The effects of external lagging on low frequency sound transmission through the walls of rectangular ducts

Previous work by the author [1], on the transmission of internally propagated acoustic noise through the walls of rectangular ducts, is extended here in an investigation of the effects of external “lagging” (consisting of a layer of porous sound-absorbing material, and an impervious external covering) on the duct walls; this type of treatment is commonly applied as a noise control measure. A simple theoretical model, based, as before, on a coupled acoustic/structural wave system, is devised and shown to give reasonably accurate predictions in comparison with measurements of the wall transmission loss (though not in the case of lagging in which an external covering of very non-uniform thickness is incorporated). The conclusion is reached that external lagging used as an acoustic treatment is not, in general, particularly satisfactory.     

Low frequency acoustic transmission through the walls of rectangular ducts

A simple theory is described for the transmission of low frequency sound through the walls of rectangular ducts, particularly those in air conditioning systems. The model is based on a coupled acoustic/structural wave system, and it is assumed that the duct radiates in the same way as a finite-length line source incorporating a single travelling wave. Measurements of wall transmission loss on two types of duct system are compared to theoretical predictions, and good agreement is obtained within the frequency range of validity of the theory. It is concluded that the present approach should give reliable estimates of noise transmission in practical situations.     

Acoustic energy in ducts: Further observations

The transmission of acoustic energy in uniform ducts carrying uniform flow is investigated with the purpose of clarifying two points of interest. The two commonly used definitions of acoustic “nergy” flux are shown to be related by a Legendre transformation of the Lagrangian density exactly as in deriving the Hamiltonian density in mechanics. In the acoustic case the total energy density and the Hamiltonian density are not the same which accounts for two different “energy” fluxes. When the duct has acoustically absorptive walls neither of the two flux expressions gives correct results. A re-evaluation of the basis of derivation of the energy density and energy flux provides forms which yield consistent results for soft walled ducts.     

Sound transmission at low frequencies through the walls of distorted circular ducts

A theoretical treatment of sound transmission through the walls of distorted circular ducts is given, for plane mode transmission within the duct. The transmission mechanism is essentially that of “mode coupling”, whereby higher structural modes in the duct walls are excited, because of the wall distortion, by the internal sound field. The theory is in two parts: an approximate analytical model for the structural response of the walls to the internal sound field, and a structural radiation model. Computed results, based on the theory, are compared to measurements on “long-seam” air conditioning ducts. Where the duct geometry can be reliably specified, reasonably good agreement is obtained between theoretical and experimental data. It is concluded that mode coupling effects serve to account for the discrepancies between ideal and observed behaviour in sound transmission through duct walls.     

Higher order mode acoustic transmission through the walls of rectangular ducts

As an extension of previous work on low frequency fundamental mode acoustic transmission through the walls of rectangular ducts, results are presented here on the transmission of internally propagated higher order acoustic modes through the duct walls. Subject to various assumptions, it is possible to obtain a closed form solution to the structural wave equation governing the motion of the duct's walls, and this is used to predict the response of the walls to the internal pressure field. The resultant acoustic radiation is estimated here by assuming that the duct radiates like a circular cylinder with the same surface velocity distribution. Both experimental and theoretical results are given and agreement between the two is tolerably good.     

Some general properties of the exact acoustic fields in horns and baffles

The propagation of the fundamental, longitudinal acoustic mode in a duct of variable cross-section is considered, and the “Webster” wave equations for the sound pressure and velocity are used to establish some general properties of the exact acoustic fields. The equipartition of kinetic and compression energies is shown (section 2.1) to hold at all stations only for (i) a duct of constant cross-section and (ii) an exponential horn; these are the two cases for which the wave equations for the acoustic velocity and pressure coincide. It is proved (section 2.3) that there are only five duct shapes, forming two dual families, which have constant cut-off frequency(ies): namely, (I) the exponential duct, which is self-dual, and is the only shape with constant (and coincident) cut-offs both for the velocity and pressure; (II) the catenoidal horns, of cross-section S～cosh², sinh², which, with their duals (III) the inverse catenoidal ducts S～sech², csch², have one constant cut-off frequency, respectively, for the acoustic pressure and velocity. The existence of at least one constant cut-off frequency implies that the corresponding wave equation can be transformed into one with constant coefficients, and thus the acoustic fields calculated exactly in terms of elementary (exponential, circular and hyperbolic) functions; this property also applies to the imaginary transformations of the above shapes, viz., the sinusoidal S～sin² and inverse sinusoidal S～csc² ducts, that have no cut-off frequency, i.e., are acoustically “transparent”. It is shown that elementary exact solutions of the Webster equation exist only (section 3.1) for these seven shapes: namely, the exponential, catenoidal, sinusoidal and inverse ducts; it is implied that for all other duct shapes the exact acoustic fields involve special functions, in infinite or finite terms, e.g., Bessel and Hermite functions respectively for power-law and Gaussian horns. Examples of the method of analysis are given by calculating, in elementary form, the exact acoustic fields in inverse catenoidal ducts, for all cases of (a) propagating waves above, (b) non-oscillating modes below and (c) transition fields at the cut-off frequency. The inverse catenoidal ducts consist of (A) the horn of cross-section S～sech², ressembling the “soliton” of non-linear water wave fame, and (B) the baffle of cross-section S～csch², which also matches two exponentially converging ducts, but has infinite, instead of finite, flare at the origin. The geometrical and acoustic properties of these ducts are illustrated by sets of six plots, in Figure 1(a) for the sech-horn and in Figure 1(b) for the csch-baffle; the exact acoustic fields are described by amplitude and phase decompositions of the sound velocity and pressure, plotted as functions of position along the duct, for four frequencies ranging from the cut-off condition to the ray limit (or W.K.B.J. approximation).     

Structural acoustic silencers--design and experiment

The effectiveness of introducing flexible structural layers into air conveying ducts for controlling noise is investigated through theoretical and experimental means, focusing at low frequencies where conventional passive silencing technology is least effective. Previous theoretical work has shown that using flexible rather than rigid walls has the potential to achieve high transmission losses. The physical mechanisms responsible for structural acoustic silencing, including the relation between transmission loss peaks and structural resonance corresponding to different transverse structural modes, are presented. Sensitivity of the performance to acoustic and structural boundary conditions is discussed. To eliminate radiated noise from these walls (breakout noise), a rigid walled cavity is introduced under the flexible plate. The challenge is to find means to reject plane waves in the two-duct system. Designs that overcome these issues and achieve appreciable transmission loss are investigated. Results based on three-dimensional finite element simulations are compared with experimental results.     

The rôle of surface shear stress fluctuations in the generation of boundary layer noise

The influence of unsteady wall shear stress on boundary layer noise and wall pressure fluctuations is discussed. It is argued that in the acoustic analogy theory of boundary layer noise the surface shear stress “dipole” characterizes acoustic propagation and not generation. Analytical results are presented in support of this view which, in addition, indicate that the effect of the surface dipole is to dininish rather than enhance boundary layer radiation at low Mach numbers.     

Approximate asymptotic solutions for acoustic transmission through the walls of rectangular ducts

Approximate expressions—valid at sufficiently high frequencies—are obtained for the acoustic transmission loss of the walls of rectangular ducts. Single mode propagation inside the duct, both in the fundamental mode and in higher order modes, is considered and a multimode model is also proposed. These theories lead to very simple formulae for the transmission loss, which prove to be in tolerably good agreement with measurements.     

The effects of wall discontinuities on the propagation of flexural waves in cylindrical shells

The transmission of flexural type waves through various discontinuities in the walls of cylindrical shells is investigated. Theoretical curves of transmission loss are obtained for different circumferential wavenumbers and wave types, as functions of frequency. Material stiffness and extensional phase speed, together with the relationship between radial vibration amplitude and total wave power of propagation, are important factors which are found to strongly influence wave transmission through discontinuities. Some practical results useful for predicting the performance of typical pipe isolators (in vacuo) are obtained.     

The attenuation of lined plenum chambers in ducts: I. Theoretical models

Two types of theory are described, with the purpose of predicting the acoustic transmission loss of lined plenum chambers (which are sometimes used as attenuators in air conditioning duct systems). The first kind of theory embodies a low frequency wave acoustic approach, and two separate models are evolved: one is for a single plenum chamber, and the second is for a plenum chamber incorporating one or more acoustically lined baffles. The other type of theory is valid at high frequencies, and is based upon geometrical (or “ray”) acoustics. This is applied to a single chamber and to chambers containing either one or two lined baffles. Both the high frequency and low frequency results are reasonably simple. A limited amount of experimental data is also presented, as justification for the validity of part of the theory.     

三通管路管壁声传递损失测试方法

针对汽车进气系统三通管路的特点,提出了多通管路的管壁传递损失测试方法。并以某车型的双涡轮增压发动机进气三通管道为例,采用该方法评价其用塑料代替铝后的声学性能,主要以声传递损失来评价涡轮增压器噪声通过三通连接管路管壁的辐射和透射特性。测试过程中,三通管道的两个连接涡轮增压器端口分别用声源两次发声,靠近进气歧管端口采用两种不同反射末端,然后在每段管路布置两个压力场扬声器进行测试,并基于平面波分离入射波和反射波,同时在三通管道外用声功率半球面十点分布法自由场扬声器测试,经过3次测量来计算管道管壁的声传递损失。由于声传递损失是管道本身特性决定,所以该测试方法能够准确找出塑料件和金属件在不同频率的声学特性差异。而后,在声传递损失测试结果的基础上,结合近场声全息方法和波束形成原理进行声源识别,可知该三通管路材质改为塑料后主要噪声来自焊缝薄弱处的中高频透射声和管壁结构的低频辐射声。     

Active noise control in finite length ducts

A simple technique for the active control of sound in ducts, initially suggested by Olson and May [1], is investigated in detail. A simple, “virtual earth” principle, feedback loop is used to drive the sound pressure to a minimum at a microphone placed close to a loudspeaker in the duct wall. This produces a reflection of downstream travelling plane waves. A detailed investigation of the loudspeaker near field has enabled the optimum position of the microphone to be identified. The system is shown to be especially effective at the frequencies of the longitudinal duct resonances, where the acoustic response of the duct produces a high loop gain. Results are presented which show a reduction of up to 20 dB in the amplitude of low frequency broadband noise at a position downstream of the cancelling source.     

Transmission of sound through pipe walls in the presence of flow

The transmission of sound through pipe walls was studied experimentally under no-flow conditions as well as with steady air flow velocities up to 120 m/s. The test specimens were commercial pipe and tube of diameter ranging from 0·07 to 0·3 m, and thickness-to-diameter ratios from 0·12 to 0·2. The technique involved two reverberant rooms, one traversed by the test pipe to measure externally radiated sound, and one in which the test pipe terminated to measure internally propagated sound. Vibration of the pipe wall was also monitored to determine radiation efficiency.The results show that no-flow transmission loss (TL) is higher than predicted by available theoretical expressions, but that TL decreases strongly with increasing flow velocity. A qualitative explanation for the latter is offered. Radiation efficiency was found to be independent of flow velocity. The scaling of results between “similar” specimens was moderately successful. The results are documented in sufficient detail to permit their use for forming empirical models as well as for testing future theoretical predictions.     

膨胀腔类超材料的传递损失研究*

声学超材料及结构已被广泛研究,其超结构通常利用3D打印技术实现,当结构刚度较小或者面积较大时,由声固耦合所导致的声学效果与设计不符的情况广泛存在。本文针对含有膨胀腔类的超材料,研究了声固耦合对其声学性能的影响,采用有限元计算结合阻抗管实验的方法,得到其传递损失,分析了声固耦合现象对传递损失的影响。结果表明：薄壁膨胀腔结构的作用频率范围与只考虑声学理论计算的设计不符,声固耦合现象对传递损失产生显著影响;采用增加膨胀腔壁厚、减少膨胀腔内径或选择金属材料的方式,都可以使得声固耦合现象对传递损失的影响减小;仿真结果与实验结果基本吻合。该研究结果说明：对于膨胀腔类超材料,当刚度较小或者面积较大时,对其进行声固耦合分析是完全必要的。     

Comparison of measured broadband noise attenuation spectra from circular flow ducts and from lined engine intakes with predictions

Sound propagation in lined circular ducts is investigated in the presence of uniform and sheared flow. The modal solutions are obtained by solving an eigenvalue equation which, in the case of sheared flow, is derived by using finite differences and by matching the pressure and the radial component of the particle velocity at the interface of the regions of uniform and sheared flow. For the uniform flow region, standard Bessel function solutions are used. The attenuation of acoustic energy at a given frequency and for a given liner length is computed on the assumption that at the inlet to the lined duct, the acoustic energy is equally distributed among the propagating modes. The total number of propagating modes is determined from the hard wall “cut off” condition. The failure to find some of the modal solutions on the attenuation computed in this way is discussed. It is shown that the reliability of this method of computing liner attenuation depends on the ability to successfully compute most of the modal solutions over a large range of frequencies, flow conditions and duct wall impedance values. A numerical technique is developed which uses a fraction of the total number of solutions to compute the total attenuations without appreciable loss of accuracy. Measured attenuation spectra from a flow duct facility and from lined intake ducts of the RB.211 engine are compared with predictions. In general very good agreement between predictions and measurements is obtained.     

Calculation of the thicknesses and elastic properties of thin-film coatings using atomic-force acoustic microscopy data

The feasibility of applying atomic-force acoustic microscopy to measure the elastic properties of thin-film coatings and their thickness in the range from several to several hundreds of nanometers is studied. In practice, this technique can be used to study diamond-like coatings. The key point of our method is application of “flat” tips, which provide a constant tip-surface contact area and, hence, a constant contact stiffness. The reason for using such tips is that experimental data for thin-film structures gained with standard (rounded) tips cannot be given an adequate quantitative interpretation. Numerical simulation is used to evaluate the thickness and indentation modulus of a coating from contact stiffness k _cont measured by atomic-force acoustic microscopy.     

声学超材料对低频噪声的消声特性

针对低频噪声较难消除的问题,设计了亥姆霍兹共振腔与声学超材料薄膜耦合的消声结构,在利用有限元软件进行屈曲分析薄膜的临界状态得知声学超材料薄膜结构临界失稳力为0.087 N·m,利用COMSOL声固耦合模块研究薄膜形态对传递损失峰值频率的影响。结果表明:薄膜扭转角度由0°增加到30°时,薄膜总体刚度增加,传递损失峰值对应频率向右偏移了30 Hz,变化并不明显。为了扩大频率偏移范围,在扭转30°的基础上,对扭矩棒施加垂直向下的压力,压力由0 kPa增加到2 kPa,薄膜预应力增大,系统刚度增加,使得传递损失峰值向右偏移了170 Hz。最后搭建实验平台,验证了薄膜在扭转时的频率偏移与仿真基本吻合,在不同压力时频率偏移一致,进而可以实现较大范围的低频率噪声控制。为声学超材料的设计和控制提供有效的依据。     

Development of a 3D finite element acoustic model to predict the sound reduction index of stud based double-leaf walls

Building standards incorporating quantitative acoustical criteria to ensure adequate sound insulation are now being implemented. Engineers are making great efforts to design acoustically efficient double-wall structures. Accordingly, efficient simulation models to predict the acoustic insulation of double-leaf wall structures are needed. This paper presents the development of a numerical tool that can predict the frequency dependent sound reduction index R of stud based double-leaf walls at one-third-octave band frequency range. A fully vibro-acoustic 3D model consisting of two rooms partitioned using a double-leaf wall, considering the structure and acoustic fluid coupling incorporating the existing fluid and structural solvers are presented. The validity of the finite element (FE) model is assessed by comparison with experimental test results carried out in a certified laboratory. Accurate representation of the structural damping matrix to effectively predict the R values are studied. The possibilities of minimising the simulation time using a frequency dependent mesh model was also investigated. The FEA model presented in this work is capable of predicting the weighted sound reduction index R_w along with A-weighted pink noise C and A-weighted urban noise C_tr within an error of 1 dB. The model developed can also be used to analyse the acoustically induced frequency dependent geometrical behaviour of the double-leaf wall components to optimise them for best acoustic performance. The FE modelling procedure reported in this paper can be extended to other building components undergoing fluid–structure interaction (FSI) to evaluate their acoustic insulation.     

Development of an optimised, standard-compliant procedure to calculate sound transmission loss: design of transmission rooms

A numerical procedure to estimate the transmission loss of sound insulating structures is proposed based upon the technology of acoustic measurements and standards. A virtual laboratory (VL), namely, a numerical representation of a real laboratory consisting of two reverberation rooms meeting certain sound field quality criteria is designed. VL is to be used for the numerical simulation of standardised measurements under predefined, controlled, acoustic conditions. In this paper, the design and optimisation of VL is investigated. The geometry of the transmission rooms is designed following first principles, in order for diffuse field conditions and sufficiently smooth primary mode distribution in the low frequency to be achieved. A finite element-based optimisation procedure, introduced by the author in previous work, is extended to arbitrarily shaped rooms. It is used to predict the appropriate local geometric modifications so as for improved mode distribution and smoother sound pressure fluctuations of the transmission rooms in the low-frequency range to be achieved and low-frequency measurement reproducibility and accuracy to be increased. Steady-state acoustic response analysis is performed in order to quantify the acoustic field quality of the virtual transmission rooms in the frequency range of measurements. A method to calculate the total absorption, A, of the receiving room is introduced by simulation of the reverberation time measurement procedure using Transient acoustic response analysis. The acoustic performance of VL is overall considered and is shown to meet in a sufficient degree, relative laboratory measurement standards in the frequency range of 100÷704 Hz.