Helmholtz resonator lined with absorbing material

A closed-form, two-dimensional analytical solution is developed to investigate the acoustic performance of a concentric circular Helmholtz resonator lined with fibrous material. The effect of density and the thickness of the fibrous material in the cavity is examined on the resonance frequency and the transmission loss. With the expressions for the eigenvalue and eigenfunction in the cavity, the transmission loss is obtained for a piston-driven model by applying a pressure/velocity matching technique. The results from the analytical methods are compared to the numerical predictions from a three-dimensional boundary element method and the experimental data obtained from an impedance tube setup. It is shown that the acoustic performance of a Helmholtz resonator may be modified considerably by the density and thickness of the fibrous material without changing the cavity dimensions.     

The impact of the neck material on the sound absorption performance of Helmholtz resonators

Helmholtz resonators with sound absorption materials filling the neck may have an improved sound absorption capacity. In this work, parallel perforated ceramics with different perforation diameters were installed into the neck of a Helmholtz resonator to improve its acoustic impedance to simultaneously achieve a better acoustic absorption coefficient and a wider absorption bandwidth. An experimental system was built to investigate the effect of the perforation diameters on the sound absorption performance of the resonator. It is found that nonlinear effects near the resonance frequency affect the resonator?s neck mouth impedance and further its sound absorption performance significantly. For frequency range 50–500 Hz, a model of the neck mouth impedance is developed based on a revised Forchheimer relationship. The experimental results are in good agreement with the theoretical model.     

Inverse Variational Problem for Nonlinear Evolution Equations

The Helmholtz solution of the inverse problem for the variational calculus is used to study the analytic or Lagrangian structure of a number of nonlinear evolution equations. The quasilinear equations in the KdV hierarchy constitute a Lagrangian system. On the other hand, evolution equations with nonlinear dispersive terms (FNE) are non-Lagrangian. However, the method of Helmholtz can be judiciously exploited to construct Lagrangian system of such equations. In all cases the derived Lagrangians are gauge equivalent to those obtained earlier by the use of Hamilton’s variational principle supplemented by the methodology of integer-programming problem. The free Hamiltonian densities associated with the so-called gauge equivalent Lagrangians yield the equation of motion via a new canonical equation similar to that of Zakharov, Faddeev and Gardner. It is demonstrated that the Lagrangian system of FNE equations supports compacton solutions.PACS: 47.20.Ky; 42.81.Dp     

谐振管形状对热声发动机内非线性声场的影响

研究了谐振管一端受活塞声源激励,另一端刚性封闭条件下,管道形状对热声发动机谐振管内部非线性声场的影响。基于流体力学基本方程建立了渐变截面谐振管内一维非线性声场的模型,考虑了黏性耗散及非线性效应的影响。利用伽辽金法数值求解了该模型的速度势方程,分析了谐振管形状、活塞振动速度及激励频率对管内声场的影响。将双曲形、指数形、锥形、正弦形等四种变截面谐振管内的非线性声场与圆柱形直管的情况进行了比较。结果反映了谐振管内声场的压力波动受活塞振动速度及谐振管形状的影响;显示了当活塞振动幅度较大时,谐振管内出现的波形畸变、频率曲线偏移、共振频率滞后等非线性现象;揭示了变截面谐振管在抑制管内的高阶谐波及提高压比等方面的优越性。      

Influence of resonator shape on nonlinear acoustic field in a thermoacoustic engine

The influence of the resonator shape on nonlinear acoustic field in a thermoacoustic engine is studied.The resonator of themoacoustic engine is boundary driving by a piston at one end,and the other end of it is rigid closed.A one-dimensional wave equation that accounts for gas dynamic nonlinearities and viscous dissipation in the resonator is established based on the governing equations of viscous hydromechanics.The nonlinear wave equation is solved using approximate Galerkin method.The nonlinear acoustic field in four different types of shaped resonators including hyperbolical,exponential,conical and sinusoidal are obtained and compared with that of a cylindrical resonator.It is found that the amplitude and waveform of the pressure are strongly affected by the resonator shape,the driving amplitude and the oscillation frequency of the piston.Waveform distortion,resonance frequency shift and hysteresis are observed,when the piston oscillation amplitude is large enough.The advantages of shaped resonator for thermoacoustic engine lie in inhibition of higher order harmonics and improvement of pressure ratio,etc.     

Electric impedance and amplitude-frequency response of a one-dimensional ultrasonic liquid-filled resonator with flat piezoelectric plates

The impedance method is used to determine the electric impedance of a resonator. The amplitude-frequency response of a one-dimensional liquid-filled ultrasonic resonator is calculated by directly solving the wave equations and piezoelectric effect equations under the corresponding boundary conditions. An analysis of the amplitude-frequency response shows that the simple analytical expression obtained from the aforementioned solution is in good agreement with experimental data. An anomalous variation of the electric current in the radiating piezoelectric plate versus the excitation frequency is theoretically revealed near the high-Q resonance peaks. This effect is confirmed experimentally. It gives rise to errors in the measured absorption coefficient and multiply broadens the resonance peaks when the measurements are performed near the resonance frequencies of the piezoelectric plates.     

Improving low-frequency sound absorption of micro-perforated panel absorbers by using mechanical impedance plate combined with Helmholtz resonators

The traditional Micro-perforated plate (MPP) is a kind of clean and non-polluting absorption structure in the middle and high frequency and has been widely used in the field of noise control. However, the sound absorption performance is dissatisfied at low frequencies when the air-cavity depth is restricted. In this paper, a mechanical impedance plate (MIP) is introduced into the traditional MPP structure and a Helmholtz resonator is attached to the MIP. Mechanical impedance plate (MIP) provides a good absorption at low frequency by using mechanism of mechanical resonance and the acoustic energy is dissipated in the form of heat with viscoelastic material. Helmholtz resonator can fill in the defect of the poor absorption effect between the Micro-perforated plate (MPP) and the mechanical impedance plate (MIP). The acoustic impedance of the proposed sound absorber is investigated by using acoustic electric analogy method and impedance transfer method. The influence of the tube’s length of Helmholtz resonator and the number of Helmholtz resonator on the sound absorption is studied. The corresponding results are in agreement with the theoretical calculation and prove that the composite structure has the characteristics of improving the low frequency sound absorption property.     

Interaction of electromagnetic waves with nonlinear substance layer under conditions of electron paramagnetic resonance in millimeter waves

Trancated equations have been obtained by the Green's functions method for a slowly varying amplitude of a transverse magnetic field component in a paramagnetic layer under conditions of the electron paramagnetic resonance (EPR). A magnetic susceptibiliti of the substence has been found from the Bloch equation for a homogeneously line breadth of the EPR. In a stationary case a solution of a nonlinear boundary-value problem is redused to a solution of two boundary problems for amplitude and phase equations. It is shown that unstable regimes of the electrodynamic system under inves tigation are possible.Electrodynamic characteristics of a nonlinear resonator of the Fabry-Pero type filled with a saturated paramagnetic medium have been analyzed numerically in a non-stationery case.     

Suppression of Helmholtz resonance using inside acoustic liner

When a Helmholtz resonator is exposed to grazing flow, an unstable shear layer at the opening can cause the occurrence of acoustic resonance under appropriate conditions. In this paper, in order to suppress the flow-induced resonance, the effects of inside acoustic liners placed on the side wall or the bottom of a Helmholtz resonator are investigated. Based on the one-dimensional sound propagation theory, the time domain impedance model of a Helmholtz resonator with inside acoustic liner is derived, and then combined with a discrete vortex model the resonant behavior of the resonator under grazing flow is simulated. Besides, an experiment is conducted to validate the present model, showing significant reduction of the peak sound pressure level achieved by the use of the side-wall liners. And the simulation results match reasonably well with the experimental data. The present results reveal that the inside acoustic liner can not only absorb the resonant sound pressure, but also suppress the fluctuation motion of the shear layer over the opening of the resonator. In all, the impact of the acoustic liners is to dampen the instability of the flow-acoustic coupled system. This demonstrates that it is a convenient and effective method for suppressing Helmholtz resonance by using inside acoustic liner.     

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.     

Mode theory of plane Fabry-Pérot resonator with inhomogeneous active medium

A new method of the eigenvalue problem solution for a resonator with inhomogeneous active medium is given. The approach is based on Maxwell's wave equation and impedance boundary conditions of resonance-type at open resonator ends. A resonator equipped with mirrors in the form of infinite long strips is studied as an example. A rigorous solutions for the cases of stepped and bounded parabolic active medium profiles are obtained. Transcendental eigenvalue equations are investigated, distributions of field amplitude of active resonator modes are found. Asymptotic behavior of rigorous solutions is investigated. A multilayer approximation method is proposed for the eigenvalue problem solution for a resonator with an arbitrary gradient profile of active medium. The testing of this method was carried out with the rigorous solutions for the bounded parabolic profile.     

The theory of planar Bragg resonators

The reflecting properties of one-dimensional planar Bragg gratings are studied. A coupled resonator model for studying the diffraction of electromagnetic waves in an arbitrarily corrugated waveguide is suggested. It is based on exact relationships that follow from the two-dimensional boundary-value problem stated in terms of the Helmholtz equation. The specific relationships for the rectangular corrugation of the grating-forming plates are presented. The reflection coefficients of the Bragg gratings vs. corrugation length and incident radiation frequency are calculated. An analytical solution for the “narrow” corrugation is obtained.     

Wave processes in media with hysteretic nonlinearity: Part 2

Nonlinear processes caused by the propagation of low-frequency and high-frequency acoustic pulses in an unbounded medium and the propagation of continuous waves in a ring resonator are theoretically studied on the basis of two hysteretic equations of state for media with imperfect elasticity. The profiles and parameters of pulses, the resonance curve and the Q factor of the resonator, and the ratio of the nonlinear resonance frequency shift to the nonlinear damping decrement are determined. For nonlinear wave processes in such media, the distinctive features that allow one to choose an appropriate hysteretic equation of state for analytically describing the experimental data are revealed.     

The factorization method in a rough-surface scattering problem: I

In the present paper the wave scattering problem on rough surface is considered for the Helmholtz equation with the Dirichlet boundary condition. An approximate solution is derived with using a factorization approach to the original Helmholtz equation. As a result, the system of two equations of parabolic type appears. The first system equation has an exact analytical solution whereas for the second one, an approximate solution, is considered in terms of perturbation series. It is shown that the obtained approximate solution is the modified classical small perturbation series with respect to small Rayleigh parameter. In Appendix A it is demonstrated that, when the derived perturbation series is converged, it is possible to summarize it and to represent the exact solution of original boundary problem in an analytical symbolical form.     

The factorization method in a rough-surface scattering problem: I

In the present paper the wave scattering problem on rough surface is considered for the Helmholtz equation with the Dirichlet boundary condition. An approximate solution is derived with using a factorization approach to the original Helmholtz equation. As a result, the system of two equations of parabolic type appears. The first system equation has an exact analytical solution whereas for the second one, an approximate solution, is considered in terms of perturbation series. It is shown that the obtained approximate solution is the modified classical small perturbation series with respect to small Rayleigh parameter. In Appendix A it is demonstrated that, when the derived perturbation series is converged, it is possible to summarize it and to represent the exact solution of original boundary problem in an analytical symbolical form.     

水下圆柱形Helmholtz共振器的声学特性分析

理论分析了水下圆柱形Helmholtz共振器的声学特性. 综合考虑壁面弹性和辐射阻抗的影响,基于电-声类比的基本原理,建立了较为完善的水下圆柱形Helmholtz共振器的低频集中参量模型. 利用电路分析的基本方法,得到了系统的输入阻抗和声压传递函数表达式. 仿真分析了主要结构参数对共振器声学特性的影响,得出了一些有意义的结论. 在充水驻波罐中对自制的Helmholtz共振器进行了测量,并对实验结果进行了详细地误差分析. 去除压电水听器对测量结果的影响后,实验与仿真结果基本一致,从而验证了理论分析的正确性. 关键词： Helmholtz共振器 共振频率 传递函数 辐射阻抗     

Tunable bulk acoustic wave resonators with the induced piezoelectric effect in a ferroelectric

An electromechanical model of the piezoelectric effect induced in an acoustic resonator based on a ferroelectric film under the action of a dc or weak ac voltage is developed. The basic equation is obtained by expansion of the free energy in a series with respect to the electric induction and the mechanical deformation. The system of electromechanical equations for variable components of the induction and the mechanical deformation involves all linear terms along with the component of the electrostriction nonlinear with respect to the mechanical deformation. These electromechanical equations made it possible to obtain a one-dimensional approximation for the effective parameters of the material: the piezoelectric modulus and the elastic modulus as a function of the strength of the electric field applied to the acoustic layer. Expressions for the controlled electromechanical coupling coefficient and resonance frequencies of the tunable acoustic resonator are found. It is shown that the most significant parameter responsible for the tuning is the nonlinear electros-triction coefficient M, whose magnitude and sign were evaluated from the available experimental data.     

Acoustic attenuation of hybrid silencers

The acoustic attenuation of a single-pass, perforated concentric silencer filled with continuous strand fibers is investigated first theoretically and experimentally. The study is then extended to a specific type of hybrid silencer that consists of two single-pass perforated filling chambers combined with a Helmholtz resonator. One-dimensional analytical and three-dimensional boundary element methods (BEM) are employed for the predictions of the acoustic attenuation in the absence of mean flow. To account for the wave propagation in absorbing fiber, the complex-valued characteristic impedance and wave number are measured. The perforation impedance facing the fiber is also presented in terms of complex-valued characteristic impedance and wave number. The effects of outer chamber diameter and the fiber density are examined. Comparisons of predictions with the experiments illustrate the need for multi-dimensional analysis at higher frequencies, while the one-dimensional treatment provides a reasonable accuracy at lower frequencies, as expected. The study also shows a significant improvement in the acoustic attenuation of the silencer due to fiber absorption. Multi-dimensional BEM predictions of a hybrid silencer demonstrate that a reactive component such as a Helmholtz resonator can improve transmission loss at low frequencies and a higher duct porosity may be effective at higher frequencies.     

Diffraction of a sound wave by the open end of a flanged waveguide with impedance walls

Diffraction of a plane sound wave by the open end of an impedance-wall waveguide connected to an opening in an impedance screen is considered. The plane wave is incident on the waveguide from a free half-space. Two versions of the problem are considered: for a semi-infinite waveguide and for a finite-length waveguide with a specified bottom impedance; the impedances of the walls, screen, and waveguide bottom can be different. The finite-length waveguide can be treated as an open cavity in the impedance screen. For the cavity of zero length, the problem is reduced to the diffraction by an impedance insert in the impedance screen. The solution in the external region determines the scattered field; the solution in the internal region allows one to determine the directional pattern of an array of receivers located in the cavity. The problem is solved using the integral Helmholtz equation with a specially selected Green’s function that provides the fulfillment of the boundary conditions. Formally, the problem is reduced to an infinite system of algebraic equations. The computational results obtained for bistatic and monostatic scattering patterns are presented.     

Resonant properties of a nonlinear dissipative layer excited by a vibrating boundary: Q-factor and frequency response

Simplified nonlinear evolution equations describing non-steady-state forced vibrations in an acoustic resonator having one closed end and the other end periodically oscillating are derived. An approach based on a nonlinear functional equation is used. The nonlinear Q-factor and the nonlinear frequency response of the resonator are calculated for steady-state oscillations of both inviscid and dissipative media. The general expression for the mean intensity of the acoustic wave in terms of the characteristic value of a Mathieu function is derived. The process of development of a standing wave is described analytically on the base of exact nonlinear solutions for different laws of periodic motion of the wall. For harmonic excitation the wave profiles are described by Mathieu functions, and their mean energy characteristics by the corresponding eigenvalues. The sawtooth-shaped motion of the boundary leads to a similar process of evolution of the profile, but the solution has a very simple form. Some possibilities to enhance the Q-factor of a nonlinear system by suppression of nonlinear energy losses are discussed.