Propagation of acoustic waves in a fluid-filled pipe with periodic elastic Helmholtz resonators

Helmholtz resonators are widely used to reduce noise in a fluid-filled pipe system. It is a challenge to obtain lowfrequency and broadband attenuation with a small sized cavity. In this paper, the propagation of acoustic waves in a fluid-filled pipe system with periodic elastic Helmholtz resonators is studied theoretically. The resonance frequency and sound transmission loss of one unit are analyzed to validate the correctness of simplified acoustic impedance. The band structure of infinite periodic cells and sound transmission loss of finite periodic cells are calculated by the transfer matrix method and finite element software. The effects of several parameters on band gap and sound transmission loss are probed.Further, the negative bulk modulus of periodic cells with elastic Helmholtz resonators is analyzed. Numerical results show that the acoustic propagation properties in the periodic pipe, such as low frequency, broadband sound transmission, can be improved.     

Transfer matrix methods for sound attenuation in resonators with perforated intruding inlets

A resonator with perforated intruding inlet (PII) is a superior silencer element, since the use of perforated inlet extensions can dramatically improve the acoustic performance. In this work, both a one-dimensional (1D) and a two-dimensional (2D) transfer matrix methods are developed to predict the transmission loss of the resonator without considering the mean flow. Based on the two groups of comparisons with tests, it is found out that the applicability of 1D method is limited by the resonator geometry even when the frequency is below the cut-off value of plane wave. Whereas the 2D approach is much more accurate while predicting the transmission losses within entire frequency range. Subsequently, five groups of resonators are chosen to determine the effects of structure parameters to transmission loss based on the 2D approach. The resonant frequency decreases and more resonant peaks appear when the length of inlet extension increases. A higher perforation rate leads to a shift of resonant peak towards higher frequencies. Besides, better acoustic performance could be obtained with the perforation being properly designed. Reducing the inlet/outlet radius can obviously improve the transmission loss without changing the frequency of resonant peak. The theories and conclusions in this study can be used for the design and optimization of resonators in various engineering applications.     

腔壁弹性对水下小型圆柱形亥姆霍兹共振器共振频率的影响

理论和实验研究了腔壁弹性对水下小型圆柱形亥姆霍兹共振器共振频率的影响.基于电-声类比理论,建立了小型共振器的简化模型,利用电路分析方法得到了便于计算的共振频率一般表达式.分别仿真分析了共振器壁面厚度和材料对共振频率的影响,得到了不同尺寸的小型共振器的近似刚性条件.在充水驻波罐中对不同壁厚、不同材料的小型圆柱形亥姆霍兹共振器的共振频率进行了测量,实验结果较好地验证了理论分析和近似刚性条件的正确性.所得结果对小型圆柱形亥姆霍兹共振器的设计和水下应用具有较好的参考价值. 关键词： 亥姆霍兹共振器 共振频率 传递函数 辐射阻抗     

General theory and design of microwave open resonators

Based on the analysis of wave propagation in a waveguide with slowly varying cross-section, a general theory is developed for microwave open resonators with rotational-symmetric waveguides; this geometry has been extensively used in the gyrotrons. The theoretical analysis is carried through under the most general conditions, that is the cross-section radius r(z) of the resonator is a slowly varying arbitrary function of z and the resonator is open. The general expressions for the resonant frequencies and the Q have been derived. A design method for this kind of open resonator has been proposed. Calculation results for two practical resonators are presented.     

Design of metaporous supercells by genetic algorithm for absorption optimization on a wide frequency band

The optimization of acoustic absorption by metaporous materials made of complex unit cells with 2D resonant inclusions is realized using genetic algorithm. A nearly total absorption over a wide frequency band can be obtained for thin structures, even for frequencies below the quarter wavelength resonances i.e., in a sub-wavelength regime. The high absorption performances of this material are due to the interplay of usual visco-thermal losses, local resonances and trapped modes. The density of resonant and trapped modes in this dissipative porous layer, is a key parameter for broadband absorption. The best configurations and critical coupling conditions are found by genetic algorithm optimization. Several types of resonators are included gradually in the studied configurations (split-rings, Helmholtz resonators, back cavities) with increasing complexity. The optimization leads to a metaporous structure with a 2-cm sub-wavelength layer thickness, exhibiting a nearly total absorption between 1800 Hz and 7000 Hz. The influence of the incidence angle on the absorption properties is also shown.     

Temperature-controlled tunable acoustic metamaterial with active band gap and negative bulk modulus

The local resonant band gap and the negative bulk modulus of the acoustic metamaterial with Helmholtz resonators are strongly affected by the temperature of water. In this paper, the acoustic transmission line method (ATLM) is introduced to investigate the influences of the temperature of water on the local resonant band gap and the negative bulk modulus of the acoustic metamaterial. Results show that the relative variations of the local resonant band gap and the negative effective bulk modulus suffering from the change of the temperature of water are approximately equivalent and are up to about 11%. The reason is that the local resonant frequency is proportional to the sound speed of fluid which is strongly effects by the temperature of water. By achieving the unambiguous relationships between these unusual properties of the acoustic metamaterial and the temperature of water, we find that the temperature-controlled acoustic metamaterial with the active band gap and the active negative bulk modulus can be realized in theory. This idea opens a new avenue for the design of the tunable acoustic metamaterial that can modulate the acoustic wave propagation.     

Nonlinear standing waves in 2-D acoustic resonators

This paper deals with 2-D simulation of finite-amplitude standing waves behavior in rectangular acoustic resonators. Set of three partial differential equations in third approximation formulated in conservative form is derived from fundamental equations of gas dynamics. These equations form a closed set for two components of acoustic velocity vector and density, the equations account for external driving force, gas dynamic nonlinearities and thermoviscous dissipation. Pressure is obtained from solution of the set by means of an analytical formula. The equations are formulated in the Cartesian coordinate system. The model equations set is solved numerically in time domain using a central semi-discrete difference scheme developed for integration of sets of convection-diffusion equations with two or more spatial coordinates. Numerical results show various patterns of acoustic field in resonators driven using vibrating piston with spatial distribution of velocity. Excitation of lateral shock-wave mode is observed when resonant conditions are fulfilled for longitudinal as well as for transversal direction along the resonator cavity.     

The design of Helmholtz resonator based acoustic lenses by using the symmetric Multi-Level Wave Based Method and genetic algorithms

Sonic crystals can be used as acoustic lenses in certain frequencies and the design of such systems by creating vacancies and using genetic algorithms has been proven to be an effective method. So far, rigid cylinders have been used to create such acoustic lens designs. On the other hand, it has been proven that Helmholtz resonators can be used to construct acoustic lenses with higher refraction index as compared to rigid cylinders, especially in low frequencies by utilizing their local resonances. In this paper, these two concepts are combined to design acoustic lenses that are based on Helmholtz resonators. The Multi-Level Wave Based Method is used as the prediction method. The benefits of the method in the context of design procedure are demonstrated. In addition, symmetric boundary conditions are derived for more efficient calculations. The acoustic lens designs that use Helmholtz resonators are compared with the acoustic lens designs that use rigid cylinders. It is shown that using Helmholtz resonator based sonic crystals leads to better acoustic lens designs, especially at the low frequencies where the local resonances are pronounced.     

A study of stepped acoustic resonator with transfer matrix method

Transfer matrix method was applied in the study of stepped acoustic resonators. Transfer matrix method was more competent in comparison with analytic method to investigate the acoustic properties of stepped acoustic resonator, especially multi-step acoustic resonator. With the help of the numerical solution, the resonance frequencies, the phase angles and the radiation impedances of stepped acoustic resonators which consisted of one to five sub-tubes were studied theoretically and experimentally. The numerical solutions were in excellent agreement with the experimental results.     

Helmholtz水声换能器弹性壁液腔谐振频率研究

针对传统Helmholtz水声换能器设计中刚性壁假设的局限性,将Helmholtz腔体的弹性计入到液腔谐振频率计算中,实现低频弹性Helmholtz水声换能器液腔谐振频率精确设计.基于细长圆柱壳腔体的低频集中参数模型,导出了腔体弹性引入的附加声阻抗表达式,得到了弹性壁条件下Helmholtz水声换能器等效电路图,给出了考虑了末端修正的弹性壁Helmholtz共振腔液腔谐振频率计算公式.利用ANSYS软件建立了算例模型,仿真分析了不同材质、半径、长度时的Helmholtz共振腔液腔谐振频率.结果对比表明弹性理论值与仿真值符合得很好,相比起传统的刚性壁理论计算结果,本文的弹性壁理论得出的液腔谐振频率值有所降低,与真实情况更加接近.本文的结论可以为精确设计低频弹性Helmholtz水声换能器提供理论支持.     

Comparison of 1D transfer matrix method and finite element method with tests for acoustic performance of multi-chamber perforated resonator

Multi-chamber perforated resonator (MCPR) is a kind of typical silencer element which can both attenuate broadband noise and satisfy specific installation requirements. The one-dimensional transfer matrix method (TMM) and finite element method (FEM) are widely used to predict the transmission loss of the resonators. This paper mainly focuses on the comparison between 1D TMM and FEM in which detailed perforation modeling is applied for the acoustic modeling of MCPRs. Five resonators with different acoustic attenuation frequency ranges are built for simulation and test. In order to verify the results of the above methods, a transmission loss test facility is designed based on two-load method. Through adjusting the distance between microphones, the facility’s effective measurement frequency can be changed. The results show that despite of the complex modeling and calculation, FEM with detailed perforation modeling shows good consistency with test results in both frequency and amplitude within entire frequency range. In comparison, TMM is limited by the cut-off frequency when calculating transmission losses. Besides, accuracy of TMM in low frequency range is also affected by perforation conditions. However, TMM is time-saving in calculation and structure optimization. In MCPRs’ development process, TMM can be used to quickly design and optimize structure parameters while FEM can be used to verify the acoustic performance before prototyping.     

Effect of Mean Flow on Acoustic Wave Propagation in a Duct with a Periodic Array of Helmholtz Resonators

Sound propagation properties of a duct system with Helmholtz resonators(HRs) are affected by mean flow.Previous studies have tended to focus on the effects of mean flows on acoustic response of a duct system with a finite number of HRs. Employing an empirical impedance model, we present a modified transfer matrix method for studying the effect of mean flow on the complex band structure of an air duct system with an infinite periodic array of HRs. The efficiency of the modified transfer matrix is demonstrated by comparison between an example of transmission response calculation for a finite single HR loaded duct and the finite element simulation result calculated using the COMSOL software. Numerical results are presented to analyze the effect of mean flow on the band structure and transmission loss of the sound wave in the duct system. It is hoped that this study will provide theoretical guidance for acoustic wave propagation of HR silencer in the presence of mean flow.     

弯曲圆盘驱动的双Helmholtz共振腔换能器

使用双面同相振动的弯曲圆盘换能器驱动双Helmholtz共振腔,既放大了弯曲圆盘换能器弯曲共振频率以下频段的声输出,又利用两个Helmholtz共振腔的同相声源辐射模型实现了在Helmholtz共振频率处的"∞"字形垂直指向性,实现了低频指向性声发射。阐述了换能器实现"∞"字形低频指向性发射的机理,研究了腔体长度、金属圆片厚度及弯曲圆盘边缘简支厚度等关键结构参数对Helmholtz共振频率的影响,求解了换能器的发送电压响应、指向性等参数。依据仿真结果制作了实验样机,在消声水池中进行了电声性能测试。测试结果显示,指向性形状及液腔共振频率与仿真结果基本相符。这种由弯曲圆盘驱动的双Helmholtz共振腔水声换能器为实现水声换能器小尺寸、低频指向性发射提供了一种技术手段。      

基模动态稳定望远镜谐振腔

本文使用等价腔和传播矩阵方法,在一般情况下推导出了望远镜腔的基模热稳条件,该表示式简单而又严格,还易于推广到含热透镜多元件腔的一般情况,并与用近似方法所得结果作了比较和讨论。     

Acoustic meta-materials in MEMS BAW resonators

This paper presents a meta-material-based design method for bulk acoustic wave (BAW) resonators with enhanced characteristics compared to those obtained with the typical bulk material implementation. We demonstrate the novel use of empty inclusions (i.e., ‘holes’) in bulk materials for engineering their acoustic (mechanical) properties (e.g. Young’s modulus E, Poisson’s ratio ν and mass density ρ) to tune and achieve optimal acoustical performance/characteristics. Inclusions have been demonstrated before to produce phononic band gaps for wave trapping. We focus on the propagation characteristics of the meta-materials brought into being by these inclusions. We implement patterns of holes with different sizes and distributions, to effectively scatter acoustic waves in bar-type BAW resonators and to devise the desired resonator properties, e.g., the resonant frequency. While the available bulk material is homogeneous and isotropic, the bar consists of an equivalent non-homogeneous material that can for example be distributed by design in order to shrink the overall resonator size, enhance electromechanical transduction coefficients or reject spurious modes. Our paper compares two extraction methods for the equivalent material properties of a periodically hole-punched material: the steady-state mechanical simulation of a unit cell and its ‘phase delay’ counterpart. We discuss their validity and practical use for the design of bar resonators.     

Analysis of underwater decoupling properties of a locally resonant acoustic metamaterial coating

This paper presents a semi-analytical solution for the vibration and sound radiation of a semi-infinite plate covered by a decoupling layer consisting of locally resonant acoustic metamaterial. Formulations are derived based on a combination use of effective medium theory and the theory of elasticity for the decoupling material. Theoretical results show good agreements between the method developed in this paper and the conventional finite element method(FEM), but the method of this paper is more efficient than FEM. Numerical results also show that system with acoustic metamaterial decoupling layer exhibits significant noise reduction performance at the local resonance frequency of the acoustic metamaterial, and such performance can be ascribed to the vibration suppression of the base plate. It is demonstrated that the effective density of acoustic metamaterial decoupling layer has a great influence on the mechanical impedance of the system. Furthermore, the resonance frequency of locally resonant structure can be effectively predicted by a simple model, and it can be significantly affected by the material properties of the locally resonant structure.     

多重散射方法研究轴对称空腔覆盖层的声学特性

发展了一种多重散射方法研究声学覆盖层的半数值半解析模型,分析了影响轴对称空腔结构声学性能的主要能量耗散机制。在球坐标条件下推导出轴对称空腔结构的位移和应力场基函数,通过对空腔表面基函数的数值积分,得到散射波和入射波之间的传输矩阵方程,结合分层介质声传播理论计算了周期性空腔结构覆盖层的反射、透射和吸声性能。研究结果表明;空腔共振是低频能量耗散的主要形式,边界条件对材料空腔结构的谐振特性影响很大,利用双空腔耦合共振可以拓宽材料的低频吸声频带;背衬对材料的高频吸声影响较小,材料的高频能量损耗取决于空腔的散射和波型转换特性。      

Acoustic scattering from a submerged cylindrical shell coated with locally resonant acoustic metamaterials

Using the multilayered cylinder model, we study acoustic scattering from a submerged cylindrical shell coated with locally resonant acoustic metamaterials, which exhibit locally negative effective mass densities. A spring model is introduced to replace the traditional transfer matrix, which may be singular in the negative mass region. The backscattering form function and the scattering cross section are calculated to discuss the acoustic properties of the coated submerged cylindrical shell.     

Noise control in enclosures: modeling and experiments with T-shaped acoustic resonators

This paper presents a theoretical and experimental study of noise control in enclosures using a T-shaped acoustic resonator array. A general model with multiple resonators is developed to predict the acoustic performance of small resonators placed in an acoustic enclosure. Analytical solutions for the sound pressure inside the enclosure and the volume velocity source strength out of the resonator aperture are derived when a single resonator is installed, which provides insight into the physics of acoustic interaction between the enclosure and the resonator. Based on the understanding of the coupling between the individual resonators and enclosure modes, both targeted and nontargeted, a sequential design methodology is proposed for noise control in the enclosure using an array of acoustic resonators. Design examples are given to illustrate the control performance at a specific or at several resonance peaks within a frequency band of interest. Experiments are conducted to systematically validate the theory and the design method. The agreement between the theoretical and experimental results shows that, with the help of the presented theory and design methodology, either single or multiple resonance peaks of the enclosure can be successfully controlled using an optimally located acoustic resonator array.     

Experimental studies on perturbed acoustic resonant spectroscopy by a small rock sample in a cylindrical cavity

A measurement system for acoustic resonant spectroscopy (ARS) is established, and the effects of resonant cavity geometry, inner perturbation samples and environmental temperature on the ARS are investigated. The ARSs of the small samples with various sizes and acoustic properties are measured. The results show that at the normal pressure, the resonant frequency decreases gradually with the increase of liquid temperature in the cylindrical cavity, while the resonant amplitude increases. At certain pressure and temperature, both the resonant frequency and the amplitude decrease greatly when there exist air bubbles inside the cavity fluid. The ARS is apparently affected by the sample porosity and the sample location in the resonant cavity. At the middle of the cavity, the resonant frequencies reach their maximum values for all of the measurement samples. The resonant frequencies of the porous rock samples are smaller than those of the compacted samples if other acoustic parameters are the same. As the sample is moved from the top to the middle of the cavity along its axis, the resonant amplitude increases gradually for the compacted rocks while decreases for the unconsolidated rocks. Furthermore, the resonant amplitude increases firstly and then decreases if the porosity of the rock sample is relatively small. In addition, through the comparisons between the experimental and theoretical results, it is found that the effects of the acoustic parameters and sizes of the samples and the size of the cylindrical cavity on the laboratory results agree well with the theoretical ones qualitatively. These results may provide basic reference for the experiment study of rock acoustic properties in a low frequency using ARS.