elastomeric nature of the material makes the transformation in both structural pattern and phononic band gap a reversible and repeatable process, creating a phononic band gap switch.

Larger acoustic band gaps obtained by configurations of rods in two-dimensional phononic crystals

A method is introduced to optimize the acoustic band structures of two-dimensional phononic crystals (PCs). The acoustic band gap (ABG) between any two bands could be maximized by introducing an additional rod at a suitable position in a unit cell. The results also show that the symmetry reduction of the system is more efficient in creating the ABGs except for the first one. This additional rod method will be useful in designing the ABGs of PCs.

Acoustical properties of two‐dimensional sonic crystals

The transmittance and reflectance spectra of two‐dimensional sonic crystals made of hexagonal arrays of cylindrical bars in water is reported. The balance of sound fluxes is analyzed by studying angle‐resolved spectra of transmitted and reflected sound for all directions in space. A characterization of the acoustic band structure along the two high symmetry directions in the lattice (i.e., ΓX and ΓJ directions) in a large region of frequencies will be presented. It is shown that, for example, the shape of the stop band features in different types of acoustical spectra is interlinked and must be studied jointly to understand acoustical phenomena in these materials. Moreover, the dependence of acoustical properties as a function of the number of layers is also studied. The experiments are compared with theoretical simulations based on multiple scattering theory. [Work supported by MEC of Spain.]

Publisher’s Note: Imaging Ripples on Phononic Crystals Reveals Acoustic Band Structure and Bloch Harmonics [Phys. Rev. Lett.97, 055502 (2006)

Publisher’s Note: Imaging Ripples on Phononic Crystals Reveals Acoustic Band Structure and Bloch Harmonics [Phys. Rev. Lett.<b>97</b>, 055502 (2006)