Noise reduction with coupled prismatic tubes

Transcription

1 Noise reduction with coupled prismatic tubes Frits van der Eerden

2 This research was supported by the Dutch Technology Foundation (STW). Project TWT De promotiecommissie is als volgt samengesteld: Voorzitter en secretaris: Prof.dr.ir. H.J. Grootenboer Promotor: Prof.dr.ir. H. Tijdeman Leden: Dr.ir. W.M. Beltman Prof.dr.ir. W.F. Druyvesteyn Dr.ir. A. Hirschberg Prof.dr.ir. H.W.M. Hoeijmakers Prof.dr.ir. J.W. Verheij Universiteit Twente Universiteit Twente Intel, USA Universiteit Twente Technische Universiteit Eindhoven Universiteit Twente Technische Universiteit Eindhoven/TNO-TPD Paranimfen: ir. T.G.H. (Tom) Basten ir. M.E. (Marten) Toxopeus Eerden, van der, Fredericus Joseph Marie Title: Noise reduction with coupled prismatic tubes PhD thesis, University of Twente, Enschede, The Netherlands November ISBN: Subject headings: acoustics, sound absorption, resonators Copyright by F.J.M. van der Eerden, Enschede, the Netherlands Printed by Ponsen & Looijen bv., Wageningen

3 NOISE REDUCTION WITH COUPLED PRISMATIC TUBES PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus, prof.dr. F.A. van Vught, volgens besluit van het College voor Promoties in het openbaar te verdedigen op vrijdag 4 november te 6.45 uur. door Fredericus Joseph Marie van der Eerden geboren op 3 december 97 te Tubbergen

4 Dit proefschrift is goedgekeurd door de promotor: Prof.dr.ir. H. Tijdeman

5 Summary With so little time available nowadays for design and development, there is a strong incentive to simulate the behaviour of designs in virtual reality, rather than to perform expensive and time-consuming measurements on expensive prototypes. For noise reduction, which has become an important topic in acoustics, the same trend can be observed. To predict the sound level in an open space or cavities, important characteristics such as the effect of the flexibility of panels and the effectiveness of sound absorbing materials must be known accurately. The present investigation focuses on an accurate description of sound absorption. Within this research a new technique to create sound absorption for a predefined frequency band has been developed. Additionally, a simple and efficient numerical model for conventional sound absorbing materials, such as glass wool or foams, has been formulated. It is also demonstrated that the newly gained insights are useful in applications not directly related to sound absorption. As a basis for the research a description of pressure waves in a single narrow tube or pore has been used. In such a tube the viscosity and the thermal conductivity of the air, or any other fluid, can have a significant effect on the wave propagation. This so-called viscothermal wave propagation results in energy being dissipated and the effective speed of sound inside the tube can be considerably reduced. This principle of energy dissipation has been applied to configurations consisting of a manifold of tubes, the so-called coupled tubes. A design strategy was developed to create broadband sound absorption for a wall with configurations of coupled tubes. These broadband resonators can be optimally designed so that they absorb incident waves for a predefined frequency range. Experiments in an impedance tube, or Kundt s tube, proved that both the viscothermal effects on the wave propagation and the design tool are very accurate.

6 On a micro-scale it has been demonstrated that a network of coupled tubes can be used to represent conventional sound absorbing material. The network description is simple and efficient compared to existing descriptions for sound absorbing materials such as empirical impedance descriptions and the Limp and Biot theory. Further successful applications of the coupled tubes model are: an improvement of an inkjet array, a newly developed test set-up for a voice producing element, increased damping of viscothermally damped flexible plates, and a design strategy for optimal reflection in ducts with side-resonators. The resonators of the latter application cause propagating noise in a duct to be reflected, not to be absorbed, so that the sound level beyond the position of the resonators is reduced. As an alternative to the well-known impedance tube technique with two microphones, the use of a particle velocity sensor, the microflown, has been investigated experimentally in the course of the project. With the use of a combination of a microphone and a microflown direct information on the acoustic impedance, the sound intensity and the sound energy density was obtained.

7 Contents Summary. Introduction. Background.... Aim of the investigation Outline Optimised sound absorption with coupled tubes 9. Introduction Viscothermal wave propagation in prismatic tubes....3 Coupled tubes Recursive formulation Transfer matrix formulation Broadband sound absorption Sound absorption of a single tube Sound absorption of coupled tubes Conclusions Impedance tube techniques to measure sound absorption Introduction Impedance tube techniques The p method The u method The p/u method The p u method Experimental results Comparison of the p and the u methods Comparison of the p, the p/u, and the p u methods Conclusions Experimental verification of the coupled tubes model Introduction Sound absorption of single tube resonators... 9

8 4.3 Sound absorption of coupled tubes resonators Conclusions Sound absorbing material represented by a network of tubes 5. Introduction A random network of tubes Comparison to an empirical impedance model Comparison to the Limp theory Comparison to the Biot theory Conclusions and remarks Further applications of the coupled tubes model 7 6. An inkjet array A test set-up for a voice producing element A viscothermally damped flexible plate Reflection of sound in ducts with side-resonators Conclusions 73 List of Symbols 75 References 79 A. Sound absorption mechanism of a wall with resonators 87 B. Propagation coefficient for different cross-sections 97 C. Calibration of the microflown D. The Limp theory for fibrous sound absorbing materials 3 E. Derivation of Biot s equations of motion 7 Samenvatting (in Dutch) Nawoord 3 Levensloop 5

9 Chapter Introduction. Background. Aim of the investigation 4.3 Outline 5. Background A typical example of an experience of noise could be the sound of a truck passing by while enjoying a quiet sleep early in the morning. The nuisance caused by noise is seen in situations at home as well as at work. One can think of noisy domestic appliances, such as vacuum cleaners and washing machines, traffic noise caused by cars or trains, and industrial noise caused by machines. To solve noise problems the noise radiated by the source must be either reduced, shielded or insulated. These strategies require a thorough understanding of the complete noise problem. For example important noise paths or vibration levels of panels need to be known. A well-known method for the insulation of noise is the use of sound absorption techniques or sound absorbing materials. The acoustic behaviour of a wall with sound absorbing resonators, and to be more specific, a numerical technique to predict its behaviour, is the topic of the present investigation. A sound field Sound can be described as a sensation produced at the ear by very small pressure fluctuations in the ear (Bies 996). Pressure and pressure fluctuations can be measured relatively easily in engineering. The unit for pressure is the pascal ( Pa = N/m ). It is known that the ear responds approximately logarithmically to sound energy input which is proportional to the square of the sound pressure. The

10 ear can detect pressure fluctuations ranging from µpa, for young persons, to 6 Pa before pain is experienced. Note that the mean pressure of the atmosphere is about. 5 Pa. The high dynamic range for the square of the pressure is compressed into logarithmic units for convenience termed decibels or db. The dynamic range thus varies from to 3 db. Sound pressure is usually measured in db s with a sound level meter. A sound field can be described as small perturbations of steady state variables which describe a medium through which the sound is transmitted. For a fluid such as air or water the variables are: the pressure [Pa], the particle velocity [m/s], the temperature [K] and the density [kg/m 3 ]. The total value of these variables can be seen as a small perturbation superimposed on the mean value (the steady state). The pressure, the temperature and the density are scalar quantities whereas the particle velocity is a vector. The word particle is used here for a very small part of the medium and not for the molecules of the medium itself. The speed with which a disturbance propagates through air or water is called the speed of sound. The disturbance may be described as the sum of small harmonic perturbations. So in general a sound wave contains several frequencies. Figure. A foam sample (left) and a glass wool sample (right) with a close-up above.

11 3 The influence of the viscosity of air is normally very low and shear forces hardly effect the propagation of sound waves in the open air. However, the viscosity of air becomes much more important when the air is trapped in sound absorbing materials. Due to the viscosity, sound energy is dissipated as heat inside the material. In fact the energy density of sound is small, even for sound levels of about 3 db, so the temperature rise in the material is negligible. Sound absorbing materials A close-up of foam and glass wool is shown in Figure.. These materials are used for their good sound absorbing behaviour, although glass wool was originally used for thermal insulation. A typical graph which depicts the sound absorption coefficient α of foam and glass wool as a function of the frequency is given in Figure.. A value of α =. indicates that incident waves are completely absorbed whereas α =. means complete reflection of sound waves..8 α [ ].6.4. Glass wool (5 mm thick) Glass wool (5 mm thick) Foam (5 mm thick) Center frequency of octave band [Hz] Figure. Absorption coefficient as a function of the frequency. Data obtained from literature (Bies 996). Indication of the density: wool 6 kg/m 3, foam 7 kg/m 3. The absorption coefficient of materials such as fibreglass, rockwool or polyurethane foam is usually obtained empirically. However, it would be more efficient if the effectiveness of sound absorbing materials could be predicted. Therefore it is not surprising that a variety of empirical and analytical models have

12 4 been developed to predict the sound absorbing qualities of materials (see for instance: Allard, Attenborough, Biot, Bolton, Cummings, Delaney, Ingard, Lauriks). Moreover, such methods offer the opportunity to improve the sound absorbing qualities or to design new ones via numerical simulations instead of performing extensive series of experiments. For the present investigation the acoustic behaviour of air inside a single pore or narrow tube will be taken as a basis for studying sound absorption. In Figure. small cavities and pores in the foam can be seen. This explains the assumption that sound absorption is created inside a set of small tubes (see also Biot). Zwikker and Kosten (949) presented a model for wave propagation in cylindrical tubes which included the effects of viscosity and thermal conductivity of the medium. This model for the so-called viscothermal wave propagation proved to be complete and accurate for both narrow and wide tubes (Tijdeman 975). It is noted that in the present investigation the effect of a mean flow through the tubes or ducts is not taken into account. However, for low flow velocities the wave numbers of the forward and backward travelling waves can easily incorporate a small mean flow component. It is also noted that the use of sound absorbing material can be characterised as a passive method, as opposed to active methods. An alternative technique for noise reduction is active noise control (ANC). This technique uses active elements in a room or a structure. Examples are loudspeakers which generate anti-noise or actuators such as piezo-materials which are attached to or imbedded in a structure to reduce radiated or transmitted noise.. Aim of the investigation Following the considerations of the previous section the aim of the investigation is formulated as follows: Develop and validate a numerical tool for absorptive structures, based on the viscothermal wave propagation in narrow tubes, so that the sound reduction can be calculated accurately and efficiently.

13 5 In order to reach this aim a number of basic aspects and questions need to be investigated: What is the sound absorbing behaviour of a single pore or narrow tube and what are the effects if such a tube is scaled to a larger size for more convenient acoustic experiments. What is the acoustic effect when tubes are connected or coupled to each other. Can a detailed network of coupled tubes be used to represent sound absorbing materials. Are there other type of acoustic problems which can benefit from the developed numerical tool..3 Outline First, viscothermal wave propagation in prismatic tubes is described according to the so-called low reduced frequency model. An important parameter in this model is the non-dimensional shear wave number which is a measure for the ratio of the inertial and viscous forces. It can be seen as an acoustic Reynolds number. The prismatic tubes are coupled via a mass and momentum balance. Two formulations for coupled tubes are presented: a recursive formulation and a transfer matrix formulation. For a single tube the model predicts the acoustic impedance, and therefore the sound absorption coefficient, at the entrance of the tube. When the other end of the tube is acoustically hard a quarter-wave resonator is obtained. It is demonstrated that a wall with a uniform distribution of such resonators can absorb incident sound waves completely for a small frequency band. For this specific frequency band the viscothermal losses in the resonator equal the incident energy. Next, resonators have been constructed with axially coupled tubes. The geometry of the coupled tubes was designed in order to obtain sound absorption for a broader

14 6 frequency band. The design tool for an optimal sound absorbing wall for a predefined frequency range is described in more detail in Chapter. To validate the design tool measurements need to be performed. A relatively simple and efficient method is the impedance tube as described in Chapter 3. It is shown how three alternative techniques make use of a new particle velocity sensor. The combination of a particle velocity sensor and a microphone provides direct information on the acoustic impedance, the sound energy flow and the sound energy density. The actual validation of single tube resonators and coupled tubes resonators is described in Chapter 4. A very good agreement between the experimental and predicted results is observed. For example the effect of the number of resonators per unit area has been investigated. The experiments are performed for frequencies up to 4 Hz. A random network of small narrow tubes is used in Chapter 5 to model the acoustic behaviour of fibrous and porous materials. With a relatively simple and compact set of parameters different materials can be characterised. A comparison with empirical and analytical models validates the network approach. The numerical tool for coupled tubes is successfully used in four applications, as described in Chapter 6. In two cases sound absorption is not the issue but merely the technique to predict the acoustic behaviour of coupled tubes. One case deals with noise reduction in duct-systems such as air-conditioning systems. A design strategy for optimal reflection of sound in ducts with side-resonators is described. The side-resonators cause propagating noise in a duct to be reflected, not to be absorbed, so that the sound level downstream of the position of the resonators is reduced. The other applications are: A test set-up consisting of a number of coupled tubes has been developed to measure the acoustic behaviour of a voice producing element. In an inkjet array constantly propagating waves are created in a short channel with ink because at one end of the channel complete sound absorption is realised with a broadband resonator.

15 7 The vibrations of a flexible plate have been reduced by withdrawing extra energy from a trapped air-layer via resonators. The flexible plate and the airlayer are strongly coupled. Finally the conclusions of the present thesis are presented in Chapter 7.

16 8

17 Chapter Optimised sound absorption with coupled tubes. Introduction 9. Viscothermal wave propagation in prismatic tubes.3 Coupled tubes 4.3. Recursive formulation 4.3. Transfer matrix formulation 9.4 Broadband sound absorption Sound absorption of a single tube Sound absorption of coupled tubes 46.5 Conclusions 55. Introduction Two classes of sound absorbing structures can be distinguished in general: porous materials and resonance absorbers (Heckl 995). The material of the first class has a micro-structure of pores or fibres. Well-known examples are glass wool and synthetic foam. These materials show a broadband sound absorbing behaviour above a certain frequency. When the material is backed by an acoustically hard wall one can use the rule of thumb that acoustic waves are well absorbed for frequencies for which a quarter of the corresponding wavelength is smaller than the thickness. This means that for lower frequencies the thickness of the applied porous material has to be increased. Special porous materials that consist of ceramics or metal have been developed for applications where conventional materials cannot be used (Koketsu 98, Banhart 994). The sound absorbing performance is less than that of conventional porous materials but their strength and use in high temperature or aggressive environments can be advantageous. The prediction of the sound absorbing behaviour is not easy for porous materials. When soft materials are applied, for example, the uncertainty concerning the actual thickness or porosity can be a cause for large variations in the sound absorbing characteristics. Furthermore, a range of parameters, as given for example by the

18 theory of Biot, have to be measured for different operational conditions. It may also be difficult to interpret the significance of each parameter with respect to the acoustic behaviour. In the second class of sound absorbing structures the resonance of air or the resonance of panel-like structures is used. As an example a perforated panel and a panel absorber are illustrated in Figure.. porous liner m k s V Perforated panel absorbers Figure. Perforated panel absorbers and a panel absorber. Panel absorber A perforated panel can be seen as a row of Helmholtz resonators. A Helmholtz resonator basically can be seen as a mass-spring system with a resonance frequency of ω = π f = k s / m. The spring stiffness k s is represented by a volume V of air and the mass m is the mass of a small column of vibrating air in a perforation of the panel. The vibrating air dissipates sound energy. Extra sound energy can be dissipated by placing a liner of porous material in the volume. Perforated panels are used for the absorption of a small band of low and medium frequency sound. The panels use a relatively small volume. There is a considerable number of publications on sound absorbing materials with perforated and microperforated facings. For details reference is made to for instance: Ingard 95, Heckl 995, Kang 999. The same remark applies for panel absorbers. These are used for low frequency sound and also show a mass-spring resonance behaviour. A thin panel or foil is backed by a narrow air layer that acts as a spring. The mass is represented by the mass of the resonating panel. Extra porous material is used to create sufficient

19 energy dissipation. Panel absorbers are generally used to absorb particular low frequency noise. Besides Helmholtz resonators also so-called quarter-wavelength resonators can be used. A wall or section with a number of these tube-like resonators is capable of absorbing sound of a specific frequency. Quarter-wavelength resonators, or with a shorter notation: quarter-wave resonators, are applied in general for low frequency sound. In a quarter-wave resonator the mass and spring functionality is continuously distributed in the tube. As a results also higher order modes can be observed. For a Helmholtz resonator a volume with a uniform pressure distribution is required whereas a slender tube can be used for a quarter-wave resonator, which may be an advantage for specific applications. In this chapter it will be shown that the effect of these type of tube resonators can be considerably improved. The small frequency band for which the quarter-wave resonators absorb sound can be widened considerably by coupling tubes. A simple and well-defined sound absorbing wall with an accompanying efficient and accurate model is presented including a strategy to design such a wall. To be more specific: the coupled tubes inside the wall are designed in such a way that at the surface an optimal acoustic boundary condition is created. For a predefined low, medium or high frequency range the optimal configuration can be calculated. An advantage of this wall is that a high sound absorption can be realised for a wide frequency band. Furthermore, the resonators can be constructed in materials that withstand high temperatures or aggressive environments. The mechanism for a broadband sound absorption is the dissipation of sound energy and the cancelling of the incident acoustic waves due to a broadband resonance of air, or any fluid, in the coupled resonators (see Appendix A). A single tube resonator shows a significant absorption for a specific first resonance frequency (and in general less noticeable for number of higher harmonics). Wave propagation in a single tube or resonator forms the basis for the acoustic model. It is shown that the viscous and thermal effects play an important role in the wave propagation. As a next step a system of coupled tubes is described. Two techniques, one analytical and one numerical, are developed to calculate the acoustic behaviour of a system or network of coupled tubes and ducts. The last section deals with broadband sound absorption of coupled tubes. The effect of the different parameters on the sound absorption is shown and an efficient technique to

20 compute the optimal configuration with coupled tubes for a given frequency band is presented.. Viscothermal wave propagation in prismatic tubes Sound propagation in prismatic tubes has been investigated thoroughly by many authors. A number of analytical models was presented on the basis of the following equations: the linearised Navier-Stokes equations, the equation of continuity, the equation of state for an ideal gas and the energy equation. For an overview see Tijdeman (975) and Beltman (999a). In the following sections three analytical solutions for wave propagation in prismatic tubes are presented: the Helmholtz equation which neglects viscothermal effects, the low reduced frequency solution which includes viscous and thermal effects, and the Kirchhoff approximation which is a first order approximation of the low reduced frequency solution. Next, boundary conditions of a prismatic tube and reflection and absorption coefficients in the tube are presented. Helmholtz equation For the most simple situation, for example the propagation of plane waves in free space without effects of steady state temperature differences, a main flow (such as the wind) or attenuation due to for example atmospheric absorption, the resulting wave equation for ~ p ( x, y, z, t) reads: ~ ~ ~ p p p ~ p + + = x y z c t (.) In equation (.) p is the pressure perturbation with respect to the mean pressure p in the air. It is a function of time t and position, where x, y and z form the rectangular Cartesian co-ordinate system. The coefficient c is the speed of sound i t in the quiescent space. For the special case of a harmonic time dependence e + ω, i.e. ~ p( x, y, z, t) = p( x, y, z) e iω t, the equation for the pressure perturbation reads: p p p k x y z p = (.)

21 3 where k is the wave number defined as ω /c with ω the angular frequency. Equation (.) is the so-called Helmholtz equation. In the present investigation the pressure perturbation is assumed to be harmonic for convenience. For a long prismatic tube with rigid walls (see Figure.) equation (.) reduces to the onedimensional Helmholtz equation which has the solution: p ik x ik x ( x) = pˆ A e + pˆ B e (.3) The sound field of (.3) consists of a plane wave with a complex amplitude pˆ B travelling in the positive x-direction with speed c and a plane wave with amplitude pˆ travelling in the negative x-direction also with speed c. A x pˆ B ll pˆ A Figure. A prismatic tube. The amplitudes are determined by the boundary conditions on both ends of the tube. The travelling waves are assumed to be plane for frequencies lower than the cut-off frequency of the tube. For frequencies higher than the cut-off frequency the acoustic wavelength is smaller than about half the characteristic dimension of the tube cross-section (for a circular cross-section the cut-off frequency is determined by f = c /.7d, with d the diameter of the tube). With the use of the linearised momentum equation (Newton s second law and ~ p ( x ) a function of location and of time via i t e + ω ), ~ p x = ρ u t (.4) and omitting the time dependence obtained: i t e ω + the particle velocity distribution is

22 4 ik x ik x ( pˆ e p e ) u( x) = A ˆ c ρ B (.5) In (.5) ρ is the undisturbed or mean density. The quantity ρ c is termed the characteristic specific acoustic impedance of the fluid and represents the impedance of a freely travelling plane wave. The acoustic impedance is the ratio of the pressure perturbation and the velocity perturbation (in the case of a porous material the velocity is usually directed into the surface and out of the fluid). The term impedance stems from the field of electrical engineering (by O. Heaviside in 886) and was later introduced in mechanics where the ratio of the force and the velocity is referred to as impedance (i.e. something impeding motion ). In 94 A.G. Webster introduced impedance in acoustics. A sudden change in the acoustic impedance causes the reflection of acoustic waves. This property of acoustic wave propagation will be used in section.4. Low reduced frequency model In small tubes or layers the wave propagation is affected by the viscosity and the thermal conductivity of the fluid. There is a variety of literature on viscothermal wave propagation of which Tijdeman (975) and Beltman (999) give an extensive overview of the different analytical solutions. Of the many approaches the one of Zwikker and Kosten (949) has proven to be an efficient and accurate appraoch for a large number of acoustical problems. They assumed a constant pressure across the tube cross-section and included the effects of inertia, compressibility, viscosity and thermal conductivity of the fluid. Their solution for the wave propagation in cylindrical tubes, the low reduced frequency model is written in a dimensionless form by Tijdeman and is characterised by four dimensionless parameters: ρ ω s = l, the shear wave number (.6) µ k r ω = l, the reduced frequency (.7) c C p µ σ =, the square root of the Prandtl number (.8) λ

23 5 C p γ =, the ratio of the specific heats (.9) CV The shear wave number s is a measure for the ratio between the inertial and viscous forces and can be seen as an unsteady or acoustic (square root of the) Reynolds number. It is a function of, amongst others, the dynamic viscosity µ and the characteristic length of the cross-section l, i.e. for a layer this is half the layer thickness h/ and for a tube with a circular cross-section this is the radius R. For small values of s the viscous effects are dominant. In that case a tube is called narrow. For s», when inertia effects are dominant, the tube is called wide. The reduced frequency k r represents the ratio of the characteristic length of the cross-section and the acoustic wavelength. It is noted that the viscous effects become less important for a wider tube: a times wider tube, R, results in times lower frequencies, k r /, so k r R remains constant whereas the resulting shear wave number s increases a factor For a given ideal fluid σ and γ can be considered as constants; in the case of air the Prandtl number equals about.7 (so σ.84) and γ.4. In (.8) and (.9) C p and C V are the specific heats at constant pressure and volume, respectively, and λ is the thermal conductivity. Tijdeman and Beltman show that the different models for viscothermal wave propagation in tubes and layers, respectively, can be put into perspective by using the dimensionless parameters k r and s. The range of validity of the models is governed by the range of k r and k r / s. Furthermore Beltman demonstrates that for most acoustic problems the low reduced frequency model is sufficient. Compared to more complex models, such as the simplified Navier-Stokes model, the low reduced frequency model is a relatively simple one. The reduced frequency model is derived from the basic equations for the propagation of sound waves in absence of mean flow. These are: the linearised Navier-Stokes equations, the equation of continuity, the equation of state for an ideal gas and the energy equation. In general they can be written as: u 4 ρ = p + µ + η µ t 3 ( u) ( u ) (.) ρ ρ t ( u ) + = (.)

24 6 p = ρ RT (.) T p ρ C p = λ T + (.3) t t where the over-bar indicates the total quantity, i.e. the mean value plus the small perturbation and a bolt symbol represents a vector. R is the gas constant (R = C p C V ), T is the temperature and and are the gradient and Laplace operators, respectively. For monatomic gases applies the bulk viscosity η =, for air η =.64µ. Furthermore the following assumptions are used: no mean flow; no internal heat generation; a homogeneous medium; small harmonic perturbations; laminar flow. For the low reduced frequency model two additional assumptions are introduced: the acoustic wavelength is large compared to the length scale l, i.e. k r «; the acoustic wavelength is large compared to the boundary layer thickness, i.e. k r / s «. The direction in which the waves propagate is separated from the other directions. For a tube this is the axial direction, i.e. for a cylindrical tube, for example, u is the velocity perturbation in the axial x-direction and v the velocity perturbation in the radial y-direction. The solution for the acoustic variables p, u, v, T and ρ can now be found but in our case the solutions for the variables p and u are sufficient. The resulting complex pressure and velocity perturbations are given as: p Γ k x Γ k x ( x) = pˆ A e + pˆ B e (.4) k x k x ( pˆ Γ Γ e p e ) G u( x) = A ˆ ρ c B (.5) These expressions are quite similar to the ones derived for the Helmholtz equation. The coefficient G is explained later. The main difference is the viscothermal wave propagation coefficient Γ. It is a complex quantity, Γ = Re(Γ ) + i Im(Γ ), where c / Im(Γ ) represents the phase velocity and Re(Γ ) accounts for the attenuation of a propagating wave. It is noticed that Γ is a function of the shear wave number s and therefore a function of the frequency. The velocity u(x) is averaged over the cross-

25 7 section so that a one-dimensional model arises. The following boundary conditions are applied: at the tube wall the axial and normal velocity is equal to zero as well as the temperature variation (isothermal wall); the velocity at the tube axis perpendicular to this axis is zero (symmetry condition). The effect of the shear wave number on the velocity profile for a circular tube can be seen in Figure.3 (Tijdeman 975). For low values of s the viscous effects are dominant and the velocity profile becomes parabolic and approaches the Poiseuille flow. For large values of s the inertial effects are dominant and for the velocity perturbation an almost plane wave front results in the tube. It is noted that the expression for the velocity is complex (see Appendix B) which indicates that not all points pass their equilibrium position at the same time. s = 5 s = Scaled radius [ ] s = s = s =.5. Scaled velocity [ ] Figure.3 Velocity profile (magnitude) in a circular tube. The wave propagation coefficient Γ depends on the geometry of the cross-section of the tube or layer. Stinson (99) formulated a general expression for the propagation coefficient. For a circular, a rectangular and a triangular cross-section Γ is given in Appendix B. Also the coefficient G, which depends on the type of the cross-section, in the expression for the average velocity is presented there. For a cylindrical tube these parameters are:

26 8 J( i i s) γ Γ = (.6) J ( i i s) n i γ G = (.7) Γ n J and J are Bessel functions of the first kind of order and, respectively. The coefficient n can be interpreted as a type of polytropic coefficient which relates the pressure and density, integrated over the cross-section, according to: p n ρ = constant (.8) n is a complex number and is a function of sσ: ( ) γ J i i sσ n = + (.9) γ J ( i i sσ ) For small values of sσ the polytropic coefficient n reduces to one, i.e. isothermal wave propagation, and for large values of sσ the polytropic coefficient n becomes equal to γ which corresponds to adiabatic wave propagation (see Figure.4) n [ ]. arg(n) [ o ] 6 4. sσ [ ] sσ [ ] Figure.4 Polytropic coefficient (magnitude and phase) for a cylindrical tube.

27 9 Kirchhoff approximation For large values of the shear wave number s, i.e. for wide tubes, the propagation coefficient Γ can be estimated by a first order approximation of the low reduced frequency solution. In the present investigation large values of the shear wave number frequently appear. Therefore, the Kirchhoff approximation is applied, which is more efficient in terms of computing time than the low reduced frequency solution. The propagation coefficient according to Kirchhoff can be written as: i Γ = i + + γ + σ sσ (.) The associated coefficient G is defined by: i G = (.) Γ Both viscous and thermal effects are included in (.) via s and sσ, respectively. The propagation coefficient for a cylindrical tube according to the low reduced frequency model, the Kirchhoff approximation, and the Helmholtz equation without viscothermal effects, are compared in Figure.5. Re(Γ ) [ ] Low reduced freq. Kirchhoff No viscothermal eff. s [ ] /Im(Γ) [ ] Low reduced freq. Kirchhoff No viscothermal eff. s [ ] Figure.5 Propagation coefficient for a cylindrical tube (real part and inverse of imaginary part). Standard air conditions are used as given in the List of Symbols. For large values of s the low reduced frequency model and the Kirchhoff approximation approach the non-viscothermal wave propagation coefficient Γ = i.

28 It can be seen that the Kirchhoff approximation can be used for s >. The real part of Γ represents the attenuation of the propagating wave whereas the reciprocal of the imaginary part represents the effective phase velocity according to: c c eff = (.) Im( Γ ) Boundary conditions The amplitudes of the forward and backward travelling waves in a tube are determined by the boundary conditions at both ends of the tube. The boundary conditions can be expressed in terms of pressure, velocity or acoustic impedance. In Figure.6 the possible boundary conditions are shown. For a closed end, i.e. for an acoustically hard wall, the velocity is zero. Note that in the remaining part of this thesis the term closed end is used for an acoustically hard termination of a tube. For a tube terminated with a sound absorbing material an impedance condition is used. For pressure measurement systems, for example in a wind tunnel model where a thin tube connects the pressure transducer to the actual input pressure, the boundary conditions are the pressures at the entrance and at the end of the tube. p x p - pressure perturbation u u - velocity perturbation L - scaled acoustic impedance Figure.6 One-dimensional representation of a prismatic tube with applicable boundary conditions. The specific acoustic impedance is usually denoted by Z a and relates the pressure, at a specific position, for example at a surface, to the normal velocity at that same position. In the present investigation the impedance is scaled to the characteristic impedance of a travelling plane wave. The characteristic impedance of a freely propagating plane wave is usually ρ c. However, in a duct where the viscothermal wave propagation is included, the characteristic impedance is ρ c /(-G), with G the coefficient for the type of the cross-section of the tube considered. Therefore the scaled (dimensionless) impedance ζ becomes:

29 G G p( x) ζ ( x) = Za ( x) = (.3) ρ c ρ c u( ) x For sound absorbing materials the acoustic impedance indicates how incident waves are reflected at the surface. It will be shown later that the real part of the impedance can only be larger than or equal to zero. If the real part is larger than zero the surface absorbs energy (Pierce 994). This principle will be used in section.4 Broadband sound absorption for the impedance at the entrance of a tube. If the boundary conditions p() = p and p(l) = p are applied for a tube of length L, the impedance at x = and x = L can be rewritten with equation (.4) and (.5) as: sinh( Γ k L) sinh( Γ k L) ζ () = ; ζ ( L) = (.4) p cosh( ) p Γ k L cosh( Γ k L) p p In (.4) the ratio p / p can be seen as a transfer function of the tube. As an example the case of a tube terminated by an acoustically hard wall is considered. By applying the boundary conditions p and u = the transfer function can be shown to be: p p = cosh( Γ k L) (.5) In section.3. the transfer functions of tubes are used in a recursive formulation for coupled tubes. For more complex tube systems this is a convenient way to relate the measured pressure to the input pressure. A novel approach however is to use the coupled tubes as a resonator to absorb sound in a specific frequency range. This will be shown in section.4 Broadband sound absorption. More convenient quantities for the sound absorption are the reflection and sound absorption coefficient. These are related to the acoustic impedance and are presented in the next section. Reflection and sound absorption coefficient For sound absorbing materials such as foam or glass wool the parameter of interest is the sound absorption coefficient α as a function of the frequency. The sound absorption coefficient is usually measured in an impedance tube which is

30 terminated with a small sample of the sound absorbing material, see also Chapter 3. Briefly, the operation can be described as follows: the transfer function between two pressure transducers is measured from which the acoustic impedance, the reflection coefficient and the absorption coefficient can be derived. The reflection coefficient R is the ratio of the reflected and incident wave. Using the same notation as in (.4) one has: R k x pˆ Γ A e ( x) = (.6) Γ k x p ˆ B e Sometimes the amplitude reflection coefficient is used which does not include the phase information of the two acoustic waves: Rˆ = pˆ A / pˆ B. The acoustic energy in a plane wave is proportional to the squared amplitude of the wave. Therefore the squared reflection coefficient is the ratio of the reflected energy and the incident energy. In the case of a sound absorbing surface the energy balance at the surface where x = gives: pˆ B pˆ A α ( x= ) = (.7) pˆ B where the sound absorption coefficient α is the fraction of the incident energy that is dissipated. Using (.6) this results in: α = R (.8) With known amplitudes pˆ A and pˆ B the impedance ζ can be derived and with the use of the definition of the reflection coefficient R this becomes: ζ ( x) R( x) = ζ ( x) + (.9) Three separate cases can be distinguished. First, for an impedance approaching zero the reflection coefficient R becomes -, i.e. the acoustic waves are reflected at the surface with a 8 degree phase shift. Such a boundary condition is called a pressure-release surface and is applicable for waves propagating in water which reflect from a water-air interface. The acoustic energy is completely reflected so the sound absorption coefficient α =. Secondly, if ζ = then R = which indicates that the incident plane wave is not reflected. In this case there is no jump in the impedance so it seems that the incident wave travels to infinity. In that case

31 3 α = which indicates that all of the incident energy is absorbed. And finally ζ corresponds to R = and represents an acoustically hard surface, i.e. the velocity perturbation u = at the surface. The acoustic energy is completely reflected, so α =. The relation between the absorption coefficient and the real and imaginary parts of the impedance is: 4Re( ζ ) α = (.3) ( Re( ζ ) + ) + ( Im( ζ )) It follows that the real part of ζ can only be positive because α. Example The impedance at the entrance of a tube was given in (.4). To demonstrate the differences between the low reduced frequency model, the Kirchhoff model and the Helmholtz model the impedance at the entrance of a cylindrical tube is shown in Figure.7. The tube is terminated by an acoustically hard wall. No viscothermal eff. Kirchhoff Low reduced freq. No viscothermal eff. Kirchhoff Low reduced frequency π/ π/4 ζ [ ] arg(ζ ) [rad]. π/4 4 6 Frequency [Hz] π/ 4 6 Frequency [Hz] Figure.7 Acoustic impedance at the entrance of a cylindrical tube (magnitude and phase) with R=.5 mm and L=5 mm. Tube is terminated by an acoustically hard wall. Standard air conditions are used (see List of Symbols). When the viscous and thermal effects are neglected the first resonance frequency of the tube is 73 Hz which corresponds to a minimum of the impedance. Viscous and thermal effects cause the speed of sound to be lower which explains the shift in Figure.7 of the first resonance frequency to 49 Hz. The imaginary part of the

32 4 propagation coefficient is then.6, so the effective speed of sound becomes c eff = c /.6. The shear wave number s ranges from.5 to and it can be seen that the differences between the low reduced frequency model and the Kirchhoff model are not very large. It is noted that the resonance frequency of a half open tube, i.e. one end closed and the other end open, is in fact somewhat lower because of the inlet effects. This is explained in more detail in section.4 where the effective length of a resonator is introduced. Furthermore it is noted that the low reduced frequency model has also been successfully used in some recent projects (Ommen 999, Rodarte )..3 Coupled tubes In this section the acoustic behaviour of a number of coupled tubes is derived. Two techniques are presented: a recursive formulation with coupled transfer functions and a transfer matrix formulation. Both methods include the viscothermal wave propagation which cannot be neglected for sound absorbing resonators as will be shown. Practical examples of the use of these techniques for networks of tubes are (provided that the air velocity is low): air conditioning ducts, gas transportation systems and mufflers. The first technique uses a recursive formulation for the dynamic response of pressure measuring systems with cylindrical tubes as presented by Bergh and Tijdeman (965). It is based on a mass balance for a volume with a number of tubes connected to it. The second method is based on a transfer matrix formulation. It can also be seen as a finite element type of formulation where the linear or quadratic shape functions are replaced by the analytical expressions for sound propagation in a tube. The degrees of freedom are the pressure perturbation and the mass flow perturbation. Examples are described below to clarify the method..3. Recursive formulation Expressions for the one-dimensional pressure and average velocity perturbations including the viscothermal wave propagation for narrow and wide tubes were presented in the previous section. These are used here to set up a mass balance for

33 5 a volume with a number of tubes connected to it. Figure.8 shows two tubes, labelled J and J+, connected to volume V j. Note that at the ends of the tubes a small index j is used and that for the tubes themselves a capital index J is used. Each tube has its own co-ordinate system x J. a) xj pˆ B pˆ A xj+ A J+ L J+ b) Volume Vj p j- p j+ p j Tube J Tube J+ Figure.8 A geometric (a) and schematic (b) view of a volume with two tubes connected to it. For the volume V j one assumes that the mass variation M V due to the change in density of the volume V j must be equal to the difference between the mass leaving tube J and that entering tube J+: d M d t V iωt ( Q x = L ) Q ( x ) ) e = J ( J J J + J + = (.3) where Q indicates mass flow. Furthermore, the pressure perturbation in the volume is assumed to be uniform: p p ( x = L ) = p ( x = ) (.3) V = j J J j J + For small values of the pressure perturbation p and the density perturbation ρ the polytropic relation applies: ρ p γ = (.33) c n V So the mass M V can be written as: MV = V j p j γ iωt + e c n (.34) V ρ

34 6 where n V is a polytropic coefficient in the volume V j. which can be estimated according to (.9) or measured. In the paper of Bergh and Tijdeman the instrument volume V can also be corrected for the elastic deformation of the volume. The mass flows entering and leaving the volume are, respectively: QJ ( xj = LJ ) = AJ ρ u( xj = LJ ); QJ + ( xj + = ) = AJ + ρ u( xj + = ) (.35) The complex amplitudes of the reflected and incident travelling waves in the tubes, pˆ AJ, pˆ B J, p ˆ AJ + and p ˆ B J + are determined by the boundary conditions at both ends of the tube. Following the same approach as Bergh and Tijdeman the boundary conditions for the pressure at both ends are used to determine these amplitudes. In the case of Figure.8 one has: pˆ pˆ AJ AJ + p = e p = e p j j ( Γ k L) J p e e j+ j ( Γ k L) J + ( Γ k L) J ( Γ k L) J e ( Γ k L) J + e ( Γ k L) J + pˆ pˆ B J B J + = e ( Γ k L) p + J j p j e ( Γ k L) J ( Γ k L) J p = e e + p j+ j ( Γ k L) J + e e ( Γ k L) J + ( Γ k L) J + (.36) Finally, with (.3), (.35) and (.36) the recursive formulation for the coupled tubes system of Figure.8 results in: p p j j = cosh ( Γ k L) J ( Γ k L) sinh + A G J J J ikγ V n V j A + sinh J + G J + ( Γ k L) J + cosh ( Γ k L) J + p + j p j (.37) Note that the transfer function p j / p j- of tube J depends on the transfer function p j+ / p j and the ratio of the cross-sectional areas A J+ / A J. In Figure.8a volume V j is drawn with a dotted line to indicate that a direct coupling of the two tubes is also possible. In that case the volume V j in (.37) is equal to zero. For the complete branch of Figure.8, a series of two tubes and a volume, the transfer function is: p p p p j+ j+ j = j p j p j (.38) The boundary condition at the end of the branch needs to be known. This could be, for example, a rigid wall or a volume. For tube J terminated by a volume one has:

35 7 p j sinh( Γ k L) = cosh( ) + J ikγ Γ k L J V p j j- p j (.39) j AJ GJ nv p And for a closed tube J one has: p p [ cosh( k L) ] j = Γ p J j- p j (.4) j The general boundary condition is an acoustic impedance condition as shown in the previous section. The transfer function of a tube J with an impedance ζ(l) at x = L J can be derived from (.4): p j = cosh( Γ k L) J + sinh( Γ k L) J (.4) p j ζ ( L) If there is a third tube connected to volume V j, say tube J+ with pressures p j and p j+ at the ends of the tube, then an additional term arises in (.37). Rewriting only the term between the large round brackets gives: ikγ A p + + j+ p J GJ AJ + GJ + j+ V + ( ) + ( ) j cosh Γ k L J + cosh Γ k L J + n ( ) ( ) V sinh Γ k L J + p j sinh Γ k L J + p j (.4) In this way it is possible to create a network of tubes and volumes with multiple branches. When each branch has a known end condition every transfer function in the network can be calculated. This is demonstrated in the following example in which the acoustic impedance at a junction with two branches is calculated (see Figure.9). One branch is terminated by a volume, the other branch has an acoustically hard termination. By locating the minima of the acoustic impedance at the junction at p the resonances in the system, consisting of the two branches, can be detected. A minimum in the impedance indicates that the system can be easily excited acoustically.

36 8 V 3 p 3 p 4 3 p p Junction 4 Tube nr. p 5 5 p 6 } } Branch Branch R =. m L =.5 m R =.5 m L =. m R3 =.5 m L 3 =.4 m V =.8 m R4 =.5 m L 4 =.8 m R5 =.5 m L 5 =. m Figure.9 A simple example of a network of coupled tubes. The transfer function p /p is calculated with (.37) in a recursive way. The acoustic impedance at p is calculated with (.4). In Figure. the results are presented. The propagation coefficient for air is calculated in three ways: no viscothermal effects, the Kirchhoff solution for wide tubes, and the general solution of Zwikker and Kosten (low reduced frequency) which is valid for narrow and wide tubes. According to the low reduced frequency solution a narrow tube corresponds to a low shear wave number whereas a wide tube corresponds to a large shear wave number. The analytical model can easily be used to investigate resonance frequencies in separate branches by switching off the successive branches. In this case the first resonance frequency at Hz belongs to the upper branch with the volume. This branch can be seen as a Helmholtz resonator. The second and third resonance frequencies at 4 and Hz belong to the lower branch and correspond to the organ pipe resonance formula for a closed pipe. The shear wave numbers range from.5 to 5 for the various tubes and it can be seen from Figure. that for large shear wave numbers, i.e. high frequencies, the viscothermal effects cannot be neglected. Furthermore, for low shear wave numbers the resonance of the upper branch is highly damped because of viscous and thermal effects in the air.

37 9 Low reduced frequency Kirchhoff No viscothermal eff. ζ [ ] 5 5 Frequency [Hz] Figure. Impedance (magnitude) at junction in the network of tubes as depicted in Figure.9. The recursive formulation presented here is an efficient and modular technique to calculate the acoustic behaviour of a network of tubes and volumes, i.e. the pressure perturbation, the impedance and the velocity perturbation can be calculated directly. This technique will be used in the following sections. However, if a cross-link between branches is present the recursive formulation can no longer be applied. In that case the system needs to be solved in a direct way rather than in a recursive way. The transfer matrix formulation is used for this purpose to set up the equations for the unknown pressure and velocity perturbations at the ends of the tubes. This is shown in the following section..3. Transfer matrix formulation In mechanics the quantities force and displacement can be related to each other via a transfer matrix. When the velocity is used an impedance matrix results (or a mobility matrix which is the inverse of the impedance matrix). An analogue approach can be applied in acoustics. Here the mass flow Q and the pressure perturbation p are used, where Q = ρ A u. The final set of equations can be formulated as:

38 3 [ ]{} p { Q} M = (.43) In the vector {Q} the forcing terms per node are ordered and in the vector {p} the unknown nodal pressure perturbations are ordered in the same way. The system matrix [M] contains mobilities of the acoustic elements such as tubes, volumes and impedance boundary conditions. The different acoustic elements will be derived in the following sub-sections. Element matrix for a prismatic tube For a prismatic tube the co-ordinate system as given in Figure. is used. Note the convention for the direction of the mass flow. As a result the element matrix for a tube will be symmetric. L J L J p j p j+ Q j u j A J x Q j+ u j+ Figure. Notation for a prismatic tube. The pressure in the tube p(x) is given by (.4). So for the pressures at the nodes j and j+ one has: p p j j ( Γ k L ) ( k L ) e J Γ e J pˆ A = ( Γ k L ) ( k L ) e J Γ e J pˆ + B Inverting this matrix gives the pressure amplitudes mass flow one can write: Q (.44) pˆ A and pˆ B for tube J. For the j = AJ ρ u j ; Q j+ = AJ ρ u j+ (.45) and by using the velocity u(x) as shown in (.5) the element matrix for a tube results:

39 3 = + + ) cosh( ) cosh( ) sinh( j j j j J J J J J Q Q p p k L k L k L c G A Γ Γ Γ (.46) The element matrix for a tube has four components which are indexed as J M, J M, J M and J M for tube J. It is noted that when the viscothermal effects are neglected the element matrix shows a singularity if kl = π (mod π), i.e. when an exact multiple of half-waves is present in the tube. Element for a volume: M V The element for a volume M V connected to one or more tubes is derived by considering two tubes as shown earlier in Figure.8. The mass balance for the volume, taking into account the directions of the mass flow, is: = j V j j p V n c ik Q Q γ (.47) The system matrix for the two tubes and the volume then becomes: ( ) = j j j j j J J J V J J J J J Q Q p p p M M M M M M M M M (.48) where V V n c V ik M γ =. Thus adding a volume at the junction of a number of tubes simply results in an extra term in the system matrix at the node where the volume is present. Element for an impedance boundary condition: M ζ An impedance condition at the end of a tube J can be written as: = j j J J j p G c A Q ζ (.49)

40 3 Using this relation and by eliminating Q j+ the element matrix for a single tube terminated with an impedance ζ j+ becomes: M M J j J J M p j Q = J M + Mζ p j+ (.5) where M ζ equals A J G J / (c ζ j+ ). Consequently a known impedance condition at the end of a tube can be taken into account with an extra term in the element matrix of that tube. Element for a pressure boundary condition Finally, the boundary condition for the pressure is treated. The same procedure as in the finite element method for a prescribed displacement is followed. For a prescribed pressure p j the j th column is subtracted from the {Q} vector, the j th row and column of the system matrix are set to zero and the element M jj is set to. In the case of two tubes connected to a volume and a prescribed pressure p at node j- one has: p j p J J + J + J M + M + MV M p j = M (.5) J + J + M M p j+ Q j+ To solve the nodal pressures and nodal velocities of a given network of tubes and volumes the first step is to solve the nodal pressures by solving the system: {p} = [M] - {Q}. In the second step the nodal velocities are solved through: {u} = [M J / ρ A J ] {Q}, where the vector {u} is arranged per tube as follows: + + { u u u u... } T J J J J j j j j + (.5) The index J indicates that for a tube J a cross-section A J has to be used.

41 33 The advantage of the transfer matrix formulation described above is that the analytical solutions are used whereas in standard finite element formulations linear or quadratic functions are usually used. Hence, there is no need for a discrete number of elements per wavelength; a single element can be used for a tube regardless of the number of wavelengths that fits in the tube. As a result the system matrix remains small. It is not surprising that with the transfer matrix technique the same results are obtained as with the recursive formulation technique. Both techniques can be applied very efficiently for the acoustic behaviour of pipe systems provided that the mean velocity is low. The present investigation considers sound absorbing material as a network of holes and volumes which is present inside the material. First the sound absorbing behaviour of a single hole is investigated, in this case a tube. This is done in section.4.. Secondly, the absorbing behaviour of coupled tubes is described in section.4.. It is shown that with a combination of tubes sound absorption over a wide frequency range can be obtained. This gives the opportunity to design special purpose sound absorbing material. For instance the coupled tubes can be applied in the casing of a noisy device to absorb a range of annoying frequencies..4 Broadband sound absorption Sound absorption is directly related to acoustic impedance. Hence, impedance is a key quantity in this section when studying the sound absorption with coupled tubes. Reflection of acoustic waves occurs when the impedance somewhere in the path of the propagating wave shows a sudden change. The most simple case is a rigid wall which has an infinite impedance. Reflection will also occur for changes of the cross-sectional area of a tube because the jump in the cross-section causes a jump in the acoustic impedance. The special case of an open-ended tube can be characterised by a frequency dependent impedance boundary condition, namely the radiation impedance. For low frequencies the radiation impedance approaches zero so that a reflection coefficient of results (see equation (.9)). For porous materials the impedance, in the ideal case, is close to the characteristic impedance of a plane propagating wave. In that case the surface of the porous material acts as if it is the entrance of an infinite volume so that the

42 34 incident waves continue their propagation without being reflected. Thus the nondimensional impedance has to be chosen close or equal to to create a sound absorbing wall. This will be shown for a sound absorbing wall with a number of orifices or tubes, see Figure.. The tubes in the wall act as resonators for specific frequencies. When the tubes are closed at one end, for example, resonance occurs when a quarter of a wavelength fits in the tube. Hence the term quarter-wave resonators, In fact this is a short notation for quarter-wavelength resonators. The geometry of these orifices is known in advance so the wave propagation in these holes or tubes can be described with the theory presented in the previous section. If the wall has a regular pattern of orifices, i.e. the porosity over the wall is constant, then the sound absorption coefficient of the wall can be calculated by considering the impedance at the entrance of a single tube and the porosity. Figure. A sound absorbing wall with quarter-wave resonators. In the next section the sound absorbing behaviour of a single tube will be discussed. The effects of the effective length, the radius, the viscous and thermal effects, the cross-sectional shape and the boundary conditions at the end of the tube will be described..4. Sound absorption of a single tube The goal is to create a sound absorbing wall with an impedance equal or close to. The impedance of the wall ζ wall can be related to the impedance at the entrance of a

43 35 single tube ζ tube by assuming that the waves are plane at a short distance from the wall, i.e. kδ «with k = ω / c (see Figure.3) and the pressure perturbation and the harmonic mass flow in a reference frame across the wall are constant. wall tube Figure.3 Cross-section of a wall with resonators. The surface porosity of the wall is defined as: wall N Atube Ω = (.53) A where A wall is the total area of the wall, N is the number of identical tubes, each with a cross-sectional area A tube. So with the assumptions discussed earlier concerning the reference frame the impedance of the wall can be written as: ζ ζ wall = tube (.54) Ω This is an essential result for a sound absorbing wall with resonators. If the impedance of the tubes and the porosity are matched in such a way that the ratio is. for a specific frequency then the sound absorption is maximal for that frequency. It is noted that in a narrow tube ζ wall needs to be scaled according to (.3). As an example the calculated impedance and the sound absorption coefficient of a wall, with tubes which are closed on one side, is shown in Figure.4 and Figure.5, respectively. The standard conditions for air are used.

44 36 4 ζ [ ] ζ tube ζ wall α [ ] α wall Frequency [Hz] Figure.4 Impedance (magnitude) of a wall L tube =.6 m, R tube =.5 m, Ω= Frequency [Hz] Figure.5 Sound absorption coefficient of a wall with quarter-wave resonators. The geometry of the tube is given in the caption of Figure.4 and with the propagation coefficient according to Kirchhoff (s > ) the impedance at the entrance of the tube can be calculated. The accompanying optimal porosity is Ω =.35. The % sound absorption for an incident wave at 4 Hz is physically interpreted in the following way. At the first resonance frequency the pressure perturbation in the tube is amplified due to the incoming waves. For a harmonic situation the amplitude of the pressure in the resonator is determined by the amount of damping in the tube which is described by the wave propagation coefficient. Exactly at the resonance frequency the accompanying incident sound energy is completely dissipated and the pressure amplitude of the reflected wave in the resonator is in anti-phase with the incident wave (see also Appendix A). In a passive way anti-sound is created for a specific frequency. The same effect could have been obtained with an active anti-sound source. The reason that there is sound absorption for frequencies close to the first resonance frequency is that due to the viscous and thermal effects a broader (and lower) resonance peak in the transfer function for a quarter-wave resonator results. This peak corresponds to a minimum in the magnitude of the impedance at the entrance of the resonator. When the wall contains tubes with different dimensions the impedance of the wall is obtained by summation of the contributions of the different tubes:

45 37 ζ wall = Ω ζ j j (.55) where Ω j is the porosity for the tubes with impedance ζ j. Effective length of a resonator Due to inlet effects the effective length of the resonator is larger than the actual length, see also Figure.3. Therefore an end correction is added to the geometrical length of the tube. This effect has been studied extensively for different configurations of the entrance of the tube. For example, flanged and unflanged pipes have been studied (see for instance: Levine and Schwinger). The end correction depends on the local geometry at the entrance and termination of the tube. The effective length L eff is the geometrical length L increased by a small increment d. According to Rayleigh (945) the increment d for a single tube with the opening in an infinite baffle is equal to: 8R d = (.56) 3π where R is the radius of the tube. This can be derived by considering the acoustic force on a vibrating piston. The fluid moved by the piston with a radius R has an apparent mass which corresponds to the fluid in a cylinder of area πr and length d (see also Pierce 994). If the resonator is open at both sides then the end correction has to be applied for both sides. For a number of perforations in a panel equally spaced by a distance a the end correction for each side is (see for instance Bies 996): (.44 R ); with a > R 8R d = (.57) 3 π a In the next section resonators with different cross-sections are coupled. The end correction for a single tube centrally located in a tube of circular cross-section with radius R is given by (Bies 996): 8R d =.5 R ; with R 3π R R <.6 (.58)

46 38 When R or a tends to infinity the value of the end correction converges to the value for a piston in an infinite baffle. When the cross-section of the resonator is not circular then the radius R can be approximated by: R A (.59) D provided that the ratio of the orthogonal dimensions is of the order of unity and where A is the cross-sectional area of the resonator and D its perimeter. For larger ratios of the dimensions reference is made to ASHRAE 993 (see Bies 996) to determine the effective radius. It is obvious that the length is the most important geometrical parameter to tune the resonator for a specific frequency. The effect of the end correction on the sound absorption coefficient is shown in Figure.6 for a wall with equally spaced resonators with a geometrical length of.6 m (see also Figure.)..8.6 Length = L Length = L + d R L d a =.35 =.5 m =.6 m =.4 m =.76 m α [ ] Frequency [Hz] Figure.6 Inlet effect for a wall with resonators. Standard air conditions are used. Evidently the inlet effects cause a significant shift of the resonance frequency. The maximum absorption is tuned for the geometric length but the shift in the frequency hardly affects the maximum absorption. Thus in this case the sensitivity of the maximum absorption coefficient for the length is small.

47 39 Effect of different radii The dimensions of the cross-section of a resonator determines amongst others the ratio of the viscous and inertial effects and thus affects the absorption coefficient. The effect on the absorption coefficient is shown in Figure.7. A comparison is made for single cylindrical tubes with different radii. The porosity and the length for the different tubes are tuned in such a way that the absorption coefficient is maximised at the same frequency..8 Radius =.m Radius =.5m Radius =.m Radius =.5m α [ ] Frequency [Hz] Figure.7 Effect of different radii for a quarter-wave resonator tuned at 34 Hz. Standard air conditions are used. In Table. the accompanying combinations of the radius R, the geometric length L and the porosity Ω are given. The effective length was obtained by using the length increase of equation (.56). In the fifth column the effective speed of sound in the tube is shown for 34 Hz and between brackets the corresponding shear wave number is given.

48 4 Radius R [m] Length L [m] Porosity Ω [-] Tubes per m [-] Speed of sound c eff [m/s] (s [-]) (36) (8) (3.6) (.8) Table. Data for a cylindrical resonator optimised at 34 Hz. For small radii the width of the absorption peak is much wider due to viscous losses in the resonator. The required porosity is much higher in that case so that more resonators per unit area are necessary. Even more than due to the smaller cross-sectional area of the narrow tubes. The viscosity effects in the resonator cause the effective speed of sound to decrease considerably. This effect is directly related to the propagation coefficient and therefore to the shear wave number s. α [ ] =.75 R =. mm L =.6 m =.5 R =. mm L =.6 m Radius =.mm. Radius =.mm Glass wool (45mm) Glass wool (5mm) 4 6 Frequency [Hz] Figure.8 Absorption coefficient of a wall with narrow resonators and a high porosity. Standard air conditions are used. For very small values of the radius the dimensions of the pores in sound absorbing foams are approached. Figure.8 shows that the sound absorbing behaviour of small resonators, with the porosity correctly chosen, looks much like the behaviour of conventional sound absorbing material; here glass wool. For the latter material,

49 4 with a thickness of 5 or 45 mm, measurements were performed up to 3 Hz. The material with quarter-wave resonators has a thickness of 6 mm so that low frequency sound waves are more absorbed than for the glass wool. For conventional sound absorbing materials the volume porosity is usually.95 < Ω <.99. It is noted that for a homogeneous material the same values for the surface porosity can be used. In Chapter 5 a statistical distribution of tubes in a wall is used to approach the sound absorption of conventional sound absorbing materials. Effect of the viscosity and thermal conductivity The effects of the viscosity and thermal conductivity on the sound absorption coefficient is determined by the non-dimensional parameters s and sσ, where s is the shear wave number and σ is the square root of the Prandtl number. For large values of s and sσ, i.e.», the effects of the viscosity and the thermal conductivity are low. These effects are numerically shown in Figure.9. Just as in the previous sections the wall with resonators is optimised for a single frequency for each configuration of different parameters. So the geometrical length is adjusted to get the same frequency and the porosity is tuned for maximum absorption..8 Viscous and thermal effects Viscous effects only (high sσ) Thermal effects only (high s) No viscothermal effects α [ ] Frequency [Hz] Figure.9 Effect of the viscosity (s) and the thermal conductivity (sσ). R= mm.

50 4 It can be seen that for this situation, where standard air conditions are used for the parameters other than s and σ, the effect of the viscosity on the absorption coefficient is larger than the thermal effect. The absorption coefficient without viscothermal effects is in fact only non-zero at the resonance frequency, but for clarity s and sσ are chosen relatively large so that a very sharp peak is shown in Figure.9. Evidently the viscothermal effects cannot be neglected for resonators if the sound absorption has to be calculated. The effective speed of sound for the resonator with a radius of mm is shown in Table.. Effects (at 34 Hz) Length L [m] Porosity Ω [ ] Speed of sound c eff [m/s] s (3.6) and sσ (.) large sσ (. 3 ) large s (3.6 3 ) large s and sσ Table. Data for air-like media in a cylindrical quarter-wave resonator. Optimised at 34 Hz. Effect of cross-sectional shape The results for the absorption coefficient of resonators with a circular (with radius R), an equilateral triangular (with sides d), and a rectangular cross-section (with sides a and b) are presented in Figure.. Also the propagation coefficient for a layer is calculated. In Appendix B these propagation coefficients are given. It is noted here that the calculation time for a rectangular cross-section is larger due to the series solution. For a comparison the cross-sectional areas are kept constant in the left-hand side figure. In the right-hand side figure the shear wave number is kept constant at 34 Hz except for the layer geometry where the shear wave number is twice as large. To obtain the same absorption coefficient for the layer geometry it is noted that the viscous effects in one direction of the cross-section are present but absent in the infinite direction. Therefore the viscous effects are doubled in the layer. Furthermore, the cross-sectional area is per unit width.

51 α [ ].6.4 a = b d a = b. R h 4 6 Frequency [Hz] α [ ].6.4 a = b d a = b. R h (s) 4 6 Frequency [Hz] Figure. Sound absorption coefficient of a wall with resonators for which the crosssection of a resonator is constant (left) and the shear wave number is constant (right). The ratio of the viscous and inertial effects determines the width of the peak of the sound absorption coefficient. It can be seen in the left-hand side figure that for larger perimeters the viscous losses are larger. The geometry and dimensions are shown in Figure., Table.3 and Table.4. Cylindrical Square, Rectangular R L b a L Triangular Layer d L h L Figure. Resonators with different cross-sectional shapes.

52 44 Cross-section Length Porosity Speed of sound [m] L [m] Ω [-] c eff [m/s] Cylindrical R= Square a=b= Rectangular a=b= Triangular d= Layer h= Table.3 Data for quarter wave resonators with the same cross-sectional area optimised at 34 Hz. Cross-section [m] Length L [m] Porosity Ω [ ] Cylindrical R= Square a=b=r.6.69 Rectangular a=b=r/.6.7 Triangular d= R.6.69 Layer ( s) h=r Table.4 Data for quarter wave resonators with the same shear wave number of s = 3.6 at 34 Hz. Effect of an open tube; the radiation condition In the previous sections the resonators had a closed end, i.e. an acoustically rigid termination. In other words the impedance at x = L is infinite. In general the resonators may have any impedance boundary condition. A special case are resonators with an open end. These kind of resonators may be used in a wall when a fluid needs to be transported through the wall or when one needs to be able to see through the wall. The pressure perturbation at x = L, or in fact somewhat outside the resonator, is approximately zero. For a tube with the end in an infinite baffle an analytical expression for the impedance at the end, the so-called radiation impedance, can be obtained from the literature. It is based on the acoustic load which is imposed on a vibrating circular piston in an infinite baffle (see Pierce 994 and Bies 996). The result is given here (ζ rad is not scaled with the coefficient G):

53 45 ζ rad = R + i X rad rad (.6) The real and imaginary part of the radiation impedance are: R X rad rad = 4 ( k R) ( k R) ( k R) 4 4 k R = π ( k R) ( k R) L 8 5 L 7 (.6) As an example the corresponding sound absorption coefficient of the end of the tube is plotted in Figure. for several radii. The absorption coefficient is calculated with the use of equation (.3). Figure. shows that at low frequencies the sound waves propagating in the resonator are not absorbed at the end but are reflected back into the resonator due to the mass reactance at the free end, i.e. the reflection coefficient R = - and α = (note the frequency range in Figure.). For higher frequencies the waves are absorbed due to radiation into infinity. The radiation impedance approaches the impedance of a freely travelling plane wave, i.e. ζ rad =..8 Radius =.m Radius =.5m Radius =.m Radius =.5m α [ ] Frequency [Hz] Figure. Sound absorption coefficient at the open end of a tube resonator located in an infinite baffle. Pierce uses ζ rad = Rrad i X rad as a result of the time convention i t e ω.

54 46.4. Sound absorption of coupled tubes It has been shown that it is possible to design a wall with a specific number of resonators to absorb sound energy for a narrow frequency band. These kinds of walls can be applied for noise problems where for example an engine motor runs at a constant number of revolutions per minute or in the case of an annoying whistle. The quarter-wave resonator is frequently applied in practice. In general noise problems are characterised by a wider frequency band due to, for example, variations in engine rpm. Therefore a design tool for a more complex configuration of resonators is presented. By means of coupling it is possible to absorb noise within a wide frequency band. The geometry of the coupled tubes, i.e. the cross-sectional areas and the lengths, are important design parameters to create a maximum absorption coefficient. It will be shown that it is efficient to calculate the absorption coefficients for a set of cross-sectional areas and lengths and subsequently choose the cross-sectional areas and lengths for which the absorption coefficient matches the design requirements for the sound absorbing wall best. The two formulations based on the basic theory for coupled tubes described in section.3 are used to calculate the impedance at the entrance of the coupled tubes. Given the porosity of the wall the sound absorption coefficient of the wall can be calculated. The most simple configuration of coupled tubes is shown in Figure.3. In the following sections five configurations are described:. Two tubes coupled in series,. Two single tubes in parallel, 3. Two tubes coupled in series with R >R, 4. Three tubes coupled in series and 5. Multiple coupled tubes. In the examples the frequency range is chosen from to 6 Hz. In this way one is able to compare the results to those of the previous section for quarter-wave resonators. Furthermore it is an interesting frequency range for noise problems. It is noted that in the examples: The value for the shear wave number s is high (s > ) so that the wave propagation coefficient of Kirchhoff can be used. Cylindrical tubes are used. The effective lengths as mentioned in section.4. are used in the calculations, i.e. L eff = L + d, whereas the geometrical lengths L are given in

55 47 the examples. For the first tube equation (.56) is used and for the second tube this is equation (.58). Two coupled tubes in series The geometry of two axially coupled tubes is given in Figure.3. Because of the assumption of one-dimensional waves the second tube does not have to be connected concentrically to the first tube. The entrance of the first tube is located in the surface of the sound absorbing wall. Tube R Tube L R L Figure.3 Two axially coupled tubes. It was seen that the impedance at the entrance of the resonator, in this case two coupled tubes, needs to be close to one in order to achieve optimal sound absorption. The magnitude and the phase of the impedance is shown in Figure.4 for a closed end of tube. Approximately the same results can be obtained for an open end of tube if the length L is approximately twice as large.

56 48 Tube : open end Tube : open end π/ π/4 ζ [ ] Coupled tubes Tube : closed end 6 5 Frequency [Hz] Coupled tubes Tube : closed end 6 5 Frequency [Hz] Figure.4 Impedance at the entrance of two axially coupled tubes with a closed end (magnitude and phase). Also depicted is a single tube with an open or closed end. Dimensions: L =.6 m, L =.6 m, R =.5 m, R =. m. The impedance characteristic of the coupled tubes is dominated by the first tube for most frequencies (for f < Hz and f > 5 Hz). Due to the presence of the second tube the first tube behaves as an open tube in the neighbourhood of the resonance frequencies of the second one ( Hz < f <5 Hz). The magnitude of the impedance at the entrance of the coupled tubes now shows two minima instead of one. Proceeding in the same way as in the previous section the porosity of the wall is chosen so that for at least one resonance frequency the absorption coefficient is.. The corresponding absorption coefficient is depicted in Figure.5. arg(ζ ) [rad]. π/4 π/

57 49.8 α [ ].6.4. Single tube Coupled tubes 4 6 Frequency [Hz] Figure.5 Absorption coefficient of a wall with resonators. Each resonator consists of two axially coupled tubes. Ω=.5. The width of the frequency band for which noise is absorbed looks promising but the question arises from Figure.5 whether the absorption around 3 Hz, where it has a local minimum, can be increased. The five parameters to be varied in this case are L, L, R, R and Ω. The search procedure for a suitable set of parameters was programmed by using the following guidelines: The centre frequency of the frequency range of interest is determined by the length of the first tube as can be seen in Figure.4 and Figure.5. Therefore the parameter L is set to a fixed value. The parameter R can be set to a convenient value. A discrete set of parameters is chosen for f, Ω, R and L, i.e. f j, Ω j, R,j and L,j. In general the discrete sets have a different number of elements but only the index j is used here for brevity. The sound absorption coefficient is calculated for each parameter according to the theory presented in section.3. A multi-dimensional array α( f j, Ω j, R,j, L,j ) results. To reduce the calculation time the Kirchhoff approximation for the propagation coefficient is used. A minimum absorption coefficient α min is specified so that α > α min. From the array α( f j, Ω j, R,j, L,j ) the widest frequency range and the accompanying parameters Ω, R and L are selected for which α > α min.

58 5 It proved to be efficient, because of the shortness of calculation times, to calculate the absorption coefficient for a large range of the parameters and to search for a frequency band for which the absorption coefficient exceeds the specified value α min. In this way Figure.6 was obtained. α [ ] α >.8 α >.9 α >.95 α >.99 single tube. 4 6 Frequency [Hz] Figure.6 Absorption coefficient of a wall with resonators which consist of two axially coupled tubes. Figure.6 shows that the region of high absorption for two axially coupled tubes can cover a much wider frequency band than for a single tube. Furthermore a larger value of α min results in a somewhat narrower frequency band, i.e. the two peaks in Figure.6 lie closer together for a higher averaged sound absorption coefficient. In Table.5 the applied parameters are listed. Parameter α >.8 α >.9 α >.95 α >.99 L [m].6 (same) R [m].5 L [m].65.6 R [m] Ω [-] Table.5 Geometrical parameters for axially coupled tubes with a high level of absorption ( WXEHVSHUP ).

59 5 Two single tubes in parallel The sound absorption coefficient of a wall with a distribution of two types of single tubes is shown in Figure.7. It can be seen that in this case the bandwidth corresponding with a high sound absorption is much smaller than for the coupled situation. Moreover there is a local minimum in the absorption coefficient which can only be removed by moving the peaks even closer, i.e. the resulting absorption coefficient is a simple linear addition of the two absorption coefficients of the single tubes. L R R.8 Tube Tube Two tubes Wall =.4 R =.5 m L =.57 m L α [ ].6.4. =.35 R =.5 m 4 6 L =.63 m Frequency [Hz] Figure.7 Absorption coefficient of a wall with two types of single tube resonators. Two tubes coupled in series with R >R When the second tube has a larger radius than the first tube then the sound absorption coefficient does not show the broadband behaviour, see Figure.8. Obviously there is no strong interaction between the two tubes. The first tube can be considered to be open for most frequencies at the junction with the second tube, i.e. it works as a half-wave resonator. Therefore it is chosen twice as long compared to the previous examples in order to obtain the same frequency range.

60 5 The second tube on the other hand can be seen as closed at both ends so that it also works as a half-wave resonator. In this case it is only the length of the first tube that needs to be corrected for the inlet effects, but now on both sides. Tube R Tube R L L α [ ] R L R L =.4 =. m =. m =.5 m =. m. R > R R > R 4 6 Frequency [Hz] Figure.8 Absorption coefficient of a wall with resonators which consist of axially coupled tubes. Three coupled tubes in series The results of the search procedure for three axially coupled tubes leads to the sound absorption coefficients presented in Figure.9. The bandwidth has increased drastically compared to a double tube configuration. Also in this case the maximum bandwidth depends on the specified minimum value for the sound absorption coefficient.

61 53 α [ ] triple α >.8 triple α >.9 triple α >.98 double α >.99 single tube Frequency [Hz] Figure.9 Maximum absorption coefficient of a wall with resonators which consists of three axially coupled tubes. Parameters as given in Table.6. Parameter α >.8 α >.9 α >.98 L [m] R [m] L 3 [m] R 3 [m] Ω [-] Table.6 Geometrical parameters for three axially coupled tubes. L =.6 m, R =.5 m ( WXEHVSHUP ). Multiple coupled tubes In the search for wide band sound absorption with coupled tubes the configuration in Figure.3 also provides good results. An advantage of this configuration is that, compared to the three axially coupled tubes, tube number 3 is now directly coupled to tube number which results in a more efficient use of space. As a result the diameter of tube 3 can be chosen somewhat larger (see Table.7).

62 54 L 3 L.8 α >.8 α >.9 α >.98 L α [ ] Frequency [Hz] Figure.3 Absorption coefficient of a wall with resonators which consist of two tubes connected to a single tube. Parameter α >.8 α >.9 α >.98 L [m] R [m]..8.7 L 3 [m] R 3 [m]...8 Ω [-] Table.7 Geometric parameters for three coupled tubes. L =.6, R =.5 ( WXEHVSHUP ). In general a large variety of combinations can be used to get the desired absorption behaviour of a wall. Figure.3 presents an overview and some examples of combinations of resonator configurations.

63 55 Single type resonators Multiple type resonators (examples) Figure.3 Cross-section of various resonators and combinations of resonators. A comparison of predicted and experimental results for a number of resonators will be presented in Chapter 4..5 Conclusions In this chapter a strategy for the design of a sound absorbing wall with tuned resonators was presented. By applying resonators consisting of coupled tubes, a wall can be designed in such a way that absorption is possible for a considerable frequency bandwidth when compared to other resonance absorbers. The coupling principle and the viscothermal effects are responsible for the broadband absorption. With the design tool presented in sections.4. and.4. an optimal distribution of resonators can be found for a predefined frequency range. The numerical basis was presented in section.3 and resulted in two efficient and accurate models for coupled tubes in general. It was seen that the viscothermal effects in tubes have a significant influence on the wave propagation and that these effects are essential to create the wide band resonance and sound absorption. As a result of the viscothermal effects in the narrow tubes the effective speed of sound is lower. This was demonstrated for different cross-sectional shapes. Advantages of the presented sound absorbing wall with resonators are: A high sound absorption for a broad frequency band can be realised. The frequency band can also be located in the low frequency range. The frequency band can be fine-tuned for a specified sound spectrum.

64 56 The resonators can be constructed in any material to prevent problems with for example: ageing, extreme temperatures or aggressive environments. The wall can be constructed with perforated resonators to enable a fluid to pass through or for visual inspections. The implementation of resonators with coupled tubes is not limited to axially coupled straight tubes. As long as the wave propagation is one-dimensional by approximation, also flexible tubes or labyrinth-like structures can be applied in the sound absorbing wall in order to reduce the wall thickness (see Figure.3). cover plate plate with resonators Figure.3 Resonator configurations to reduce the total wall thickness. In the next chapter the experimental verification of the acoustic behaviour of the resonators with coupled tubes is described. An impedance tube (Kundt s tube) is

65 used in which samples of the sound absorbing wall are placed. The transfer function of two transducer signals in the impedance tube is used to calculate the reflection coefficient, the impedance and the sound absorption coefficient. The recursive formulation with transfer functions can be seen as the basis for this measurement technique, see section

66 58

67 Chapter 3 Impedance tube techniques to measure sound absorption 3. Introduction Impedance tube techniques The p method The u method The p/u method The p u method Experimental results Comparison of the p and the u methods Comparison of the p, the p/u, and the p u methods Conclusions Introduction To verify the coupled tubes model and the predicted sound absorption behaviour of broadband resonators, as presented in Chapter, experiments were performed. In this chapter four techniques are described to measure the sound absorption coefficient in an impedance tube. Three of the presented techniques are new and make use of a particle velocity sensor, the so-called microflown. It will be shown that a combination of a microphone and a microflown provides direct information on acoustic impedance, sound intensity and sound energy density. Experimental results of the four methods are compared to each other. To be able to repeat the measurements in a reliable way a test sample with a quarter-wave resonator is used. For the experimental verification of the coupled tubes model the reader is referred to Chapter 4 where the experimental results of quarter-wave and more broadband resonators are presented. A number of measurement techniques are available to quantify the sound absorbing behaviour of porous materials. These techniques can also be used for a wall with

68 6 resonators. In this section a brief overview of measurement techniques for sound absorbing materials is given. For a more extensive background the reader is referred to the literature and standards presented in subsequent sections. It will be argued that the impedance tube method is the most convenient and efficient technique for experiments with coupled tubes resonators. In general one is interested in the sound absorption coefficient α, which is the fraction of the incident sound power which is dissipated in the porous material, the reflection coefficient R, or the normal surface impedance Z n. These quantities are usually measured for normal incident waves. The incident sound field can be classified into three types: normal incidence, oblique incidence (i.e. at angle θ), and random incidence, see Figure 3.. Normal Oblique Random / Diffuse Figure 3. Three types of incident waves. Typically, the absorption coefficient increases with increasing angle of incidence, up to a certain angle of θ. Beyond this angle, a decrease is usually observed. The explanation for this is the contribution of the so-called shear waves which propagate in the flexible porous material. This is in addition to the two dilatational waves in the porous material, see also Chapter 5. As a result, the absorption coefficient at normal incidence α n is slightly less than the absorption coefficient measured at random incidence α r for porous materials. Usually α n is measured in an impedance tube and α r in a room. The acoustic measurement techniques can be divided into three categories: Reverberant field methods Free field methods Impedance tube methods (Kundt s tube)

69 6 Reverberant field method The so-called reverberant field method is a well-known technique to measure the sound absorption coefficient for random incident waves. The experiments are performed in a reverberation chamber in which a diffuse sound field is generated (see for example Bies 996, ISO 354). There is a number of standards available for the procedures as well as for the geometry and dimensions of the test chambers. Usually a sound pressure field is generated with a uniform energy density. This is achieved with loudspeakers which are placed in the corners of the chambers and a number of diffusers to prevent the presence of standing waves in the chamber. A relatively large sample of the sound absorbing material (several m ) is placed in the chamber and for a given frequency band the reverberation time T 6 is measured. T 6 is the time it takes the sound pressure level to drop 6 db after shutdown of the loudspeakers. The same procedure is performed without the sample and the difference is a measure for the (Sabine) absorption coefficient. For highly sound absorbing materials the absorption coefficient can exceed the value of one because of extra energy loss due to edge effects and diffraction. This can also be the case if the sound field is non-diffuse. Various standards state that at least modes of vibration in the chamber are required in the lowest frequency band. As a result the room volume must be quite large. Nevertheless considerable differences have been observed for measurements on the same test materials in different reverberation chambers. It is therefore concluded that the reverberant field method is less suitable for accurately testing samples with broadband resonators, although it is the only method that applies diffuse sound fields. Free field method The free field method is commonly used for radiation measurements of sources of sound. The free field condition indicates that waves only propagate directly from the source of sound to the point of measurement. This condition can be approached in an anechoic chamber. In practical situations there is usually reflection from the ground. For these situations outdoor measurements above a reflecting plane can be made or a semi-anechoic chamber can be used where the floor is a reflecting plane.

70 6 A number of authors have proposed methods to measure the acoustic properties of sound absorbing materials at free field conditions (see for example Tamura 995 and 99, Allard 989a and 989b). In general the methods are suited for measurements with oblique incident waves. One technique is for example the pulse technique. A short signal is generated and the direct and reflected waves are separated to calculate the reflection coefficient. It is noted that the sample has to be placed outside the near field, which can pose a lower limit on the frequency band of interest, and on the dimensions of the samples (several m ). Another technique uses two microphones placed close to the sound absorbing surface. With this method it is possible to calculate the normal impedance at the surface for oblique incident waves. The area of the test material can be much smaller ( m ). For lower frequencies (about 5 Hz) however the size of the anechoic chamber may be a restricting factor because the source should be placed outside the near field. The possibility to measure the acoustic behaviour of sound absorbing materials at oblique incident waves is a strong advantage of the free field method. It was already mentioned that for oblique incident waves the shear waves which propagate in the sound absorbing material itself cause a different acoustic behaviour. However, for the material tested with tube resonators no shear waves were present. Furthermore, the resonators are locally reacting. It will be explained that it is sufficient to use the impedance tube technique to measure the normal impedance of a wall equipped with a number of resonators. Impedance tube method The most common technique used for measurements on sound absorbing material for normal incident waves makes use of an impedance tube. In Figure 3. a sketch of two techniques is shown.

71 63 Standing wave ratio technique Two-microphone technique movable microphone microphones pˆ B pˆ A pˆ B pˆ A loudspeaker test sample rigid tube backing plate Figure 3. Schematic representation of two measurement techniques in an impedance tube. At the left-hand side a loudspeaker is placed and at the opposite side a sample of the test material is placed. In the tube a standing wave pattern is formed: the result of a forward (or incident) travelling pressure wave with amplitude pˆ B, and a backward (or reflected) wave with amplitude pˆ A. The frequency of the sound waves is kept lower than the cut-off frequency (see section.) to ensure the generation of plane propagating waves in the tube. Earlier techniques made use of the measured standing wave ratio (SWR) for a specific frequency in the tube. By means of a movable microphone the ratio of the pressure maximum to the pressure minimum is determined. This ratio is then used to calculate the reflection coefficient and the acoustic impedance. An advantage of this method is that it is not necessary to calibrate the microphone. Drawbacks are the complex set-up with a movable probe and the time needed to find the maximum and minimum pressure for each frequency of interest. In 98 Chung and Blazer (Chung 98) presented a technique that is based on the transfer function of two fixed microphones which are located at two different positions in the tube wall (see right-hand side of Figure 3.). The standing wave pattern in this case is built up from a broadband stationary noise signal. With the measured transfer function the incident and reflected waves are separated mathematically. This leads to the reflection coefficient of the sample for the same frequency band as the broadband signal. The impedance and absorption coefficient can be derived as well. The method is as accurate as the SWR method and considerably faster. They also presented two techniques to improve the measured transfer function (see section 3..). The transfer function method has proven to be reliable and has been standardised (ISO 534-, 998). For sound absorbing materials the impedance measured with the method as described above strongly depends on the thickness of the material because sound

72 64 waves reflect at the backing plate. Therefore some authors advise the use of acoustic properties which are independent of the test configuration such as the characteristic impedance and the propagation coefficient in the material (see for example: Delany 97, Minten 988, Lauriks 989, Utsumo 989, Voronina 998, Iwase 998, Song ). One technique to derive these two coefficients is to measure the surface impedance of the material with two different thicknesses. For low frequencies the impedance tube method may not give accurate results because an airtight fit of the sample is needed and at the same time the sample has to be able to vibrate freely. This may also be a problem for higher frequencies when laminated materials or materials covered with a screen (for example a perforated sheet) are used. Furthermore, for a non-zero transverse contraction ratio (Poisson s coefficient) it is unlikely that a small sample is representative for a large area. For rock and glass wool, however, Poisson s coefficient is approximately zero. In an impedance tube normal incident waves are generated. As a result, only dilatational waves are generated in the sound absorbing material, whereas in general three types of waves are presen: two dilatational ones mainly in the air and in the flexible material and a shear wave. In the special case of a sample with a number of resonators it is sufficient to use only normal incident waves because the resonators are locally reactive. This means that the behaviour of one resonator is hardly influenced by the adjacent ones. To conclude: the impedance tube method is a well suited and simple method to verify the numerical results of Chapter. The two-microphone technique as mentioned earlier can be related to sound intensity measurements. This works as follows. Sound intensity is a vector quantity and represents the propagation of sound energy. It is the product of the pressure perturbation p and particle velocity perturbation u (see for instance Fahy 995). Note that the acoustic impedance is the ratio of the pressure and the velocity. The measurement of sound pressure is relatively straightforward but the particle velocity perturbation is usually estimated indirectly by using two closely spaced microphones. Via the momentum equation one can show that from the pressure gradient the particle velocity is approximated. The gradient is estimated using the two microphones (Euler s method). Measuring sound intensity has a number of advantages compared to sound pressure measurements. For instance the radiated sound power of a source can be

73 65 measured in situ instead of in a reverberation or anechoic chamber because it is possible to perform measurements with stationary background noise. A large number of new techniques on sound intensity measurements is listed in the literature and a number of standards has been developed (see for example Bies 996, Isaksson 998). 3. Impedance tube techniques In this section the impedance tube technique will be described for the twomicrophone technique as well as three new techniques which make use of a particle velocity sensor. It was explained that for the measurement of the acoustic impedance and the sound intensity the particle velocity perturbation is an important quantity. With the use of a particle velocity sensor, the so-called microflown, the impedance and the sound intensity can be measured directly (see also Druyvesteyn ). Four methods are presented: The p method. This is the standard technique which uses the transfer function of two microphones. The u method. Instead of two microphones two microflowns are used. The p/u method. The impedance is measured directly at the sensors. The p u method. The sound intensity and the energy density is measured and from these quantities the reflection coefficient of the sample is calculated. The working principle of the microflown is explained in section 3... Furthermore it will be shown that for the p/u and the p u method a special procedure is needed to calibrate the microflown with respect to the microphone. To test the four methods two types of impedance tubes are used. For convenience a tube with a circular cross-section and a tube with a square cross-section are used as shown in Figure 3.3.

74 R x 3 x x 3 x 66 transducers a sample a x 3 x x speaker Figure 3.3 Two impedance tubes, one with a circular and one with a square crosssection. Length of impedance tubes is approximately. m. For the impedance tubes several positions for the transducers can be used. Each separation distance between two sensors is chosen such that for a given frequency range less than half a wavelength fits between the two sensors. Table 3. gives the separation distances and the corresponding frequency range. Separation distance [m] Frequency range [Hz] x = x = x 3 = (cut-off freq.) Cross-section [m] Cut-off frequency [Hz] R =.5 c / (.7 R) = 4 a =.4 c / ( a) = 48 Table 3. Separation distances, cross-sections and frequency ranges for the impedance tubes. A frequency range above 4 Hz only becomes important for sound absorbing materials with a small thickness of approximately cm. In many applications such a material thickness is not a problem so that the sound absorption is sufficiently high above 4 Hz, i.e. approximately percent. For the present impedance tubes the cut-off frequencies are high enough to investigate the acoustic behaviour of common porous sound absorbing materials. For special material such as resonance absorbers the frequency range of interest corresponds in general to the range for porous materials. However, below that range extra sound absorption can be gained, within a limited volume, with resonance absorbers, i.e. an interest in the

75 67 low frequency range originates from the fact that resonance absorbers can provide an additional narrowband sound absorption where porous materials are usually less efficient (Heckl 995). For the impedance tube with the square cross-section additional sensors are used. In this way it is possible to measure for instance the transmission coefficient of a sample which is placed halfway in the impedance tube. The two large transducers represent a half-inch pressure microphone and a packaged half-inch microflown respectively. For the p method ¼ inch Kulite microphones are used. For the p/u and p u method a ½ inch B&K microphone and a ½ inch ICP probe from Microflown Technologies are used. It is noted that for standing waves with frequencies up to 4 Hz the average pressure at a ½ inch microphone is a good representation of the actual pressure at a point (at for instance x=l) because the area of the microphone is at most 5% of the wavelength. The microflowns are positioned inside the impedance tube but the size of the sensors is much smaller than the acoustic wavelength so that the effect on the wave propagation is negligible. It is noted that for accurate measurements the wall of the impedance tube has to be rigid otherwise the signals of the microphones may be affected by vibrations of the wall. Therefore the impedance tubes are constructed of 5 mm thick aluminium. Furthermore the distance from the sample to the sensors is kept larger than 4 times the width of the cross-section so that plane waves are predominant.

76 68 The complete measurement set-up to determine the transfer function is sketched in Figure 3.4. PC with FA Analyser card Function generator B&K 8 Oscilloscope Amplifier D-Tac software Amplifier B&K 76 Speaker Sensors Low-pass filter Impedance tube Figure 3.4 Measurement set-up. 3.. The p method In the p method the transfer function between two microphones is used. Figure 3.5 shows the one-dimensional representation of the impedance tube. p x x, x, x 3 p L sample Figure 3.5 Schematic drawing of the impedance tube. The measured transfer function is: p S p p H p = = (3.) p S p p where p and p are the pressures in the frequency domain, S pp is the crossspectrum and S pp is the auto-spectrum. The complex transfer function H p can be

77 69 measured directly with a two-channel FFT analyser, see Figure 3.4. With the theory for the one-dimensional wave propagation as presented in section. one can derive the reflection coefficient at the surface of the sample, i.e. at x = L: Γ k x H k L p e Γ e R( x= L) = (3.) Γ k x k L H p e Γ + e where the separation distance x is used. In the impedance tube the propagation coefficient Γ according to the theory of Kirchhoff is used because the impedance tube can be seen as wide, i.e. the shear wave number s». The absorption coefficient α and the impedance ζ at the surface of the sample are: + R H p sinh( Γ k L) sinh( Γ k( L x )) α = R ζ = = (3.3) R H cosh( Γ k L) cosh( Γ k( L x )) p The two microphones need to be calibrated to determine the gain and phase characteristics. However one can use sensor-switching to avoid this possible cause of errors (Chung 98). Measurements are performed a second time with the sensors exchanged. Furthermore it is possible to use a third sensor to improve the accuracy of the measurement of the transfer function. This technique makes use of three coherence functions between the signals of three sensors. In the experimental results presented in section 3.3 the transfer function improvement technique was not needed. 3.. The u method The u method is similar to the p method. However instead of two microphones two microflowns are used (see also Van der Eerden 998). First the microflown is briefly introduced and then the utilisation of the two microflowns in the impedance tube is described. Finally the governing equations for the u method are given. The microflown: an acoustic particle velocity sensor The microflown (or µ-flown) was developed at the department of Electrical Engineering of the University of Twente (de Bree 996). Instead of sound pressure the microflown measures the acoustic particle velocity, i.e. averaged over a small

78 7 finite volume of air. The microflown consists of two cantilevers of silicon nitride with a platinum pattern on top, see Figure 3.6. The size of the cantilevers is 8 4 µm (l w h). A large number of microflowns can be created on a single wafer. The measurement principle of the microflown is based on the temperature difference between two resistive sensors which are 4 µm apart as shown in Figure 3.6 and Figure 3.7. Two sensors on top (4 µm apart) S Acoustic waves S Acoustic wave Acoustic wave present Stationary situation Mass Power supply Die Signal Breaking groove S S Position Figure 3.6 Photograph of the microflown. Figure 3.7 Photograph of the sensors on top and estimated temperature profile. A travelling acoustic wave causes motion of the air and as a result heat is transferred from one sensor to the other (harmonically for a single frequency). This results in a temperature difference between the two sensors. This temperature difference causes a differential electrical resistance variation between the two sensors, which is measured. To realise a temperature difference which is high enough to determine the resistance variation the sensors are heated by a DC current up to about 4-6 K. The sensitivity of the microflowns shows approximately a so-called first order low pass behaviour. The corner frequency, above which the sensitivity drops with 6 db per octave, is between 3 Hz and khz. For high frequencies more averaging and a higher signal-to-noise level may be needed. Estimated temperature

79 7 More practical characteristics of the microflown are: The directional sensitivity varies cosine-like (as in a figure of eight). This means the sensors can be quite accurately aligned. No need for moving parts. Low cost for the transducer due to simplicity and batch size. The ability to measure the particle velocity in the near field where the sound intensity technique with two closely spaced microphones fails. The vulnerability of the two sensors. A protective package may be used (see also Figure 3.9 and Figure 3.). The microflown has to be positioned outside a boundary layer, whereas a microphone is usually flush-mounted in the impedance tube wall. An alternative design is shown in Figure 3.8. The cantilevers of the two sensors are supported on both sides for extra stability. Furthermore the sensors are smaller, to provide extra sensitivity. In Figure 3.8 a microflown with three sensor pairs is also illustrated to measure the three orthogonal velocity components. By combining the latter sensor with a microphone the measurement of the sound intensity vector becomes relatively easy. S S S S S S S S Figure 3.8 A bridge-type microflown (with sensors S and S). For one and three directions. Application in the impedance tubes The microflowns with the two sensors on top are used for the u method. They are used at the same positions as the microphones. The microflowns are mounted in a hollow bolt with the same thread as the Kulite microphones (see Figure 3.9). The

80 7 sensing cantilevers of the microflowns are placed about 5 mm from the tube wall. To avoid damage of the cantilevers a protective wire frame is used. Wire Frame ½ inch Microflown M5 bolt Figure 3.9 Photograph of the microflown placed in a bolt. Microflown Figure 3. A ½ inch microphone and a ½ inch packaged microflown (with enlarged view). For the p/u and p u method a ½ inch microphone and ½ inch microflown are used. The microflown is protected with a package which consists, among others, of two cylinders (see Figure 3.). As a result of the amplified acoustic flow between the two cylinders the sensitivity of the microflown is significantly increased (approximately 5 db). Data processing The measured transfer function is: u S uu H u = = (3.4) u Su u With the use of H u and the theory presented in section. the reflection coefficient R of the sample in the impedance tube becomes: Γ k x k L H u e Γ e ( x= L) = (3.5) Γ k x k L H u e Γ e R

81 73 See Figure 3.5 for the parameters s and L and note the difference with equation (3.). The absorption coefficient α and the impedance ζ are given in equation (3.3). The measurements were performed for sound pressure levels (SPL) below 3 db (ref. µpa). This implies that the theory of linear acoustics is still valid (see for instance Bies 996). If only microflowns are used in the impedance tube one can use the particle velocity level (PVL) to determine the level for which the linear theory breaks down. In Table 3. the PVL, the SPL and the sound intensity level (SIL) are listed. The standard reference levels for air are chosen in such a way that for a plane wave with a characteristic impedance of approximately 4 kgm - s - the three levels give the same result. For instance: 94 db PVL (ref. 5 nm/s) corresponds to a SPL of 94 db (ref. µpa). Sound level [db] Reference level Sound pressure level (SPL) prms SPL = log p ref = µpa pref Particle velocity level(pvl) Sound intensity level (SIL) urms PVL = log u ref = 5 nm/s uref Irms SIL = log I ref = pw/m I ref Table 3. Acoustic sound levels and corresponding reference levels The p/u method When a microphone and a microflown are positioned at the same cross-section of the impedance tube the acoustic impedance at that position can be measured directly. It is noted that the shear wave number is large in the impedance tube which is an indication for the wave front to be plane. Therefore the particle velocity can be measured at one point in the cross-section (outside the thin boundary layer). The position of the ½ inch probes is indicated by the large sensors in Figure 3.3. The transfer function H p/u is measured according to:

82 74 p S p u H p / u = = (3.6) u Su u where S pu and S uu are the cross-spectrum and auto-spectrum in the frequency domain. Measurements of the acoustic impedance with the p/u method were first performed by Schurer in 996 directly in the throat of a horn. However if the dimensions of the resonators and the dimensions of the sensors are considered then it is more convenient to measure the impedance of the sample some distance away. If the microphone and the microflown are located at x = then with the theory for the one-dimensional wave propagation the impedance at the sample, i.e. at x = L, is: H p / u cosh( Γ k L) ρc sinh( Γ k L) ζ ( x= L) = (3.7) ρ c cosh( Γ k L) + H sinh( Γ k L) p / u The absorption coefficient and the reflection coefficient of the sample can be derived from ζ. The amplitude and phase characteristics of the combination of the two different types of sensors need to be known. In Appendix C a calibration procedure for a microflown and a microphone is described. The calibration is performed in the same impedance tube with the reference microphone at the end of the tube. This method is used to calculate the experimental results presented in section 3.3. In the next section sound intensity and energy density are measured with the p u method. It will be shown that besides the transfer function also the auto-spectra of the microphone and the microflown are needed. Therefore both the reference microphone and the combination needs to be calibrated The p u method The magnitude of the reflection coefficient can be determined by measuring sound intensity and sound energy density. First the sound energy density and the sound intensity are described.

83 75 Sound energy density When viscothermal effects are negligible the problem of the propagation of small perturbations in air can be treated as a conservative elastic process. In general this is a reasonable approximation for audible frequencies in air. As a result the summation of kinetic energy of a fluid per unit volume T and potential energy per unit volume U gives the total energy per unit volume (Fahy 995): p( x, t) E = T + U = ρ u( x, t) + (3.8) ρ c where p and u are time- and space-dependent perturbations. E is also called the sound energy density. The time-averaged energy density E is obtained by replacing the pressure and the velocity by the root mean square values. In the frequency domain, and by knowing that p rms = ½ p p *, we find: * * p p E = ρ u u + (3.9) 4 4ρ c where the superscript * indicates the complex conjugate. By using the root mean square values the energy density is independent of the position (the viscothermal losses are neglected). Sound intensity The instantaneous normal sound intensity I n (t), where the subscript n indicates the normal direction, is defined as the work rate per unit area δ S normal to that area and can be written as: ( dw dt) I n ( t) = = p( x, t) u( x, t) δ S n (3.) For one-dimensional plane waves the relation between the instantaneous pressure and velocity perturbations is: + + = p( x, t) ρ c and u( x, t) = p( x, t) ρ (3.) u( x, t) c

84 76 where the superscripts + and - refer to the components propagating in the positive and negative x-direction, respectively. So the sound intensity in the plane waves is (with the x- and t-dependence implicit): + ( p ) ( p ) ) ρc I = (3.) The time-averaged sound intensity I is obtained by using the root mean square pressures. For a pure progressive wave, i.e. when no reflections are present, one can derive: prms I = = c E (3.3) ρ c while for a standing wave the time-averaged intensity I = and I / E c. In this case the instantaneous sound intensity represents a purely oscillatory flow of sound energy. In general the sound intensity may be split into two components (Fahy 995): an active component of which the time-averaged value is non-zero and a reactive component of which the time-averaged value is zero. The active component indicates that there is a local net transport of sound energy and it is this value that is usually measured. The time-averaged reactive component represents a local oscillatory transport of energy. If the complex amplitudes for the pressure p and the velocity u are used in the frequency domain one has: * I = pu = I + i active I reactive (3.4) where the time-averaged active or mean intensity I active and the amplitude of the reactive intensity I reactive are: I * = Re{ pu } and Ireactive Im{ pu * } (3.5) active = Again it is noted that the mean intensity in a one-dimensional plane wave is independent of position. The amplitude reflection coefficient Fahy (Fahy 995) showed that in a tube the mean speed of the energy transport e c is the ratio of mean intensity to the mean energy density:

85 77 I R = = c (3.6) c e E + R where R is the amplitude reflection coefficient (see equation.6 in section.) of the end of the tube, see Figure 3.. Equation (3.6) forms the basis of the p u method, but now we are interested in the amplitude reflection coefficient. microphone amplitude reflection coefficient R pˆ A pˆ B microflown Figure 3. Set-up of the impedance tube for the p u method. The pressure and the velocity in the impedance tube are given as (note that in contrast to Chapter here Γ = i): p( x) = pˆ B pˆ u( x) = B ρ c ik x ik x { e + R e } ik x ik x { e R e } with R = pˆ pˆ A B (3.7) where pˆ B is the amplitude of the incident wave. The mean intensity and the sound energy density can now be written as: I active = Re{ E = ρ u u 4 * * p p + 4ρ c * pˆ pu } = B ρ c pˆ B = ρ c ( R ) (+ R ) (3.8) With the equations (3.6) and (3.8) the amplitude reflection coefficient and the sound absorption coefficient can be calculated: E c Iactive R = and α = R (3.9) E c + Iactive

86 78 However, in a long impedance tube the viscothermal losses during the wave propagation become important. This is especially the case for higher frequencies via e Γ k x. Therefore, if I and E are measured at x =, then the reflection at x = L becomes, with the help of (3.7): R ( x Γ k L e = L) = R ( x= ) (3.) Γ k L e The acoustic impedance at the surface of the sound absorbing sample cannot be derived from the sound intensity and the energy density because of their spatial independent character. For ordinary impedance tube measurements meant to determine α or R this is not a drawback. However, in finite element or boundary element calculations the full complex impedance, as derived by the experiments, is often required as a boundary condition for sound absorbing surfaces. In that case one of the other measurement methods has to be used. The energy density is measured to eliminate the amplitude dependence pˆ B in the mean intensity. The auto-spectra in E require a calibration of both sensors (see Appendix C) and the sensor-switching method cannot be applied here. Data processing With a two-channel FFT analyser, Iactive Iactive = Re{ S pu} and E are measured as follows: E = ρ Suu + S p p ρc (3.) where S pu is the cross-spectrum and S uu and S pp are the auto-spectra. For the sake of completeness it is noted that I reactive = Im{S pu }. The p method is the most commonly used technique to estimate the particle velocity. For sound intensity measurements two microphones are separated by a distance δ with a spacer. In a one-dimensional field the mean intensity is: I = Im{ ρ ω δ (3.) active S p p }

87 Experimental results This section presents the experimental results for the different measurement techniques. First the results of the p and the u methods are compared. A sample with an acoustically hard wall is initially used because its behaviour is well known. Next the sound absorption of a quarter-wave resonator is measured. It is shown that both methods provide accurate results. In fact the latter results are already an indication that the model on viscothermal wave propagation in coupled tubes is correct. In section 3.3. the p method, the p/u method and the p u method are compared. For this purpose measurements with a quarter-wave resonator are performed. The measured reflection coefficient of the sample serves as a comparison Comparison of the p and the u methods An acoustically hard wall A loudspeaker was set up to generate a random signal in frequency bands of 4 Hz as shown in Table 3.3. The overlap between the frequency bands is used because the FFT analyser makes use of a Hanning window. The first and last 5 Hz of each band are omitted to obtain the complete transfer function from 45 to 395 Hz.

88 8 Lower bound [Hz] Upper bound [Hz] Sensor spacing [m] 4 8 x = x 3 = Table 3.3 Set-up of the frequency bands and accompanying sensor spacing. In Figure 3. and Figure 3.3 the measured transfer functions with the p and the u methods are shown and compared to the theoretical ones. H pp [ ] H uu [ ] Microphones Theory (p) Frequency [Hz] Figure 3. Transfer function for an acoustically hard wall using the p method. Microflowns Theory (u) Frequency [Hz] Figure 3.3 Transfer function for an acoustically hard wall using the u method. The standing wave pattern for different frequencies in the impedance tube has a number of pressure nodes and velocity nodes located at the positions of the sensors. A maximum in the transfer function corresponds to a node (pressure or velocity) at the first sensor and a minimum to a node at the second sensor, see also Figure 3.5. In this case the location of a pressure node differs from the location of a velocity

89 8 node by a quarter of a wavelength, which explains the differences between the two transfer functions. With the use of these peaks and corresponding frequencies the lengths in the impedance tube, L, x and x 3, can be determined very accurately, provided that the mean speed of sound c is known in advance, as c is a function of the temperature. A comparison of the theory and the measurements resulted in more accurate lengths of: L =.99 m, x =.449 m and x 3 =.5 m. These lengths were again used in the theory presented in Figure 3. and Figure 3.3. It is also noted that the peak heights of the theory match reasonably well with those of the measurements. A large number of data points are used in the figures to provide an accurate comparison. The peak heights are determined by the viscothermal losses which are predicted well. The viscothermal effects can also be seen in the phase angle of the transfer function, see Figure 3.4 and Figure Microphones (p) Theory (no viscothermal eff.) 9 Microflowns (u) Theory (no viscothermal eff.) arg(h pp ) [deg] 9 arg(h uu ) [degrees] Frequency [Hz] Figure 3.4 Phase of the transfer function for an acoustically hard wall using the p method Frequency [Hz] Figure 3.5 Phase of the transfer function for an acoustically hard wall using the u method. The jump in Figure 3. and Figure 3.3 at 8 Hz is caused by the use of two sensor distances x and x 3. The measured and theoretical reflection coefficients are derived via the transfer functions. In Figure 3.6 and Figure 3.7 the results for the p and the u method are presented.

90 8.8.8 R [ ].6.4 R [ ].6.4. Microphones Theory (p) Frequency [Hz] Figure 3.6 Magnitude of the reflection coefficient of an acoustically hard wall (p method).. Microflowns Theory (u) Frequency [Hz] Figure 3.7 Magnitude of the reflection coefficient of an acoustically hard wall (u method). For both methods the experimental results correspond to the theoretical reflection coefficient of R =, except for the higher frequencies where some deviation occurs. It was seen that the loudspeaker cannot generate the same pressure levels in the tube for these frequencies so that the signal-to-noise ratio for the transducers is less. The oscillating variation is the result of a frequency dependence to calculate R. To be more specific: with two sensors the two unknown forward and backward travelling waves are determined. The condition number of the resulting matrix is frequency dependent and shows a maximum which corresponds to the maximum deviation of R. Also, from equation (3.3) follows that for high values of the impedance ζ the denominator approaches zero. This causes numerical problems and incorrect values of the impedance. It was seen that an inaccurate transfer function can lead to a negative real part of the impedance which is physically impossible. It is remarked that the both the microphones and the microflowns as used in the measurements have an almost equal sensitivity. As a result no calibration factors were used. A single quarter-wave resonator The same procedure for the measurements is followed but now an aluminium sample with a single quarter-wave resonator is placed in the impedance tube. The length of the resonator is 6 mm and the radius is 5 mm. Due to the inlet effects the

91 83 effective length of the resonator is 63 mm (see section.4.). The porosity Ω of the sample is.4. In Figure 3.9 and Figure 3. the measured and theoretical absorption coefficient α is given for the p and the u method, respectively. It can be seen that both methods provide accurate results for the frequency range of interest, i.e. from to 7 Hz. Furthermore the absorption coefficient can be predicted exactly for each frequency which indicates that the model for the viscothermal wave 5 mm 6 mm Figure 3.8 Sample with a quarter-wave resonator. propagation is accurate. For the shear wave number one has s > so that the Kirchhoff approximation is used. 5.8 Microphones Theory (p).8 Microflowns Theory (u) α [ ].6.4 α [ ] Frequency [Hz] Figure 3.9 Absorption coefficient of a single resonator (p method). 4 6 Frequency [Hz] Figure 3. Absorption coefficient of a single resonator (u method). The quarter-wave resonator consists of a single tube which is closed at one end. It appeared that the closed end can be approximated more accurately by an impedance condition of ζ = instead of ζ =. The fact that not exactly all the energy was reflected is caused by the presence of a wall which is almost acoustically hard. The small disturbances or peaks in the figures are the result of the in-series connected frequency bands of 4 Hz. It is noted that the repeatability of the experimental results is very good and that both the p and the u methods are very well suited to obtain the absorption

92 84 coefficient of sound absorbing samples. Also, the microflowns can be seen as a good alternative to microphones in an impedance tube. By considering the similarity of the theoretical results and the experimental results for the sound absorption coefficient it is concluded that the theoretical model for viscothermal wave propagation in two axially coupled tubes, the impedance tube and a resonator, is accurate for frequencies ranging from to 7 Hz Comparison of the p, the p/u, and the p u methods Results of the p/u method and the p u method are compared to the ones of the p method in this section (see also: de Bree ). The experimental results are again presented for a sample with a quarter-wave resonator because its acoustic behaviour is well known. The impedance tube with the square cross-section as shown in Figure 3.3 is used for both the p, the p/u, and the p u method. The single quarter-wave resonator has the following dimensions: the radius R = 4.55 mm and the effective length L+d = 7.5 mm. The first resonance frequency of this resonator lies at Hz and absorbs approximately 8% of the incident energy at this frequency. For the measurement set-up broadband noise is generated by a DSP SigLab - 4 box. This box is also used as a front-end for the two input signals. The SigLab box is connected to a PC which runs the SigLab software under MatLab. The following figures show the results for the amplitude of the reflection coefficient R for both the measurements and the theory. The frequency range of the generated noise is 5 to 45 Hz. However in the figures a limited frequency range is depicted from 5 to Hz around the first resonance frequency of the resonator. It is noted that R is calculated at the surface of the sample.

93 85.8 R [ ].6.4. Measurements (p) Theory α 5 5 Frequency [Hz] Figure 3. Reflection coefficient (magnitude) using the p-method..8 R [ ].6.4. Measurements (p/u) Theory 5 5 Frequency [Hz] Figure 3. Reflection coefficient (magnitude) using the p/u method.

94 86.8 R [ ].6.4. Measurements (p.u) Theory 5 5 Frequency [Hz] Figure 3.3 Reflection coefficient (magnitude) using the p u-method. The figures show that the three methods provide identical results for the frequency range where the quarter-wave resonator is effective, i.e. from to 4 Hz. Furthermore the theory predicts the acoustic behaviour of this quarter-wave resonator in a sample very well. The new p/u and the p u measurements show some oscillating inaccuracies. The results may be further improved with a more accurate correction function as obtained from the calibration. In Appendix C it is shown that the correction function is only a third order polynomial which is fitted through the experimental results of the calibration. For the frequencies below Hz it can be seen that the correction function deviates somewhat from the calibration results. The oscillating variation is the result of a frequency dependence to calculate the correction function (the condition numbers of the matrix for the p/u and the p u method itself do not depend on the frequency). In this respect it has to be remarked that the calculation of R for an acoustically hard wall is very sensitive to measurement inaccuracies. For the p u method the imaginary part of the reflection coefficient cannot be measured. However when only the absorption coefficient α (α = - R ) or the magnitude of the reflection coefficient R is needed this is not a drawback. For the other two methods the complex reflection coefficient is calculated from which the acoustic impedance can be derived. The complex results for R are shown in Figure

95 The frequency range of 5 to 4 Hz is used so that also the second resonance frequency at 36 Hz can be seen (three-quarters of a wavelength). R [ ] Measurements (p/u). Theory (at x = L) Theory (at x = ) Frequency [Hz] arg(r) [deg] Measurements (p/u) Theory (at x = L) Theory (at x = ) Frequency [Hz] Figure 3.4 Reflection coefficient (magnitude and phase) of a quarter-wave resonator. If R is calculated at x = instead of at the surface of the sample, at x = L, then the viscothermal wave propagation in the impedance tube causes an extra energy loss over the length L. In the case of Γ = i the difference is negligible. 3.4 Conclusions It was shown that the impedance tube technique is an efficient and inexpensive method to quantify the acoustic properties of sound absorbing material for normal incidence. Moreover, it is sufficient to measure the sound absorption with only normally incident waves because the resonators in the samples are locally reacting. The frequency range of interest is between 5 and 4 Hz which is large enough to cover most practical problems in noise control engineering. Besides the standardised p method with two microphones three additional methods have been presented and tested. These new methods make use of a new acoustic particle velocity sensor: the microflown. The u method corresponds to the p method but uses two microflowns instead of two microphones. It gives the same accurate results. With the p/u method the impedance at a cross-section in the impedance tube is directly measured. Next, one can calculate the impedance and sound absorption coefficient at the surface of the sample with the model for the one-dimensional viscothermal wave propagation. However, a more accurate

96 88 calibration for the combination of a microphone and a microflown, compared to the one as used in Appendix C, needs to be used. The p u method measures the acoustic energy density and the sound intensity in the impedance tube. The combination is a measure for the reflected energy and so the amplitude reflection coefficient of the sample can be calculated. The results for the reflection coefficient of an acoustically hard wall and the sound absorption coefficient of a sample with a quarter-wave resonator were presented in section 3.3. It was seen that the four measurement techniques give the same results. Furthermore, the experimental results agree very well with the theoretical results. So the viscothermal wave propagation is correctly modelled for this simple case of an impedance tube coupled to a single quarter-wave resonator. For a more profound validation of the coupled tubes model the reader is referred to the results presented in Chapter 4.

97 Chapter 4 Experimental verification of the coupled tubes model 4. Introduction Sound absorption of single tube resonators Sound absorption of coupled tubes resonators Conclusions Introduction In Chapter coupled tubes in a wall were presented. In the coupled tubes a broadband resonance is present so that broadband sound absorption is created. It was explained that the incident waves from a source of sound are cancelled by the waves in the resonator as well as damped by the viscothermal losses in the resonator. The present chapter describes the experimental verification of the coupled tubes model. The experiments were performed in an impedance tube. Reference is made to Chapter 3 for a description of the impedance tube technique. Some results are obtained by using particle velocity sensors, so-called microflowns, instead of microphones. The microflown is also described in Chapter 3. To verify the predictions of the coupled tubes model a number of samples with resonators were constructed. In Figure 4. the measurement set-up and a number of samples is shown. In section 4. the sound absorption coefficients of samples with single tube resonators are presented. The first sample consisted simply of a single quarter-wave resonator. Next the number of resonators per unit area was varied to examine the effect of the porosity parameter Ω of a wall with resonators. The effect of different lengths of the resonators was briefly investigated. The quarter-wave resonators were closed at one end and as a comparison the sound absorption coefficient for resonators with an open end was also measured. The latter resonators can be described as half-wave resonators.

98 9 Filters Oscilloscope Impedance tube Amplifiers Speaker Figure 4. Photograph of the measurement set-up and of a number of samples. As a next step three samples with coupled tubes resonators were tested. These samples show the broadband sound absorption as predicted in Chapter. The resonators in the first sample simply consisted of two axially connected tubes and operated between 5 and Hz. The second sample was designed to absorb at least 8 percent of the incident energy over an even broader frequency band. The resonators consisted of a more complex configuration of coupled tubes. Finally, the third sample was designed for a higher frequency range, i.e. from 7 to 4 Hz. 4. Sound absorption of single tube resonators A single quarter-wave resonator An aluminium sample with a single quarter wave resonator was placed in the impedance tube. The length of the resonator was 5 mm 6 mm 5 6 mm and the radius 5 mm. Due to inlet effects the effective length of the resonator was 63 mm (see section.4.). The porosity Ω of the sample was.4. In Figure 4.3 the measured and predicted absorption coefficients α are given for Figure 4. Sample with a quarter-wave resonator. the two-microphone technique (p method). The measurements and the theory agree very well. It can be seen that indeed the quarter-wave resonator absorbs the incident energy for a frequency range which is

99 9 much wider than the quarter-wave resonance frequency of 35 Hz. This is explained by the viscothermal effects in the resonator, see Chapter. Furthermore the measurements demonstrate that the absorption coefficient can be predicted very well for the whole frequency range indicating that the model for viscothermal wave propagation is accurate..8 Measurements (p) Theory α [ ] Frequency [Hz] Figure 4.3 Absorption coefficient of a single quarter-wave resonator (p method). Ω =.4 The quarter-wave resonator consists of a single tube which is closed at one end. Due to a not perfectly hard wall it is expected that not exactly all energy is reflected from the closed end. Therefore in the application of the theory for practical calculations the closed end is approximated with an impedance condition of ζ = instead of ζ =. The effect of the surface porosity Ω The sound absorption coefficient α for samples with a different number of resonators per unit area is investigated. The length of the resonators is 5 mm and for the effective lengths the theory of section.4. is used.

100 9.8 Microphones Theory =.4 α [ ].6.4. Ω =.4 Ω =.6 =.6 =.36 Ω = Frequency [Hz] Figure 4.4 Sound absorption coefficient of samples with a different porosity Ω (p method). Figure 4.4 shows that the predicted results correspond very well with the experimental results. So the theory predicts the effect of the porosity correctly. Moreover, the peak in the absorption coefficients shifts to the right when more resonators are used. So the effective length, which depends on the distance between the resonators, is also correctly predicted by the theory, i.e. more resonators per unit area lead to higher effective resonance frequencies. Figure 4.4 also shows that in this particular case more resonators do not lead to a higher sound absorption coefficient. To realise α = at a specific frequency one needs to tune the porosity, i.e. the number of resonators per unit area. Then the amount of dissipated sound energy is equal to the energy of the incident waves, and the harmonic mass flow at the entrance of the resonators balances the flow of the incident waves (see section.4.). The effect of different lengths In Figure 4.5 the sound absorption coefficient of a sample with two quarter-wave resonators of different lengths is depicted. As expected the effect of the each individual resonator can clearly be seen. Also here the results of the theory correspond very well with the experimental results.

101 93 It was demonstrated in Chapter that the combination of two separate quarterwave resonators does not provide the broadband resonance of coupled resonators. This was not the objective of the present sample as the values of the two resonance frequencies are too far apart. The somewhat noisy experimental results at low absorption coefficients are a minor effect because then the sample acts approximately as an acoustically hard wall. It was demonstrated in section 3.3. that the impedance tube technique is relatively sensitive to measurement errors in that case..8 Microphones Theory 6 5 α [ ] Frequency [Hz] Figure 4.5 Sound absorption coefficient of a sample with two resonators of different lengths (p method). A half-wave resonator The next sample contains two tubes which are open at both sides. These tubes can be seen as half-wave resonators. An advantage of resonators which are open at both ends is that they can act as an acoustic filter and yet can let pass sound, a mass flow or light. In Figure 4.6 the sound absorption coefficient is shown with an absorption peak at about 6 Hz. It is noted that the effective length of the resonators consists of the geometrical length plus two small end corrections d at both ends of the resonators. At the open end the impedance ζ = is prescribed and the correction length for a piston was used as an estimation. The total correction length was 4. mm.

102 94 α [ ] Microphones Theory 5 d 5 d. =. d = 4. mm Frequency [Hz] Figure 4.6 Absorption coefficient of a sample with two half-wave resonators (p method). The measurements prove again that the viscothermal model for coupled tubes is accurate. Also the resonance frequency is predicted well via the application of an effective length. The results for the single tube resonators demonstrate that a very good agreement exists between the predicted acoustic behaviour and the measurements. The viscosity and the thermal conductivity of the air in the resonators play an important role when sound absorption is considered. Evidently the treatment of viscothermal effects was quite adequate over the complete frequency range in this case. The same test strategy was followed for samples with resonators consisting of coupled tubes. 4.3 Sound absorption of coupled tubes resonators The measured sound absorption coefficients of three samples with coupled resonators are compared to the predicted absorption coefficients. The first sample has a high absorption coefficient for frequencies between 5 and Hz. The second sample contains a more complex configuration of coupled tubes and is effective in the same frequency range with an absorption coefficient of α >.8. The third sample is optimised for frequencies between 7 and 4 Hz.

103 95 Sample : 5 Hz The sample has three equal resonators which consist of two coupled tubes. The sample is designed for the frequency range of 5 to Hz. Therefore the tube at the entrance has a diameter of mm and the coupled tube has a diameter of.5 mm. The experimental and analytical results are given in Figure 4.7. α [ ] Frequency [Hz] Microphones Microflowns Theory Theory: =. L = mm L = mm = Figure 4.7 Sound absorption coefficient of a sample with three coupled tubes. It is seen that the sample is indeed effective; the theory agrees well with the measurements for the specified frequency range. This indicates that the viscothermal effects and the coupling of tubes are correctly modelled. The value of the sound absorption coefficient is below the maximum of.. It is noted that the optimal porosity for a sample with these resonators can be higher or lower than the arbitrary value as used for this sample. The impedance of the wall can be used to determine the optimal porosity. The small deviations for α in the specified frequency range are probably due to errors in the geometry and the estimated inlet effects. The effective length L is obtained by using the end correction for a wall with equally spaced resonators (see equation (.57) with a = 6 mm). For L the end correction of a single tube centrally located in a larger tube is used (equation.58). The end condition for the smaller tube is estimated using ζ =.

104 96 It is also shown that the p method and the u method provide similar results. However, there is some noise in the results at higher frequencies. The sample acts as an acoustically hard wall for these frequencies and therefore the measurement methods provide less accurate results. Sample : 3 Hz The second sample as shown in Figure 4.8 was designed for a broader frequency range than the previous sample, i.e. a sound absorption of more than 8 percent from 3 to Hz. Hence an extra tube was coupled to the tube at the entrance. Note that the measured frequency range differs from the range of Figure 4.7. α [ ].8.6 Microphones Microflowns Theory 5.4 Theory: =.8. L = mm L = mm L 3= mm = Frequency [Hz] Figure 4.8 Sound absorption coefficient of a sample with coupled tubes The experimental results prove that the desired more broadband sound absorption is indeed achieved. It also shows that the coupled tubes model predicts the amount of absorption as well as the actual frequency band very accurately. Therefore it is concluded that the theoretical model can be applied reliably for this frequency range to design sound absorbing walls with coupled tubes resonators for a predefined frequency range.

105 97 Sample 3: 7 4 Hz The sample designed for a high frequency range from 7 to 4 Hz is illustrated in Figure 4.9. The demands on the level of sound absorption are stringent for this sample: a maximum sound absorption of % at two frequencies and around these frequencies 9%. The sample was constructed from a perforated brass cylinder connected to a back-plate with capillary tubes. The sample had 8 tubes with a diameter of mm. Three of the tubes were used to connect the backplate to the sample by means of three pins. The remaining 5 capillary tubes in the back-plate were closed at one end with beeswax. As a result the length of the capillary tubes was somewhat reduced. In Figure 4.9 the results are presented. The word gap in the legend indicates that in that case there is a small gap between the back-plate and the brass sample of. mm. The latter results indicate the effect of leakage of the coupled resonators. brass sample.8 α [ ].6.4. Microphones (beeswax) Microphones (gap) Theory Frequency [Hz] =.48 capillary tube 4 backplate Figure 4.9 Sound absorption coefficient of a sample with coupled tubes. Dimensions as indicated in the sketch in mm. The experimental results indicate that the level of sound absorption meets the design requirements well. The sample indeed shows a high sound absorption over the specified frequency range. Some differences can be seen with the predicted results. These are attributed to minor deviations of the geometry of the resonators. For instance the small amount 5 5.5

106 98 of beeswax inside the capillary tubes causes the higher resonance peak (at 36 Hz) to shift somewhat to the right. From the construction point of view it is noted that a small gap between the connected tubes has little effect on the sound absorption coefficient, i.e. around the resonance frequency of the capillary tubes there is a pressure minimum at the entrance and this is precisely the location of the small gap. So fortunately a small gap has little effect on the sound absorption behaviour. It is noted that the repeatability of the experimental results is very good for the samples presented in this section. In the next chapter the coupled tubes model will be used for a comparison with models for porous materials. It will be shown that the acoustic behaviour in the pores can be approximated well with the coupled tubes model. The numerical results of the models found in the literature will be compared to the results of the coupled tubes model. 4.4 Conclusions The experimental and the theoretical results for the sound absorption of samples with single tubes, i.e. quarter-wave and half-wave resonators, and samples with coupled tubes were presented in sections 4. and 4.3, respectively. The different samples demonstrate the effects of different lengths, a varying porosity, half-wave resonators, and coupled tubes. It was also shown that the samples with coupled tubes resonators can create a more broadband sound absorption. Moreover, the measurements provided test-cases for the theoretical results. It was seen that the coupled tubes model predicts the sound absorption of the samples very accurately for the complete frequency range. This indicates that the model for the viscothermal effects in the tubes and the coupling between tubes represents physics well. Finally, it is concluded that the coupled tubes model is a reliable tool for the design of a sound absorbing material with resonators for a specified frequency range. In Figure 4. simulated results for coupled tubes resonators are summarised.

107 99 α [ ] Single Double Multiple Triple Frequency [Hz] Figure 4. Numerical sound absorption coefficients of a wall with different configurations of resonators for optimal absorption (see also Chapter ). For a physical interpretation of the more broadband resonance and the high sound absorption of the resonators the reader is referred to Chapter and Appendix A.

108

109 Chapter 5 Sound absorbing material represented by a network of tubes 5. Introduction 5. A random network of tubes Comparison to an empirical impedance model Comparison to the Limp theory Comparison to the Biot theory Conclusions and remarks 5 5. Introduction The coupled tubes model of Chapter provides an interesting opportunity to predict the acoustic behaviour of sound absorbing materials consisting of a network of tubes. In the present chapter this approach is described, i.e. sound absorbing material is numerically represented by a labyrinth-like distribution of a large number of tubes. It will be shown that with a limited number of parameters the predicted acoustic behaviour corresponds well with the results of other, more sophisticated, numerical models. Conventional sound absorbing materials such as fibrous material (i.e. glass wool or rockwool) and porous material (i.e. with pores) such as foams, are commonly applied to reduce noise problems. The acoustic behaviour of these materials can be measured to determine which one is the most effective for a particular application. Obviously, it is more effective to define a set of parameters governing the sound absorbing properties of such materials and to predict the acoustic performance in advance. The comparison between a numerical network of tubes, the so-called network description, and models from the literature has been made for a simple case with a normal incident sound field. It is noted that, among other models, the network

110 description can also easily be used for oblique or random incident waves because the wave propagation within the volume of the material is captured as well. This is in contrast to an impedance description which concerns the surface of the material. An advantage of a surface description is its limited calculation time in, for instance, finite element type of calculations. Drawbacks are its restriction to locally reacting materials and normal incident waves. Furthermore, the model is limited to a class of materials, for instance fibrous rockwool materials, due to its empirical character. The volume description actually takes into account the interaction between the frame of the material and the fluid trapped in the frame. The frame can be seen as the skeleton of the sound absorbing material and often physical parameters are given for the frame instead of for the solid material of which the frame is constructed. In Table 5. a survey of volume descriptions as used in the present investigation is given. A more extensive description of the Limp and the Biot theory, including some appropriate references, is given in sections 5.4 and 5.5. Rigid Limp Biot Network Approach Volume Volume Volume Volume Frame Infinitely stiff (no mass) No stiffness (with mass) Poisson s ratio Young s mod. Infinitely stiff Mechanisms Inertia of fluid Inertia of frame Viscous effects Thermal effects Wave types x dilatational x dilatational x dilatational x shear Table 5. Comparison of models for sound absorbing materials. x dilatational For the theories as described in the literature a subdivision can be made in increasing order of complexity: the so-called Rigid theory treats the frame as infinitely stiff, the Limp theory treats the frame without stiffness but includes the mass, and the Biot theory includes both the stiffness, mass and the acousto-elastic coupling between the frame and the fluid. For the volume description of Biot three types of waves with different wave numbers are present inside the material. This is the result of the extra degrees of freedom for the frame. Measurements have

111 3 indicated that indeed three types of waves propagate in materials with an elastic frame and a fluid phase (Geerits 993). Although the Biot theory is the most comprehensive one, the large number of parameters needed to describe the bulk material and some difficult techniques to obtain these parameters are drawbacks. Furthermore the computational effort in finite element calculation is high because of the number of degrees of freedom per node. In the Rigid, Limp and Network descriptions the pressure perturbation in the fluid is the only degree of freedom. It will be shown in the next section that the network description provides an easy to use volume description of sound absorbing material with a limited number of parameters. The description is based on the coupled tubes model as presented in Chapter. Furthermore, the results can easily be translated to an efficient surface description (for various angles of incident waves). In the subsequent sections the results of the network description are compared to: an empirical impedance description, the Rigid and Limp theory, and the complex Biot theory for normal incident waves. 5. A random network of tubes Two examples of a random network of tubes as applied in the present calculations are depicted in Figure 5.. Each tube is identical, has length L and radius R and is represented by a short line. The length and radius of the tubes can be seen as equivalent ones for the pores inside the material. The tubes are connected to each other (see Figure 5.) to form a type of network. Tubes which end inside the material are considered as being closed with an acoustically hard termination. At the left-hand side of the network incident waves are assumed via a constant pressure for normally incident waves or via a complex pressure distribution for oblique incident waves. At the right-hand side an acoustically hard wall is prescribed to simulate a sample of material in an impedance tube. Other boundary conditions, such as an open end, can also be prescribed. Both networks are constructed from a so-called full network of tubes (here: 3x9 tubes on a square grid). To create a labyrinth-like network tubes are randomly deleted from the full network. In this way the two networks as shown in Figure 5. were obtained. Only 85 percent of the full network is shown on the left-

112 4 hand side and on the right-hand side this is 7 percent. To eliminate tubes from the full network a uniform distribution of random numbers was used..38 Length [m].38 Length [m] Figure 5. Examples of a random network of coupled tubes (85% and 7% of full network, thickness (or length) of the samples is indicated along the horizontal axis). The thickness of the sample of the sound absorbing material is determined via the number of horizontal tubes. The height of the material is not specified. Instead the porosity Ω of the sample is used so that the acoustic behaviour per unit area is calculated. Summarising: the four main parameters for the network to describe acoustic materials are: L, R, Ω and the percentage % of the full network. These parameters describe the bulk behaviour of a sound absorbing material. In general the porosity of a material is known. The equivalent length, radius and percentage can be found by a comparison to experimental data. It will be shown that the It was found that in an independent study a similar two-dimensional network has been used for the drainage dominated flow in porous media (Aker 998). There a full network is used with a random variation of the radius of the pores.

113 5 percentage and the radius are the main bulk parameter to describe a sound absorbing material. The computed sound absorption coefficients for various samples of the type as depicted in Figure 5. are shown in Figure 5. and Figure 5.3. The total network had a thickness of 38 mm. The length and radius of each tube were chosen as L =. mm and R =. mm. The surface porosity had a value of Ω =.95. For sound absorbing materials a volume porosity of.95 is common and it is assumed here that the surface porosity equals the volume porosity. The results show a striking resemblance with the measured absorption coefficient of two samples of glass wool (5 mm and 45 mm thick) as shown earlier in Figure.7. In Figure 5. the effect of the density of the distribution of tubes is shown. A combination of the viscothermal losses in the tubes and the effect of the labyrinthlike structure is the cause for this sound absorbing behaviour (see also section 5.3 for the effect of the so-called flow resistivity)..8.8 α [ ].6.4 Full network. 85 % of full network 7 % of full network 55 % of full network 3 4 Frequency [Hz] Figure 5. Sound absorption coefficient for various networks of tubes. L=. mm, R =. mm, Ω=.95, thickness = 38 mm. α [ ].6.4 Basic configuration Thickness = 76 mm. L =. mm Ω = Frequency [Hz] Figure 5.3 Sound absorption coefficients. Basic configuration: L=. mm, R =. mm, Ω=.95, 7%, thickness = 38 mm. An interesting example to apply the network description to is aluminium foam (Banhart 994). The frame is rigid and the porosity Ω is known in advance. The main bulk parameters % and R need to be determined. See also the paper of Lu, Chen and He ().

114 6 In Figure 5.3 the three other parameters are varied:. The thickness of the sample is doubled (3x38 tubes, thickness = 76 mm). As a result also lower frequency sound waves are more absorbed.. The length of each tube is set to L =. mm. This change has only a small effect on the sound absorption. 3. The porosity Ω =.8. As a result the sound absorption is slightly lower. The result of a different radius R for the tubes is shown in Figure 5.4. The network with R =. mm and Ω =.95 is again the basic configuration (thickness = 38 mm, 3x tubes)..8 α [ ].6.4 R=.4 mm. R=. mm R=. mm R=.5 mm 3 4 Frequency [Hz] Figure 5.4 Sound absorption coefficient for different radii of the pores. The effect of the radius is significant. This is not surprising since the viscous effects are governed by the shear wave number s which is proportional to the radius R. As follows from Figure 5.4 the slope in the low frequency range is determined by the radius. At higher frequencies the viscosity effects for a radius of R =.5 mm are still too high and acoustic waves are not able to penetrate sufficiently in the network, i.e. the jump in the impedance at the interface with the numerical sound absorbing material is too large. The absorption coefficient of the samples is calculated with the transfer matrix procedure as described in section.3.. The nodal pressures p i at each junction i are solved via the inverse of the system matrix. The nodal velocities u i are solved

115 7 subsequently. These quantities provide the information for the non-dimensional acoustic impedance via: ζ px= ( x= ) = Ω ρ c n (5.) y ui ( x= ) i= where p x= is the prescribed pressure at the surface of the sample (at x = ) and n y is the number of tubes in the y-direction (vertical direction in Figure 5.). With the use of ζ the reflection and absorption coefficients are calculated. On an ordinary PC the calculation of the sample shown in Figure 5. takes one or two minutes for 4 frequencies. It is noted that a new calculation with a different random distribution but the same percentage of the full network gives slightly different results for α. This was done by starting the random generator at a different state so that a different labyrinth-like structure results with approximately the same number of tubes. As will be shown later on a finer network with more tubes reduces this effect (see Figure 5.8). A random distribution for the radius or the length of the tubes The absorption coefficient of a labyrinth-like network of tubes was presented with a uniform random distribution of identical tubes. Obviously a random parameter can also be used for the radius R and the length L of the pores. Some first results are presented in Figure 5.5 and Figure 5.6 for a normal distribution of the radius and length. A full network of 3x tubes is used. In the figures the 95% density region of the parameter to be varied is indicated. The other parameters are: thickness = 38 mm and Ω =.95.

116 8 α [ ] α [ ] R =. mm.5 < R <.5 mm. < R <. mm 3 4 Frequency [Hz] Figure 5.5 Sound absorption coefficient of a full network with a normal distribution for the radius R of the tubes.. L =. mm.5 < L <.5 mm. < L < 3. mm 3 4 Frequency [Hz] Figure 5.6 Sound absorption coefficient of a full network with a normal distribution for the length L of the tubes. It can be seen that for these random distributions of the radius the effect on the sound absorption coefficient is small. For a variation of the lengths it was already seen that this parameter is of less importance. 5.3 Comparison to an empirical impedance model In 97 Delany and Bazley (Delany 97) presented a simple power-law for the acoustic impedance of fibrous sound absorbing materials. Based on measurements on a range of these materials they showed that the impedance can be given as a function of the frequency f divided by the flow resistivity φ :.75 f. 73 f ζ = Re( ζ ) + Im( ζ ) i = i.9 (5.) φ / φ / The quantity φ in (5.) has units [Nsm -4 ] which differ by a factor of from the units as originally used by Delany and Bazley. The flow resistivity is frequency dependent but for fibrous materials usually the static flow resistivity is used. The static value is measured relatively easily from: p φ = (5.3) u L

117 9 where p is the static pressure drop across the sample. L is the thickness of the sample and u the flow velocity through it. According to Biot the resistivity is related to the dynamic viscosity of the fluid µ, for a Poiseuille type of flow, as: µ φ = (5.4) ξ with ξ being the so-called Darcy coefficient of permeability. The flow resistivity is also referred to by other authors as the static flow resistance per unit width φ. Picard describes a method to measure φ dynamically (Picard 998) and it is noted that various authors such as Zwikker & Kosten and Biot use a cylindrical pore for fibrous (!) materials to approximate the dynamic flow resistance per unit width as a function of the shear wave number s. In the present investigation the static flow resistivity is used for a more straightforward comparison. In Table 5. the range for φ for fibrous materials as measured by different authors is indicated. The range for the porosity Ω is indicated as well. Author Flow resistivity φ [Nsm -4 ] Porosity Ω [-] Delany & Bazley (97), 8, --- Picard (998) 9, 35, Allard (986), 88, Attenborough (983) 5,, Lauriks (989) 4,8 47, Table 5. Range of possible values for various fibrous sound absorbing materials. The absorption coefficients as calculated with the empirical impedance model of Delany and Bazley are given in Figure 5.7 for different values of φ.

118 .8 α [ ].6.4 φ =, [Nsm 4 ] φ =, [Nsm 4 ]. φ = 4, [Nsm 4 ] φ =, [Nsm 4 ] 3 4 Frequency [Hz] Figure 5.7 Sound absorption coefficient according to Delany and Bazley. Clearly a higher flow resistivity causes a lower sound absorption. The reason for this behaviour is that for a high flow resistivity the fibres and the fluid inside the frame are vibrating more in phase than for a small value of φ. Therefore the dissipating mechanism, i.e. the viscous drag force, is less effective. This behaviour can be shown in detail with the theory of Biot which uses both degrees of freedom for the frame and the fluid (see section 5.5). To compare the results of our network description with the prediction of the empirical impedance description, the flow resistivity of the network needs to be known. The network description provides an easy way to calculate the flow resistivity as a function of the frequency. With the help of equation (5.3) φ is calculated. The length L of the sample is known. A pressure drop of. Pa is prescribed and the resulting flow velocity is determined. The static flow resistivity is determined for a frequency approaching. Hz. It was observed that the flow at both ends of the sample is approximately the same for frequencies close to. Hz so that it is concluded that the static flow resistivity can be determined accurately by performing the calculation for a very low frequency. In Figure 5.8 the static flow resistivity is given for a number of samples. Results are given for two different thicknesses. Extra numerical samples were used with a different random distribution of tubes, i.e. by starting the random generator at a different state, to investigate the sensitivity on φ.

119 3 Width = 38 mm Width = 76 mm φ [Nsm 4 ] Percentage of full network [%] Figure 5.8 Static flow resistivity as a function of the network configuration. It can be seen from Figure 5.8 that the range for the predicted φ lies within the ranges as shown in Table 5.. Furthermore, the thickness of the sample hardly affects φ so that the flow resistivity is indeed per unit width. For a low dense network φ is rather high. For these densities there is a lot of dead material inside the network, i.e. tubes without any connection to the entrance of the sample. As a result the differences between these networks for random distributions with the same densities are large. The sound absorption coefficient as measured indirectly by Delany and Bazley are compared to the results of a network with a flow resistivity of, Nsm -4 (the density of the network is then 86%), a thickness of 76 mm (3x39 tubes), a porosity of.95 and pores of L =. mm and R =. mm. The results for the network description for α agree quite well with the experimental results of Delany and Bazley for the complete frequency range. Evidently, a network of tubes is well suited as a volume description for a sound absorbing material.

120 .8 α [ ].6.4. Random network (,) Delany & Bazley (,) Delany & Bazley (,) 3 4 Frequency [Hz] Figure 5.9 Sound absorption coefficient of a network compared to the empirical model of Delany and Bazley. A dynamic flow resistivity It is common use to measure the static flow resistivity. The network model offers also the possibility to study the effects of the frequency on the flow resistivity. A harmonic pressure perturbation is prescribed at both sides of the network to realise a pressure difference, i.e. p (x=) > p (x=l). The flow resistivity is determined with equation (5.3) and the flow velocity u is determined on both sides of the numerical sample, i.e at x = and at x = L. The numerical dynamic flow resistivity of a specific network of tubes is given in Figure 5.. The dynamic flow resistivity is clearly frequency dependent and decreases to a lower value for higher frequencies. This behaviour for a network of tubes can be ascribed to an increasing value of the shear wave number for higher frequencies. Obviously the effect of the viscosity is reduced and a lower resistivity results. The continuously higher flow resistivity at the side with the lower pressure is the result of the storage and dissipation of pressure and velocity perturbations inside the network. As a result the dynamic flow velocity at the low pressure side is somewhat lower. Furthermore at the low pressure side the effect of resonances in the sample is much smaller and therefore the velocity is lower. The high flow resistivity at Hz is a result of a distinct anti-resonance in the sample.

121 3 x 4 φ (with u at x=) φ (with u at x=l) φ [Nsm 4 ] Frequency [Hz] Figure 5. Flow resistivity φ as a function of the frequency. Density of the network: 86%, thickness = 76 mm, Ω =.95, Tubes: L =. mm and R =. mm (3x39 tubes). Measured data on fibrous rockwool samples as reported by Picard (998) show a slight increase of the flow resistivity as a function of the frequency. Obviously the acoustic behaviour of fibrous material and labyrinth-like material with tubes differs. Further study is needed to give more insight into the behaviour of the latter materials but it appears that considerable difference between the static and the dynamic flow resistivity may occur. It is noted that the network description provides a quick way to determine the dynamic flow resistivity. In the following two sections the results for the network description are compared with the results of other numerical volume descriptions. 5.4 Comparison to the Limp theory The so-called Limp theory takes into account the interaction between fibres and fluid. The fibres have no stiffness, i.e. they are limp, but the mass of the fibres and the drag force due to the presence of the fibres is accounted for. In Appendix D the Limp model is described in more detail. In this section the basic results of the Limp theory are presented. The Limp theory has been developed by Ingard (98) and Göransson (993). As a consequence of the assumption of limp fibres the application of the model is

122 4 restricted to materials for which the thickness is less than the shortest wavelength in the frequency range of interest. This imposes an upper limit on the frequency range so that resonance of the fibres, of which the elasticity is neglected, is avoided. The main parameters of the Limp model are: the volume porosity Ω (also used as surface porosity), the static flow resistivity φ, the structure factor K s, which approaches unity for fibrous materials, the density ρ s of the solid material (not the frame). The wave equation for Limp material is given as: p + kl p = (5.5) where k L is a complex wave number which can be written as: k ω iφ Ω ω ρ L = F F Ks c ~ ( ω) ; ( ω) = (5.6) iφ Ω ω( Ω ) ρ In (5.6) ~ c is the isothermal speed of sound because it is assumed that for low frequencies the thermal fluctuations in the pores are absorbed by the surrounding solid material. It is noted that the Rigid description (Morse & Ingard, 968) can easily be derived from the Limp description by assuming the mass of the fibres to be infinitely large. The fibres are unmovable (rigid). The interaction between the frame and the fluid is represented by the viscous flow across the fibres. As a consequence the Rigid description is valid for high frequencies or for materials with heavy fibres. The relation between the displacement U of the fluid, with U being a vector and averaged over a finite elemental volume, and the displacement in the pores U is given by: U = Ω U (5.7) The displacement U is not explicitly used in the Limp description so that the number of degrees of freedom is decreased. As a result a single dilatational wave propagates inside the Limp material. For the present investigation however it will be shown that it is more convenient to use the average velocity of the fluid u (u = iω U) to determine the acoustic impedance of a sample. s

123 5 To compare the acoustic behaviour of a network of tubes to a sample of Limp material a one-dimensional numerical set-up as shown in Figure 5. is used. p p p Air region Limp material pˆ A pˆ B pˆal pˆb L x x Rigid back Figure 5. Set-up for the calculation of the absorption coefficient of a sample of Limp material. At the left-hand side of the air region a harmonic pressure perturbation p is prescribed and as a result waves are travelling back and forth. In the air region the pressure and particle velocity perturbations are assumed to be: p( x, t) = u( x, t) = k x k x { pˆ Γ e Γ pˆ + e } A G ρ c k x k x i t { pˆ Γ e Γ ω pˆ e } e A B B e iω t (5.8) where pˆ B and pˆ A are the amplitudes of the forward and backward traveling waves. Reference is made to the List of Symbols for the other symbols for the sake of brevity. In the Limp material applies: p( x, t) = u( x, t) = ik x ik x { pˆ e L pˆ + e L } AL H ρ c ik x ik x i t { pˆ e L pˆ e L ω } e AL The coefficient H is given as: H F( ω) κ ( ω) ρ B L B L K ρ Ω e iω t iφ ω iφ Ω ω( Ω ) ρ s (5.9) s s = ; ( ) = (5.) κ ω where F(ω) was already introduced in equation (5.6).

124 6 In this set of equations the unknowns are the four amplitudes of the waves pˆ A, pˆ B, pˆ AL and pˆ B L. The four boundary conditions to solve the unknown amplitudes of the harmonic pressure and velocity perturbations are:. Prescribed pressure: p(x =) = p,. Continuity of pressure: p(x =L ) = p(x =), 3. Continuity of mass flow: u(x = L ) = u(x = ) f(ω), 4. Acoustically hard back: u(x =L ) =. The function f(ω) in boundary condition 3 relates the free fluid in the air region and the fluid in the pores (see Appendix D): iω ( Ω ) ρ + s φ f ( ω) = (5.) iω( Ω ) ρ Ω + s φ For low frequencies f(ω) approaches /Ω. This indicates that the fluid displacement in the pores U equals the displacement of the free fluid (see equation (5.7)). For high frequencies f(ω) approaches.. Now the averaged fluid displacement in the pores U is equal to the free fluid displacement in the air region. By solving the system of equations and application of the boundary conditions the pressure perturbation and the velocity perturbation as a function of the position and the frequency is known. This provides the information to calculate the sound absorption coefficient α of the sample of Limp material. Computations are performed for the following values of the main parameters: Ω =.95, K s =., ρ s = kgm -3. These values are representative for glass wool.

125 7.8.8 α [ ].6.4 Network (,) Limp (,). Limp (,) Limp (4,) Limp (,) 3 4 Frequency [Hz] Figure 5. Sound absorption of a sample of Limp material for different φ (thickness = 76 mm). α [ ].6.4. Network (,) Limp (,) 3 4 Frequency [Hz] Figure 5.3 Sound absorption of a sample of Limp material (thickness = 38 mm). In Figure 5. the effect of the flow resistivity φ is shown. Again a high value of φ results in less sound absorption in the material (for extreme low values of φ also less sound absorption can be seen). The resemblance between the results with the Limp material and as predicted with a sample of a network of tubes is quite good, both in Figure 5. and Figure 5.3. The network description predicts a broadband sound absorption which is comparable to Limp material although it neglects the interaction with a frame. For the present material the energy dissipation because of the inertial forces acting on the fluid in the Limp material is small (and the Rigid theory may be applied). Although in the impedance description the effect of the thickness of the sample is implicitly taken into account via the flow resistivity per unit length, it can be seen that for the Limp theory the same kind of results as for the network description are predicted if the thickness is reduced. As expected low frequency waves are less absorbed for thin samples. As for the effective speed of sound for both descriptions it is noted that the Limp theory uses the isothermal speed of sound (here: 9 m/s) whereas the network description uses the propagation coefficient Γ of the low reduced frequency model. Γ is frequency dependent via the shear wave number s. For the pore radius as used in Figure 5. and Figure 5.3 (R =. mm) the shear wave number at Hz is s =. and the effective speed of sound is only c eff = 5 m/s. A larger pore radius (R=. mm) gives: s = 4. and c eff = 8 m/s. It can be seen from

126 8 Figure 5.4 that in that case the absorption coefficient more closely resembles the results of the Limp theory, as expected. The pressure distribution in the one-dimensional Limp material is given in Figure 5.4 for a frequency of Hz. The Limp material has a thickness of 76 mm and a flow resistivity of, Nsm -4.. p [Pa] Air region Limp material.5..5 ^..5 Length [m] arg(p) [rad] π/ π 3/π Air region Limp material π.5..5 ^..5 Length [m] Figure 5.4 Pressure perturbation (magnitude and phase) along the horizontal axis (at Hz). For this situation 85 percent of the incident energy is absorbed inside the Limp material. It can be seen that the magnitude of the pressure perturbation drops rapidly inside the material due to the interaction of the fluid with the fibres. This is in contrast to a standing wave pattern as seen in resonators. There the pressure perturbation is amplified in the resonator (see Appendix A) so that sound energy is dissipated. It is noted that when the Rigid theory is applied the same kind of results are obtained. 5.5 Comparison to the Biot theory Biot (96) formulated his equations for sound absorbing material as a function of the displacements of the elastic frame U frame and the displacements of the fluid inside the frame U fluid. The current section briefly introduces Biot s volume

127 9 description. For more detail the reader is referred to the literature or the summary in Appendix E. In the second part of this section the sound absorption coefficient of Biot material is compared to the absorption coefficient as predicted by the network description. Just as in the previous section on Limp material a simple one-dimension case will be described. It is recalled here that the network description can potentially be used also in three-dimensional problems. Introduction to the theory of Biot Figure 5.5 presents a schematic view of a piece of Biot material with pores and an elastic frame. It suggests anisotropic behaviour. However, Biot s theory is isotropic. dy Uframe dz z y Ufluid Fluid x dx Elastic frame Figure 5.5 Simplified representation of a piece of porous material. Via stress-strain relations and Lagrange s equations of motion Biot derives the following equations for the elastic frame and the fluid, respectively: N U frame + ( A + N) ( U frame ) + Q ( U fluid ) = ( ρu ) ( ) frame + ρ U fluid + b U frame U fluid t t Q ( U frame) + R ( U fluid ) = ( ρu ) ( ) frame + ρ U fluid b U frame U fluid t t (5.) In (5.) N, A, Q and R are the elastic coefficients. They can be seen as the equivalent elastic constants when the frame consists of a homogeneous medium. N

128 is the shear modulus (one of the constants of Lamé) of the elastic frame which is measured in vacuum. The other constants are defined as follows: A = Kb 3 Q = ( Ω ) K f ( Ω ) N + K f Ω (5.3) R = Ω K f where Ω is the porosity, K f the bulk modulus of the fluid (K f = ρ c ) and K b the bulk modulus of the elastic frame: ( + ν b ) Kb = N (5.4) 3( ν b ) Poisson s ratio of the frame ν b is also measured in vacuum. Note the resemblance of the elastic coefficients with Hooke s law. A corresponds to one of the constants of Lamé with an extra term for the stress caused by the fluid. The coefficient Q represents the elastic coupling between the frame and the fluid. Biot s dynamic coefficients for the inertial forces are: ρ ρ = ( ) = ρ Ω ρ a s + ρ a (5.5) ρ = Ω ρ f + ρ a where ρ a is a mass coupling factor due to extra mass of the fluid and the frame that has to be accelerated because of the so-called tortuosity of the material. It is defined as: ρ a = Ω ρ ( Ks ) or ρa = ρ ( q ) (5.6) with K s the structure factor and q the tortuosity factor. These factors account for twisted pores or random directions of fibres. The viscous effects are accounted for in the viscous factor b: b = φ Ω or b = φ Ω F( ω) (5.7)

129 It is a function of the flow resistivity φ and the porosity Ω. For higher frequencies the factor b is multiplied by the function F(ω) which accounts for the assumption of a Poiseuille flow in the pores breaking down for higher frequencies. In Chapter it was seen that the low reduced frequency model, originally derived by Zwikker and Kosten, predicts a Poiseuille type of flow for the velocity profile for low shear wave numbers and a plane wave front for high shear wave numbers (see Figure. for the amplitude of the wave front, which is complex in general). It will not be surprising that Biot used the velocity profile according to Zwikker and Kosten (Biot 956). As for the thermal effects Biot also used the fluid density ρ for higher frequencies with the polytropic coefficient n(sσ) as given in Chapter. For the present investigation, for a frequency range between and 4 Hz, Biot s theory without the high frequency adjustment is used in order to have a more straight-forward comparison. Compared to the previous models Biot s theory is rather complex. However, it is the most complete one. It is capable of describing three types of waves inside the material: a so-called fast and slow dilatational (compression) wave and a shear wave. Bolton, Shiau and Kang (Bolton 996) presented compact expressions for these wave numbers. A drawback of the theory is the use of a large number of parameters which are difficult to measure in practice: Ω, K f, N, ν b, φ, q or K s and ρ s. Some need to be determined for the frame while others represent the bulk material. When one considers an uncertainty range for each parameter it may be expected that the predicted results for Biot material are rather inaccurate. A second drawback is that the finite element formulation of Biot s theory has 6 degrees of freedom per node, i.e. 3 displacements for both the fluid and the frame (or 5 degrees of freedom when the pressure and the fluid displacement potential are used (Göransson 998)). This increases the computational times dramatically in acoustic problems which normally have one degree of freedom per node, i.e. the pressure perturbation. A one-dimensional numerical set-up The numerical set-up as depicted in Figure 5. is used again, but the Limp material is replaced with Biot material. For this specific case the curl-free version

130 of the equations (5.) is used, i.e. rotations in the material are not described ( U = ): ( PU ( QU frame frame + QU + RU fluid fluid ) = t ) = t ( ρ U ( ρ U frame frame + ρ U + ρ U fluid fluid ) + b ( U t ) b ( U t frame frame U U fluid fluid ) ) (5.8) where P = A + N. Proceeding in the same way as for the Limp material one-dimensional plane waves in the Biot material are assumed. As indicated in the previous section there are two dilatational waves propagating in the material. The corresponding wave numbers are k I and k II. The backward and forward travelling waves become: U frame ( x, t) = { Uˆ AI e iki x + Uˆ B I e iki x + Uˆ AII e ikii x + Uˆ B II e ikii x } e iωt U fluid ( x, t) = { m Uˆ I AI e iki x + m Uˆ I B I e iki x + m Uˆ II AII e ikii x + m Uˆ II B II e ikii x } e iωt (5.9) In (5.9) m I and m II are the ratios between the amplitudes of the two dilatational waves of the frame and the fluid. The ratios can be calculated from the eigenvalue problem of Biot material. When in equation (5.8) a harmonic time dependence is substituted one obtains: k k P + ω ρ Q + ω ρ iωb + iωb k k Q + ω ρ R + ω ρ + iωb U ˆ ˆ iωb U frame fluid = (5.) A solution of (5.) is possible if the determinant of the symmetric matrix M is zero. The condition Det[M] = leads to a quadratic equation for k which provides the wave numbers k I and k II for the positive and negative x-directions. Substituting these wave numbers in (5.) gives the ratios m I and m II. Next the boundary conditions to solve the 6 unknown amplitudes in the air region and the Biot material are concerned. The boundary conditions are:

131 3 For the air:. Prescribed pressure: p(x =) = p, For the Free air Biot material interface:. Continuity of mass flow, 3. Continuity of fluid pressure, 4. Continuity of frame stress, For the acoustically hard wall: 5. Suppressed frame displacement: U frame (x =L ) =, 6. Suppressed fluid displacement: U fluid (x =L ) =. For the continuity of mass flow one has: U free fluid = Ω ( Ω ) U frame + U fluid (5.) The average pressure of the fluid in the Biot material is given as: p fluid ~ s U frame U fluid = with : ~ s = Q exx + R ε xx, exx =, ε xx = (5.) Ω x x where s ~ is the x-component of the fluid stress tensor s. e xx and ε xx are the accompanying components from the solid and fluid strain tensor. For the continuity of stress for the frame - free fluid interface one has: U frame U fluid σ xx = ( Ω ) p free field with: σ xx = P + Q (5.3) x x Finally the set of complex amplitudes in the air region and the Biot material can be solved as a function of the frequency. The pressure and velocity perturbations in the air region are used to calculate the acoustic impedance and the absorption coefficient at the surface of the Biot material. The parameters as used in the numerical example are listed in Table 5.3. In the table also some characteristic values are given. Note that both Allard and Dhainaut use an extra percent structural damping for the frame.

132 4 Parameter Example Allard (993) Dhainaut (996) glass wool light glass wool heavy glass wool Ω [-] φ [Nsm -4 ], 4, 4,, q [-] ρ [kgm -3 ] (-Ω)ρ s [kgm -3 ] N [Nm - ]. 5.(+.i) 6.85(+.i) 5.85(+.i) 5 ν b [-]..4.4 Table 5.3 Standard values and reference values as used for the Biot material. In Figure 5.6 the absorption coefficients of Biot material for different values of the flow resistivity are depicted. In Figure 5.7 the thickness is reduced to 38 mm..8.8 α [ ].6.4 Network (,) Biot (,). Biot (,) Biot (4,) Biot (,) 3 4 Frequency [Hz] Figure 5.6 Sound absorption of a sample of Biot material for different φ (thickness = 76 mm). α [ ].6.4. Network (,) Biot (,) 3 4 Frequency [Hz] Figure 5.7 Sound absorption of a sample of Biot material (thickness = 38 mm). A comparison of the results of a sample of Biot material with results of the network description, although both techniques are different, gives a satisfactory agreement. The Biot material behaves just like the Limp material because the same glass wool material is used and only normal incident waves are applied. The main difference is the effect of the elastic frame, via N, which is rather stiff for glass wool. In the numerical model of the Biot material it was seen that the fluid and frame displacements, U fluid and U frame, have a phase difference of 9 degrees for low frequencies which apparently causes the low frequency sound absorption.

133 5 5.6 Conclusions and remarks In the present chapter the coupled tubes model was used as an alternative technique to predict the acoustic behaviour of conventional sound absorbing material, such as glass wool or foams. This so-called network description uses a random distribution of a large number of coupled tubes (or pores) which can be seen as a representation of the interior of the material. It makes use of a small set of basic physical parameters to characterise the material: the porosity Ω, the uniform random distribution to approach the flow resistivity φ, and the pore length L and pore radius R. In general the porosity Ω of a material is known or easily measured and it was seen that the parameter L is of minor importance. Therefore only two parameters need to be established to represent the bulk material. A comparison of results with well-known models was used to validate the acoustic behaviour of the network description for a simple one-dimensional case. It proved that the network description is a simple and efficient technique. An advantage of the network description compared to the Biot theory is that it uses a relatively small set of parameters. It is also possible to estimate a dynamic flow resistivity for Limp or Biot material with the network description. In practise one needs to determine the parameters for the network description, i.e. the percentage and R, for a particular material. This can be done easily with the help of an impedance tube by matching the measured absorption coefficient with the numerical frequency dependent absorption coefficient. As a next step one can use the set of parameters obtained to select processing parameters to produce, for example, metallic foam with better sound absorbing capabilities. Also, one can predict the acoustic behaviour in different cases. For example the effect of oblique or random incident waves on the sound pressure level can be simulated for a particular design. A pressure gradient inside the material can be generated with a prescribed complex pressure across the numerical sample to simulate the angle of incidence. The effect of random incident waves is estimated with an average of incident waves at different angles. Obviously, this method can be carried out for various materials in an efficient way in order to compare the performance. It is relatively simple to calculate the acoustic impedance of a network of tubes. In Chapter the direct relation between the pressure perturbation and velocity perturbation in a tube was shown. So with the pressure known at both ends of the

134 6 tubes the velocity at only the surface of the sample, and thus the impedance, is known. A volume description such as the Biot theory is used in general in extensive finite element simulations with a complex sound field. It was seen that a more efficient technique is to use a surface description of sound absorbing material. Therefore it may be useful for some applications to calculate the acoustic impedance of materials as an average of incident waves at different angles. Such an average can be calculated in an easy way for the network description by prescribing complex pressures across the numerical sample. This may be a problem for commercial finite element codes. A major difference of a volume description with a network of tubes compared to the Limp and Biot theories is the absence of the interaction with a frame. However, the acoustic behaviour proved to be comparable for the presented fibrous material. A more extensive comparison needs to be made for different materials and for oblique and random incident waves.

135 Chapter 6 Further applications of the coupled tubes model 6. An inkjet array 8 6. A test set-up for a voice producing element A viscothermally damped flexible plate Reflection of sound in ducts with side-resonators 58 In the present chapter the coupled tubes model is used in a number of different cases. In two cases sound absorption is not the issue but merely the technique to predict the acoustic behaviour of coupled tubes. The first case is described in section 6. and uses coupled tubes to model the acoustic behaviour of ink in the inkjet array of a print head. The problem to be tackled is to reduce the differences in performance of the jets from the inkjet arrays. A comparison of numerical and experimental results proves that the coupled tubes model is well capable of predicting the acoustic behaviour of an inkjet array. Furthermore a new design with a broadband resonator shows that the performance of the arrays will be much less sensitive to small deviations in, for example, the geometry. The second case concerns the design of an experimental test set-up for measuring the acoustic behaviour of a voice producing element. This element seeks to restore speech in cases where the vocal folds have been removed due to cancer. The acoustic loads on the element are represented by the vocal tract, i.e. the mouth cavities, as well as the subglottal tract, i.e. the lungs. Both tracts are modelled with the coupled tubes model and closely reflect the physical acoustic behaviour. The designed tracts were applied in the experiments. The third case describes the application of resonators to reduce the vibration of a flexible plate. The plate is backed by a small air cavity. It is shown that by extracting energy from the air layer via the resonators the flexible plate is damped because of the acousto-elastic coupling of the plate and the air layer. Finally, a design strategy is presented to place resonators perpendicular to the axis of a duct. The resonators create a so-called acoustic mirror for acoustic

136 8 waves in the duct. Waves within a small particular frequency band are reflected. As a result the amount of noise leaving a duct can be significantly reduced. 6. An inkjet array The newly developed technique to optimise sound absorption with coupled tubes was used in a numerical study of the inkjet array in a print head (see Figure 6.). In cooperation with Stork Digital Imaging B.V. improvements in the performance of the array were sought. The deviations in performance of the jets from the inkjet arrays were mainly caused by the variable effective speed of sound in ink within the array. The speed of sound is reduced in the array because of both viscothermal effects and the presence of a flexible cover plate. The flexible cover plate with nozzles is glued onto the print head. Due to the glue and the flexible cover plate, large variations can occur in the speed of sound so that the performance between the arrays differs too much. This section shortly describes the basic principle of the inkjet array and its performance when a broadband resonator is applied. A comparison with experimental results shows that the numerical model predicts the acoustic behaviour of the array very well. In contrast to the previous sections, where the resonance of air in tubes is treated, here the acoustic behaviour of ink in a small channel is described. It is shown that the low reduced frequency model for viscothermal wave propagation can also be applied for ink. Furthermore, it is shown that the coupled tubes model is a suitable design tool for the present application. Working principle of the inkjet array Stork Digital Imaging produces inkjet printers for the professional market. One of their products is used for printing on textiles. Textile widths of.5 meters can be printed on. The print head is very small and has 6 nozzles (in Figure 6. the inkjet array is shown with only 8 nozzles). The basic printing principle is the so-called continuous ink jet which means that the ink is pumped continuously through the nozzles under high pressure. Ink is supplied via a relatively large reservoir.

137 9 Housing Cover plate Channel Nozzles Reservoir Figure 6. Schematic drawing of an inkjet array. Piezo After a short length, L B, the ink jet breaks up into drops (see Figure 6.). The drops are selectively charged so they can be directed in a magnetic field to create the required textile print. Non-charged drops are captured and re-used in the printing process. Nozzle Inkjet Drop Breaking length L B Figure 6. Breaking length for an ink jet. The drops are charged and directed within a limited distance. Hence the breaking length L B needs to be within a certain range. For the present investigation it is important to know that the length L B is controlled with a pressure perturbation which is superimposed on the average pressure in the ink. By means of a piezoelectric element this perturbation is generated for a single, optimally chosen frequency. A standing wave pattern is created in the ink channel because the opposite side of the channel is acoustically hard. At the location of each pressure maximum a relatively small nozzle is situated so that for each nozzle

138 3 approximately the same breaking length is created (see Figure 6.3). The actual dimensions and frequencies are company confidential. Piezo Reservoir Channel Nozzles Acoustically hard wall Supply Pressure amplitude Figure 6.3 Working principle of the inkjet array. The first arrays produced, however, showed a large variation in breaking lengths L B. Possibly this was caused by a lower effective speed of sound in the ink in the channel due to the glue and the flexible cover plate. Therefore an approximation for the reduced effective speed of sound in the channel is derived in the following section. Reduced effective speed of sound due to the flexible cover plate It is assumed that: the ink is non-viscous, homogeneous and isotropic, the perturbations of the variables are small, the cover plate is flexible, i.e. the cross-sectional area of the channel varies as a function of the pressure, the wave propagation is one-dimensional. If the x-direction is the direction of the propagating waves then the variables for an infinitesimal volume dv are (Lap 998): A ( x, t) = A + A( x, t), the cross-sectional area of the channel, ρ ( x, t) = ρ + ρ( x, t), the density of the ink, p ( x, t) = p + p( x, t), the pressure in the ink, u ( x, t) = u( x, t), the velocity perturbation of the ink. The mean speed u is much smaller than u.

139 3 The difference in the mass flow Q entering and leaving the infinitesimal volume dv equals: dq = (ρ A u dx x ) (6.) and the change in the mass flow within the volume dv due to the changes in density and the cross-sectional area amounts to: dq = (ρ A dx t ) (6.) So the mass flow balance for the volume dv is: ( ρ A) = ( ρ A u ) t x (6.3) Neglecting the higher order terms gives: A + t ρ A ρ u = t x (6.4) Next, with the linearised momentum equation perturbation u can be eliminated: p x = ρ u t the velocity A A + t ρ ρ = t ρ p x (6.5) The pressure perturbation p is related to the adiabatic bulk modulus K of the ink according to: p = (6.6) K ρ ρ where K = c ρ. Furthermore it is assumed that, because of the flexible cover plate, a pressure rise leads to a larger cross-sectional area: p A = K A (6.7) A where the modulus K A relates the pressure p to the relative change of the crosssectional area.

140 3 Substituting equation (6.6) and (6.7) into equation (6.5) gives the wave equation in the channel with a flexible cover plate: K + K A p = t ρ x p p = c t eff x p (6.8) Evidently, the effective speed of sound in the channel can be written as: c eff = ρ K + K A (6.9) Thus the limited stiffness of the channel causes the effective speed of sound in ink to be lower than the undisturbed speed of sound c. As a result the actual standing wave pattern in the channel differs from the predicted pattern so that the location of the nozzles is not optimal. In other words: the amplitude of the pressure perturbation at the entrance of a nozzle depends on its specific location. Hence the breaking length L B also differs among the nozzles, as is shown in the next section. Measurements for various frequencies The variation in the standing wave pattern, and therefore the difference in pressure perturbations at the various nozzle positions, can be demonstrated by varying the Normalised L B [ ] Normalised frequency [ ] Figure 6.4 Measured breaking length L B as a function of the frequency. Jet Jet Jet 3 Jet 4 Jet 5 Jet 6 Jet 7 Jet 8 Jet 9 Jet Jet Jet Jet 3 Jet 4 Jet 5 Jet 6

141 33 excitation frequency of the piezo-electric element. For a range of frequencies the breaking length L B for each nozzle is determined. A characteristic result is illustrated in Figure 6.4. The breaking lengths and the frequency are normalised between zero and one. Obviously there is a significant difference in the performance, in terms of achieved L B, for the different nozzles. At the frequency of.8 this particular inkjet array performs best because L B is almost equal for each nozzle. In practice the piezoelement of all inkjet arrays operates at a normalised frequency of.5. The experimental results serve as a test-case for the numerical model. It will be shown later on that the numerical model performs quite well (see Figure 6.7). Solution with a broadband resonator Having in mind the non-reflective wall with tuned resonators a solution was sought in a channel without a standing wave pattern. By leaving the standing wave principle in the inkjet array the problem with different pressure perturbations at the different nozzles is absent. So the design of the array should be in such a way that at the end of the channel, opposite to the location of the piezo element, a nonreflective boundary condition is created. p excitation 6 5 Nozzles p 6 p i p Inkjet array Coupled resonator Nozzle Channel p i p= Second resonator tube First resonator tube Channel Figure 6.5 One-dimensional model of an inkjet array with a broadband resonator at the end.

142 34 With the theory of Chapter a broadband resonator which consists of two coupled tubes was designed. In Figure 6.5 the numerical model for the inkjet array and the resonator is depicted. The resonator is shown on the right-hand side and the excitation pressure p excitation on the left-hand side. The amplitudes of the pressure perturbation at the entrance of the nozzles is represented by p i. The pressure perturbation at the exit of the nozzles is set to zero. Given the dimensions of the closed end of the channel an optimal configuration of smaller channels can be determined so that the absorption coefficient α of the closed end is almost. for a wide frequency band. A single channel resonator could be used for a specific frequency but because of the expected variations of c eff a broadband resonator is preferred. For the frequency range of interest a resonator which consists of two coupled tubes is sufficient. The predicted sound absorption coefficient of the wall on the right-hand side of the array channel is given in Figure α [ ].6.4. α at end of channel Normalised frequency [Hz] Figure 6.6 Absorption coefficient of the end of the array channel. It is permitted to note that the bandwidth of the frequency in Figure 6.6 is khz which is quite large. Such a broadband sound absorption is possible because the normalised frequency of.5 represents a rather high frequency. The calculated amplitudes of the pressure perturbation at the entrance of the nozzles without the resonator are shown in Figure 6.7.

143 35 Transfer function H ij [ ] Normalised frequency [ ] Figure 6.7 Dynamic pressure transfer function at the position of the nozzles without a broadband resonator. Along the vertical axis the transfer function H ij is shown which represents the ratio between the pressure p i at nozzle i and the excitation pressure p excitation. It seems reasonable to assume a direct relation between the magnitude of the pressure perturbation at the entrance of a nozzle and the breaking length L B of the jet. This means that a high value of H ij corresponds with a short breaking length. Another observation is that at the operational normalised excitation frequency of.5 the breaking lengths for all nozzles is equal (see Figure 6.7). For that situation the spacing of the nozzles and the effective speed of sound in the ink is optimal. A different excitation frequency (or a different c eff ) leads to different values of L B. Furthermore there is a striking resemblance between the results in Figure 6.7 and the measured breaking lengths of Figure 6.4. Therefore it appears justified that the numerical model of the inkjet array can be used for the channel with a resonator present at the end of the channel. In Figure 6.8 the numerical results are given for the transfer functions at the nozzles with the resonator present.

144 36 Transfer function H ij [ ] Normalised frequency [ ] Figure 6.8 Dynamic pressure transfer function at the position of the nozzles with a broadband resonator. The reduction in the variations of H ij is about an order of magnitude compared to the previous results and the effect of the resonator is clearly present, i.e. the amplitudes H ij, which remain at the level of the original arrays, are almost independent of the frequency. At the boundaries of the frequency domain the effect of the somewhat lower absorption coefficient of the resonator can be seen. The area of the cross-section of the channel and the resonator are in the order of micrometers squared. The cross-sectional area of the smallest channel of the resonator is about 5 times smaller than that of the array channel. With current manufacturing techniques this is possible to produce. It can also be shown that the low reduced frequency model is valid for the present frequency range and dimensions, i.e. k r <. and k r / s <.. Concluding remarks The numerical study of the inkjet array with a broadband resonator was a new application for optimised sound absorption. Moreover the coupled tubes model showed to be an effective design tool. It was also demonstrated that instead of air any other fluid can be used in the coupled tubes model. Alternative techniques

145 37 were used by Stork Digital Imaging to improve the performance of the inkjet array so that the technique based on the broadband resonator was not implemented in practice. 6. A test set-up for a voice producing element In this section the application of the coupled tubes model in the test set-up for a voice producing element is described. This element restores the ability to speak for patients who have undergone a removal of the vocal folds due to cancer. The study was performed in cooperation with the department of BioMedical Engineering of the University of Groningen, the Netherlands. The following topics are treated in this section: The need for a voice producing element, An in vitro test set-up, The vocal tract for the acoustic simulation of different vowels, The subglottal tract for the simulation of the acoustic properties of the lung airways. In brief: with the coupled tubes model a system of tubes was designed to create the desired acoustic impedance of the section below the voice producing element, the subglottal tract, and the section above the voice producing element, the vocal tract. The designs were built, tested and applied in the complete set-up. A voice producing element For a number of people with cancer in the larynx the only medical treatment possible is the operative removal of the larynx. In medical terms this is called a laryngectomy. This involves, among other things, the removal of the vocal folds. The vocal folds operate by means of an air flow from the lungs which causes them to vibrate. The sound generated in this way passes the vocal tract up to the lips to form the desired sound of speech. The vocal tract can be seen as an acoustic converter. It will be clear that the lack of speech is a severe handicap in the human communication process and therefore alternative ways of voice production are sought.

146 38 On the left-hand side in Figure 6.9 the situation after laryngectomy is sketched. The neck is reconstructed after the removal of the larynx. It can be seen that the trachea, the pipe from the lungs, is led outside (via a so-called stoma). Nose Tongue Lips Flexible lip Vocal tract Voice producing element Esophagus Pseudoglottis Subglottal tract Stoma Shunt valve Trachea Lungs Figure 6.9 Schematic drawing of the vocal tract, the voice producing element, and the subglottal tract. The conventional solution to restore the possibility of voice production is to put soft tissue from the esophageal entrance into vibration. This tissue is called the pseudoglottis. In earlier days the pseudoglottis was set into vibration by air injected from the mouth into the esophagus. Since about 98 a simple shunt valve is placed into the tracheo-esophageal wall as indicated in Figure 6.9. By closing the tracheostoma in the neck the air flow is directed into the esophagus. The flow causes vibration of the pseudoglottis and the resulting vibrations of the air can be articulated into speech. Unfortunately the speech quality is poor. The sound produced in this way has a very low fundamental frequency which is, especially for women, undesirable. The ability to vary the frequency is usually absent so that a typically monotone voice results. To overcome these drawbacks a project supported by the Dutch Technology Foundation (STW) was started in 996 (GGN.37) to develop a voice producing element which can be placed in a shunt valve (the Groningen button LR mm by

147 39 Medin Instruments, Groningen). The four participating groups are: the Department of BioMedical Engineering of the University of Groningen, the EarNoseThroat and Head & Neck Department of the Vrije Universiteit, Medical Centre Amsterdam, the Department of Biomedical Engineering of the University of Twente and the Fluid Dynamics Laboratory of the Eindhoven University of Technology. It was chosen to place a prosthesis, the voice producing element, directly in the shunt valve (de Vries ). The prosthesis consists of a metal tube with an inner rectangular cross-section in which a flexible silicon lip is glued (see Figure 6.). Flexible lip Air flow Shunt valve Voice producing element One cycle of the vibrating flexible lip Figure 6. A prototype of the voice-producing element, placed inside a Groningen button shunt valve. The air pressure, built-up in the trachea, causes a vibration of the flexible lip via a fluctuating air flow so that a tone is generated. The prosthesis is different for men and women: for men an average frequency of Hz is desirable and for women this frequency is Hz. Furthermore a higher air pressure causes a higher frequency of the silicon lip. This corresponds to the effect of normal speech where a higher air pressure results in a somewhat higher frequency and a higher intensity. To validate the proper action of the device an in vitro test set-up was developed. This in contrast to in vivo tests which are restricted by a medical protocol. In the test set-up (see Figure 6.) three sections are distinguished: the vocal tract which represents the speech articulation section, the voice producing element, and the subglottal tract which represents the trachea and the lungs. The subglottal and the

148 4 vocal tract were introduced in the test set-up because it was known that the effect of the voice producing element changes in different acoustic circumstances. Therefore the physical acoustic behaviour of the vocal tract and the subglottal tract were simulated with a system of coupled tubes. Experimental results of the important harmonic frequencies of vowels (Peterson 95) and the acoustic impedance of physical lungs (Ishizaka 976) were used as an objective for the design of the tracts. An in vitro test set-up The acoustic behaviour of the subglottal tract and the vocal tract on the voice producing element is measured with the set-up as shown in Figure 6.. Microphone Pressure transducer Vocal tract model Voice producing element Flow transducer Subglottal tract model Sound absorbing material Pressure vessel Figure 6. The in vitro test set-up for the voice producing element. An air flow is supplied to a pressure vessel and is controlled and measured with a flow transducer. By doing so the in vivo range of the flow rate can be simulated. The pressure built up in the vessel is measured with a pressure transducer. The air flow can escape the vessel via the system of tubes representing the lungs, the voice producing element and the model of the vocal tract. The produced sound is measured with a microphone, as indicated in Figure 6..

149 4 The design strategy to create the desired acoustic behaviour of the vocal tract model and the subglottal tract model with a number of coupled tubes is described in the following sections. The vocal tract for the simulation of different vowels In the vocal tract articulated speech is created. A variety of different sounds can be formed with the mouth of which the vowels can be seen as the basic sources of sounds. Peterson and Barney (95) investigated the acoustics of these vowels by plotting the first so-called formant frequency against the second formant frequency, see Figure 6.. The first and second formant frequencies are the first and second main frequency components of the auto-spectrum of the vowel. 4 Second formant frequency, F [Hz] 3 // i // a // u First formant frequency, F [Hz] Figure 6. Vowel triangle of Peterson and Barney representing the frequency areas of the different vowels. Three extreme vowels were selected for the simulation of the vocal tract in the test set-up: the /a/, the /i/ and the /u/. The vocal tract can be considered as a combination of resonance tubes (Mol 97). With the help of the theory on coupled tubes presented in Chapter the dimensions of the tubes were determined in such a way that the first and second resonance frequency of the coupled tubes system fall

150 4 in the regions as given by Peterson and Barney. The final geometry of the coupled tubes for the three vowels is shown in Figure 6.3. // a // i / u/ p p L =8mm R =3mm L R =6mm =5mm L =6mm R =5mm L =6mm R =7mm p L R =98mm =.5 mm L =7mm R = 3.5 mm L 3 =55mm R 3 = 7.5 mm L 4 =64mm R 4 =7mm Voice producing element Figure 6.3 Configuration of tubes for the vowels /a/, /i/ and /u/. The voice producing element is connected to the lower side of the resonator tubes. In a first approach it was assumed that this side acts as a closed side, i.e. acoustically hard. So the transfer function, being the ratio between the pressure perturbations at both ends of the coupled tubes system, for example for the vowel /a/, is derived as: p p p H tot = = (6.) p p p In Figure 6.4 the calculated transfer functions for the three vowels are shown. It is noted that in the calculations instead of the geometrical lengths, as given in Figure 6.3, the effective lengths of the tubes are used (see Chapter equation (.58)).

151 43 Vowel /a/ Vowel /i/ Vowel /u/ H tot [ ] Frequency [Hz] Figure 6.4 Transfer function of coupled tubes which represents different vowels. The peaks at the resonance frequencies clearly show up in Figure 6.4. These resonance frequencies could be distinguished by the ear when the experimental models for the vowels were used in the test set-up. Vowel Resonance frequency Measured [Hz] Predicted [Hz] /a/ st 6 7 nd 8 /i/ st 5 8 nd 4 /u/ st 9 nd Table 6. Comparison of the measured and predicted resonance frequencies of coupled tubes which represent the vowels /a/, /i/ or /u/. The performance of these acoustic filters was also measured. At the position of the voice producing element an airtight B&K condenser microphone was placed. This microphone was used as a speaker and was fed by a signal generator. Near the outlet of the vocal tract model the sound was measured with a microphone in the

152 44 same way as sketched in Figure 6.. The auto-spectrum of the latter microphone was measured. In Table 6. the measured and simulated results are given for the coupled tubes for the vowels /a/, /i/ and /u/. The measured resonance frequencies show a reasonable agreement with the predicted ones. The differences can be explained via the assumed boundary conditions at both ends of the coupled tubes. At one end the excitation is a membrane instead of the assumed acoustically hard wall and on the other side a radiation boundary condition would be more appropriate. However, at this stage the results of the measured resonance frequencies could be used very well and there was no need for a second, more accurate, simulation. The simulation of vowels with a combination of tubes and volumes is not new. For instance Fant (97) already used a lumped element technique for the coupled tubes and volumes. The lumped elements are described with equivalent circuitry formulas, where the inductance (or mass) L = ρ l/a, the capacitance (or spring) C = la/ρ c, and resistance (or energy loss) R are used. To simulate the different vowels Fant used a double Helmholtz resonator and gives the corresponding lengths, cross-sections and volumes. The resonating mass in a tube, here the socalled neck of a Helmholtz resonator, is estimated as a lumped mass with a certain length. The volumes of the Helmholtz resonators behave as springs. So the main difference is that the present model is a continuous one. This offers the freedom to use also a combination of tubes because the mass and spring functionality are continuously distributed in the tubes. Furthermore higher order modes can be calculated as well whereas this is impossible with the lumped element approach. The subglottal tract for the simulation of the lungs The acoustic load on the voice producing element on the side of the lungs is given by the acoustic impedance of the trachea and the lungs, i.e. the subglottal tract. Measurements of the impedance of the subglottal tract of Japanese laryngectomees have been reported by Ishizaka (976). The peaks in the impedance spectrum lie at 64, 4, and 3 Hz. Furthermore the geometry, i.e. lengths and crosssections, of each successive generation in the lungs is reported in the literature so that a system of tubes can be constructed. This approach is used in the Groningen project. Two models have been constructed, see Figure 6.5.

153 45 Generation Generation Generation 3 Generation 4 Model Model Closed ends Voice producing element Air flow Impedance L = 5 mm R =9mm L = 5.6 mm R = 6. mm L 3 =9mm R 3 = 4. mm L 4 = 7.6 mm R 4 =.8 mm Figure 6.5 Schematic representation of the subglottal tract: lung models and. The first model resembles the geometry of the lungs reasonably well. For convenience the last generation of tubes is terminated with acoustically hard walls. The air flow is provided via a capillary tube and numerical simulations showed that this tube has no influence on the acoustic behaviour of the model. The second model consists only of the generations and and is terminated with a thin layer of glass wool. This second model was introduced because the results for the impedance of the first model were too high compared to Ishizaka s results. The acoustic behaviour was predicted with the coupled tubes model presented in Chapter. The transfer function for a tube with a symmetric Y-junction at one end yields: p j = cosh p j sinh( Γ k L) cosh( ) + sinh( ) + J AJ G p J j Γ + J Γ k L J (6.) A J GJ Γ k L J p j ( k L) The tube is labelled J, p j- is the pressure perturbation at the entrance of the tube and p j the perturbation at the Y-junction. Γ is the wave propagation coefficient, k is the wave number, L is the length of a tube, A is the cross-sectional area and G is the corresponding coefficient for the type of the cross-section (see Appendix B). The recursive formulation of transfer functions is used to calculate the transfer function of the tube of generation. With the latter transfer function the acoustic impedance at the entrance can be calculated (see section.3.). The acoustic impedance of both models was measured in the impedance tube as described in Chapter 3. The numerical and experimental results for the first model

154 46 are given in Figure 6.6 for and generations and in Figure 6.7 for 3 and 4 generations. ζ for generation ζ for generations Measurements ( gen.) Measurements ( gen.) ζ lungs [ ] Frequency [Hz] Figure 6.6 Acoustic impedance of lung model (sections and +). ζ for 3 generations ζ for 4 generations Measurements (3 gen.) ζ lungs [ ] Frequency [Hz] Figure 6.7 Acoustic impedance of lung model (sections ++3 and ++3+4). The correspondence of the results of the numerical and the physical models is reasonable. The reason for the noisy results for higher values of the impedance is that the model consisted of a number of separate sections with tubes so that pressure leakage can occur between these sections. Also, the last section consisted

155 47 of tubes which were closed on one end with a rather flexible aluminium trip whereas the theory uses ζ = at these ends. The peaks in Figure 6.7 in the case of 4 generations lie close to the frequencies as measured by Ishizaka; however, the peaks are too high. Therefore, more damping was needed in the subglottal tract. This was solved by numerically applying an impedance boundary condition at the ends of the tubes which can be seen as the use of damping material at the tubes end. Consequently the coupled tubes model showed that the resonance and anti-resonance frequencies change. The final model, model in Figure 6.5, was designed with only generations and an acoustic impedance at the ends of approximately ζ =.. This value for ζ was realised in practise with a thin layer of glass wool. The results, depicted in Figure 6.8, show a reasonable agreement between the measured and simulated acoustic behaviour. Simulation Measurements ζ lungs [ ] 5 5 Frequency [Hz] Figure 6.8 Acoustic impedance of lung model. The damping caused by the glass wool is less in the measurements than in the simulations. The acoustic behaviour of the glass wool was not actually measured because the measurements proved that model was suited for use in the test set-up.

156 48 Concluding remarks It was demonstrated that the coupled tubes model was successful in predicting the acoustic behaviour of the vocal and subglottal tracts. For the vocal tract the acoustics of different vowels was simulated in an efficient and flexible way. This was also the case for the design of the system of tubes to simulate the acoustic impedance of the trachea and the lungs. Measurements proved that the numerical predictions were sufficiently accurate. The designs were successfully used in the in vitro test set-up for the voice producing element. 6.3 A viscothermally damped flexible plate In the present section the viscothermal effects in air are used to reduce the vibrations of a flexible plate which is backed by an air filled cavity. The damping of the plate is achieved in two ways. Firstly, the viscothermal effects in a thin air layer trapped under the plate cause dissipation of energy via the so-called acoustoelastic coupling of the plate and the air layer. Secondly, a number of tuned resonators is used to create extra damping in the air layer. The extra damping is added in a small frequency range where the damping of the plate by the air layer is low Rigid frame 4 Flexible clamped plate h Airtight volume Movable rigid bottom Figure 6.9 An airtight box with a flexible clamped plate and a movable bottom. Dimensions in mm.

157 49 The vibration of plates is a major topic in noise problems and the goal in general is to reduce the radiated sound. An example where air, or any fluid, can be used in for instance sound transmission problems are double wall panels (Beltman 998b, Basten 998). For demonstration and feasibility purposes the problem is restricted to a rectangular clamped plate with a trapped air layer as shown in Figure 6.9. The set-up was previously used by Beltman (998a) to validate a computational method with new viscothermal acoustic finite elements in acousto-elastic problems. In the following section the basic equations for both the analytical and finite element method are reviewed briefly. Next, the effects of the trapped air layer on the dynamical behaviour of the flexible plate are shown. Finally the numerical and experimental results are given for a viscothermally damped flexible plate when resonators are also present. Basic equations Consider the squeeze film damping problem as depicted in Figure 6.. z x h h Flexible plate Air layer Fixed surface Figure 6. Squeeze film damping problem. An air layer with a mean thickness of h is trapped between the flexible plate and the fixed surface. The plate performs a small harmonic oscillation around the equilibrium position h o : = iωt h h( x, y) e (6.) h ( x, y) + Sound radiation is not necessarily directly coupled to the vibration level of the plate. By using the so-called radiation modes of a plate configuration one knows the relative importance on the radiation. This topic is beyond the scope of the present investigation and reference is made to for instance Currey 995, Chen 997 and Gibbs. Note that the width of the air layer is h which results is a shear wave number of s = h (ωρ/µ).

158 5 For the air layer the so-called low reduced frequency model is used. It is emphasized that in this model there inherently are no pressure gradients across the layer thickness. The acousto-elastic coupling results in an extra forcing term, due to the squeeze motion of the flexible plate, on the right-hand side: p p ρc n + k Γ p = k Γ h x y h γ (6.3) where p represents the total pressure (p + p e ω ) and Γ is the propagation coefficient in the air layer (see Appendix B). In the propagation coefficient the viscothermal effects are accounted for. Γ is a function of the shear wave number s, which represents the ratio between the viscous and inertial effects in the fluid. A low value of s indicates that the viscous effects are dominant. For the deformation of the plate a standard plate equation can be used (for instance Kirchhoff). Also here, a right-hand side term is present where the forcing term consists of the pressure perturbation. The acousto-elastic coupled system can be solved analytically in some cases, see Beltman (999b, a one-dimensional case with an infinite flexible plate) and Basten (, a two-dimensional case with a double wall panel). However, for most cases an analytical solution cannot be found. Therefore the finite element formulation is used. The basic equations are reformulated and the acousto-elastic coupling is established by demanding continuity of the normal velocity across the interface. This leads to the following coupled system of equations, see Beltman 998a for an extensive derivation: s { } s c { } ext [ M ] [] u [ K ] [ K ] u { F } ω + = (6.4) c a [ ( )] [ ( )] { } a M s M s p [] [ K ] { p} {} The mass matrix consists of a standard structural part [M s ], a frequency dependent acoustic part [M a (s)] and a frequency dependent coupling matrix [M c (s)]. The stiffness matrix has a conventional structural part [K s ], an acoustic part [K a ] and a coupling matrix [K c ]. The system matrices in (6.4) are frequency dependent, asymmetric and complex. The finite elements for the viscothermal air layer were implemented by Beltman in the finite element code of B (Merazzi 994). He also implemented onedimensional elements for tubes which include the viscothermal effects. The latter i t

159 5 elements can be coupled to other (standard, volume) acoustic elements and are used to model the resonators. The coupling elements, to accomplish the acoustoelastic interaction between acoustic elements and structural elements, were implemented by Grooteman (994). A flexible plate coupled to a viscothermal air layer A finite element model in B was used to calculate the effects of the viscothermal air layer on the flexible plate. This was done for different widths of the air layer. In Figure 6. the eigenfrequencies of only the first three structural modes for the box as shown in Figure 6.9 are depicted versus the gap width h. It is noted that for these acousto-elastic coupled modes the structural part dominates the modes. Frequency [Hz] 3 mode mode 3 mode Calculations Damping coefficient ξ [%] mode mode Calculations Gap width h [mm] Gap width h [mm] Figure 6. Eigenfrequencies of the flexible plate (Beltman 998a). Figure 6. Damping coefficient of the flexible plate (Beltman 998a). For large gap widths the eigenfrequencies and mode shapes approach the ones in vacuum. When the gap width is reduced the influence of the air layer becomes apparent. The eigenfrequencies of the acousto-elastic coupled system differ considerably from the eigenfrequencies in vacuum. The eigenfrequency of the first mode increases initially but decreases for even smaller gap widths (the mode shapes for different gap widths are depicted in Figure 6.4). Similar results can be

160 5 seen for the 3-mode *. The -mode on the other hand shows a continuously decreasing eigenfrequency. As a result a cross-over of different modes can be seen. The finite element method (FEM) calculations were validated with experiments. The results in Figure 6. show that the low reduced frequency model and the acousto-elastic coupling are accurate. The amount of damping of the plate for each mode shape i can be calculated from the complex eigenfrequencies according to: Im( ωi ) ξ i = % (6.5) ω i where ξ i is the viscous damping coefficient and ω i the angular eigenfrequency. In Figure 6. the damping coefficients for the first and second modes are shown. Also here the experimental results agree reasonably well with the numerical ones. It can be seen clearly that the damping increases rapidly with decreasing gap width. Also, the -mode is more damped than the -mode. This is explained by the added stiffness and added mass effects on the dynamical behaviour of the plate, see Figure 6.3. mode: Added stiffness effect mode: Added mass effect V V= Figure 6.3 Added stiffness and mass effect due to the change in the cavity volume. For the -mode the air in the cavity is compressed and decompressed. Because of the acousto-elastic coupling the flexible plate behaves as if it possesses a higher stiffness compared to the situation in vacuum. The -mode on the other hand causes the air to be pumped backwards and forwards and there is no net volume change of the cavity. As a result of the coupling, the plate experiences an added mass effect, which in turn causes a large amount of damping due to viscous effects * The term 3-mode indicates that there are 3 half-wavelengths in the x-direction and half-wavelength in the y-direction.

161 53 in the air layer. It was shown that the thermal effects are of lesser importance for this kind of systems (Fox 98). The structural mode shapes in Figure 6.4 show that for a smaller gap width the symmetric modes, i.e. the - and the 3-mode, try to escape the change of volume under the plate by changing their mode shape. By doing so the added stiffness effect is reduced and the added mass effect is amplified. This explains the increase and decrease of the eigenfrequencies as shown in Figure 6.. For the asymmetric -mode the mass effect is the most important effect so that a continuous decrease of the eigenfrequency can be seen for a decreasing gap width. Mode h= mm h= 5 mm h = 3 mm 3 Figure 6.4 Structural modes of the acousto-elastic coupled flexible plate. The effect of resonators on the damping of a flexible plate For practical applications it may be desirable to reduce the vibrations of both the symmetrical and the asymmetrical modes. Obviously the symmetrical modes of a plate with a small air layer are less damped by the squeeze film effect than the asymmetrical ones. Therefore extra damping is needed for the frequencies corresponding with the symmetrical modes.

162 54 As a first attempt quarter-wave resonators are connected to the air layer to realise extra viscous damping for the frequency of the -mode. The set-up is shown in Figure 6.5. The plate is excited by a harmonic force F close to the centre of the plate. The response is measured with an accelerometer. In this way the frequency response H X/F of the plate can be determined, where X is the amplitude of the displacement (i.e. X& = ω X ). Resonators at the left side Flexible clamped plate Movable rigid bottom Excitation force F 49/ Accelerometer Ẋ. 49/ Resonators almost left Resonators in the middle Figure 6.5 Possible locations of the resonators to create extra damping in the narrow air gap. To determine the effect of the position of the resonator three locations have been investigated. The FEM model without the resonators as well as the experimental results showed that with a gap width of mm the -mode of the flexible plate is sufficiently damped. The resonance frequency of the -mode is approximately 33 Hz so the quarter-wave resonators were tuned to absorb maximally at 33 Hz (i.e. the length, the radius and the surface porosity are: L=.586m, R=.5mm, Ω=.44). The predicted effect of the resonators on the frequency response as well as the effect of the position of the resonators is given in Figure 6.6.

163 55 H X/F [m/n] 3 x 4 With resonators Without resonators Structural mode Structural mode Frequency [Hz] H X/F [m/n] 3 x 4 Structural mode no resonators r. in the middle r. almost left r. at left side Frequency [Hz] Figure 6.6 Calculated frequency response (magnitude) of the flexible plate (gap width = mm, frequency = 33 Hz). The amplitude of the response at the resonance frequency of the -mode has been reduced with more than 5 percent. Furthermore, the right-hand side shows that, as expected, the best position of the resonators is at one side of the air layer. For the -mode of the plate the corresponding pressure in the air layer shows a maximum at both ends of the gap, due to the closed ends, so that the effect of the resonators is more pronounced at that location. This can also be observed in Figure 6.7. Along the horizontal axis the length of the plate is shown and along the vertical axis the pressure perturbation. The pressure is normalised between. and. and the excitation force is maintained constant for both situations. The resonators are located 3 mm from the plate edge. The left-hand side of Figure 6.7 shows that the amplitude of the pressure perturbation in the air layer is reduced with more than 5 percent. So by withdrawing energy from the air layer, and thus from the complete coupled system, the plate is damped.

164 56 p [ ] With resonators Without resonators arg(p ) [rad] π 3/ π 5/4 π π 3/4 π/ Length [mm] With resonators π/4 Without resonators Length [mm] Figure 6.7 Calculated amplitude and phase of the pressure distribution in the narrow gap (gap width = mm, frequency = 33 Hz). Reference = F excitation. The right-hand side indicates that the resonators also introduce a phase shift. Therefore the now somewhat asymmetric pressure distribution results in a more pumping-like behaviour which produces extra damping of the plate. However, when an extra row of resonators is connected to the air layer at the right-hand side, so that the system is again symmetric, the damping is even higher (see Figure 6.8). Therefore it is concluded that the energy dissipation in the resonators mainly contributes to the damping of the plate. 3 x 4 Resonators at left side Resonators at both sides Without resonators H X/F [m/n] Frequency [Hz] Figure 6.8 Calculated frequency response (magnitude) of the flexible plate.

165 57 For the experiments the resonators were placed at one side in the rigid bottom plate 3 mm from the edge. The reason for this choice is that for a gap width of mm the resonators with a diameter of 3 mm simply cannot be placed at the far left side as indicated in Figure 6.5. Furthermore, the experimental results without resonators showed that the -mode lies at a somewhat higher frequency of 5 Hz. It was necessary to use a new flexible plate, apparently slightly thicker, which explains this higher frequency. The gap width is again mm. As a result the parameters for the quarter-wave resonators are: L=.56 m, R=.5 mm, Ω=.44. In Figure 6.9 the measured frequency response of the plate is given. x 4 5 mode 4 mode H X/F [m/n] 4 3 With resonators mode Without resonators Frequency [Hz] Figure 6.9 Measured frequency response (magnitude) of the flexible plate (gap width = mm). The response, H=X/F, with the resonators present is 35% of the original response. The magnitudes show the same behaviour as the calculated response so that it can be concluded that the effect of the extra resonators is correctly modelled, apart from the use of a new flexible aluminium plate. The asymmetric 4-mode is also shown in Figure 6.9 since the symmetric 3-mode lies in a higher frequency region due to the added stiffness effects. For this particular case the resonators have a small negative effect on the frequency response of the 4-mode.

166 58 Concluding remarks It was demonstrated in this section that resonators, connected to a viscothermal air layer, reduce the vibrations of the first symmetric mode of a flexible plate considerably. This principle can be used in applications where air-damped plates are present and in particular where a specific resonance frequency is present. As a next step one can think of the application of broadband resonators. The amount of damping can be increased in particular for frequencies for which the effect of the squeeze film damping is insufficient. For low frequencies the required length of the resonators is rather large. However, it is allowed to bend or coil the resonator to save space. This technique can be applied as long as the one-dimensional wave propagation in the resonators is preserved. It is also interesting to investigate the effect of Helmholtz resonators. 6.4 Reflection of sound in ducts with side-resonators This section deals with noise reduction in duct (or tube) systems such as air ventilation systems, gas transportation systems or exhaust pipes. This is achieved by placing tubes perpendicular to the direction of the sound propagation in the duct (see Figure 6.3). The tubes are referred to as quarter-wave side-resonators. A set of side-resonators can be considered as a reflection damper because noise is reflected in the duct whereas an absorption damper dissipates acoustic energy. Closed end Transmitted sound Duct Source of sound Sound wave Side-resonator Figure 6.3 Main duct with side-resonators.

167 59 It will be demonstrated that the coupled tubes model, including viscothermal effects, is a practical and efficient tool to predict the noise reduction in a duct downstream of the resonators. First the basic equations for a side-resonator, based on the recursive transfer function formulation, are given. The equations are used to determine, respectively, the acoustic impedance, the reflection coefficient, the insertion and the transmission loss in a duct when a row of side-resonators is applied. A design method is presented to create an optimal reflection coefficient. Finally, the numerical results are validated with experiments in an impedance tube. It was convenient to use the same impedance tube as applied earlier in the experiments described in Chapter 3 and 4. As a result the frequency range (from to 5 Hz) is rather high for normal exhaust pipes or air-conditioning ducts. For high frequencies it is common to use absorption dampers. Furthermore it is noted that the viscous effects become less important in wider duct systems: a times wider duct, R, decreases the frequency with a factor, k/, so kr remains constant whereas the resulting shear wave number s increases a factor Performance of mufflers The general term for a device that reduces noise in a duct and at the same time allows the passage of a fluid is a muffler. There are many publications on muffling devices, see for example Munjal (987) or Bies (996). A study on mufflers is beyond the scope of the present investigation. The main goal of this study is to verify the application of the coupled tubes model on side-resonators. The effect of mufflers is usually characterised by the Insertion Loss (IL) and the Transmission Loss (TL). The insertion loss is defined as the reduction (in db) of sound power transmitted through a duct with and without the muffler in place. The transmission loss of a muffler is defined as the difference (in db) of transmitted power by the muffler and incident power at the entry of the muffler. Muffling devices which are based upon reflection are called reactive while sound absorbing mufflers are called dissipative. The performance of reactive mufflers, such as devices with quarter-wave sideresonators, depends on the impedance of the source and the termination (outlet) of the duct. In the case of low impedance sources the energy reflected by the sideresonators has little effect on the generated incident energy. Typical examples of

168 6 low impedance sources, also called acoustic pressure sources, are for instance centrifugal and axial fans and impeller type compressors and pumps. However, when high impedance sources are used the reflected energy is built up in the duct and even more noise can be radiated at the outlet of the duct. High impedance sources are characterised by a fixed cyclic volume displacement and can be seen as constant acoustic velocity sources. A typical example is an oscillating piston in for instance compressors and pumps. In the experiments with an impedance tube the source can be seen as a pressure source because the loudspeaker is backed with a rather large box filled with absorption material. Basic equations for side-resonators A single side-resonator, connected flush to a duct, is considered (see Figure 6.3). It is assumed that frequencies of the propagating waves in the duct are below the cut-off frequency so that the one-dimensional coupled tubes model can be applied. The side-resonator is closed at one side, at p 4, so that a quarter-wave resonator results. At p a noise source is present and at p 3 the duct has an open end. For an open end the radiation impedance ζ rad can be applied (see Section.4.). p 4 3 p 4 3 p p 3 p p p 3 rad p Geometric Schematic Figure 6.3 Two coupled ducts and a side-resonator, geometric and schematic view. Following the notation as used in Chapter the transfer function for the sideresonator, with an acoustically hard wall at one side, becomes: p p 4 [ cosh( k ) ] = L Γ 3 (6.6)

169 6 Tube has a prescribed impedance boundary condition which gives the following transfer function: p p 3 Γ (6.7) ζ rad = cosh L ( k L) + sinh( Γ k ) Coupling these tubes gives the recursive transfer function for tube : p p cosh = ( Γ k L) ( Γ k L) sinh + A G 3 AG p cosh( Γ k L)... + sinh( Γ k L) p (6.8) A3G3 p4 N cosh( ) res Γ k L 3 sinh( ) Γ k L 3 p where N res is the number of identical resonators connected to the duct at the same axial location (see Figure 6.3). From equation (6.8) can be learned that a large number of resonators has the same effect as one resonator with a larger crosssectional area. With the transfer function p / p the acoustic impedance ζ and the reflection coefficient R at p can be calculated. The reflection coefficient is a measure for the amount of incident energy that is reflected. R and ζ can also be calculated at x = L R with the two-microphone method, at p and p, as shown in Figure 6.3. The dimensions of the duct and the resonators are of the same order of magnitude as used in the experimental set-up with an impedance tube. L R L end x L L R Lend L Rduct = 5 mm = mm =5mm =5mm pˆ A pˆ B pˆc pˆ D L resonator =6mm R resonator =5mm f resonance = 43 Hz p p rad Figure 6.3 Duct with a single row of side-resonators (second row = not active).

170 6 In the duct four plane travelling pressure waves can be distinguished with amplitudes p ˆ A, pˆ B, pˆ C and pˆ D. The sound energy of a plane wave in a duct can be estimated with: I prms p = = (6.9) ρc ρc where p is the amplitude of the plane wave. With the use of (6.9) an energy balance can be derived for a narrow control volume located at x = L R. It is noted that due to the viscothermal effects a small amount of the incident energy, α pˆ B, is dissipated in the resonators. I ( pˆ pˆ ) ( pˆ pˆ ) dissipated = Iin Iout α pˆ B = B A D C (6.) Rewriting (6.) gives the reflection coefficient R and transmission coefficient τ in a duct with side-resonators and a reflecting boundary condition (non-anechoic). pˆ ˆ ˆ α = A pd pc = R τ (6.) ˆ ˆ p p B B So in addition to the energy balance for a sound absorbing wall, where only α and R play a role, the transmission coefficient τ emerges. Obviously the transmission coefficient τ represents the fraction of the incident power that is transmitted downstream of the side-resonators. The coefficient τ is related to the transmission loss TL according to: TL = logτ [db] (6.) The insertion loss IL is defined as the reduction of transmitted power when a muffler is installed. Usually the same reduction is measured when the sound pressure level is used so that the IL can be written as: IL = SPL SPL [db] (6.3) without with where SPL without represents the sound pressure level downstream of the muffler while the muffler is inactive and SPL with the level when the muffler is active. In the numerical and experimental results presented in the following sections, the pressure was determined inside the duct so that the effect of the outlet on the IL was not taken into account.

171 63 Insertion loss can be seen as the quantity directly related to noise reduction of the source-duct-muffler system. The transmission loss is used in general to measure the effect of a muffler with an anechoic outlet, i.e. in a laboratory. In the special case of a pressure source and an anechoic termination of the duct ( p ˆ C = ) IL 7/ applies. It is noted that in the literature the performance of mufflers is often calculated with an electrical circuit to model the acoustic system. So-called lumped elements are used and acoustic analogies for electrical inductances, capacities and resistances are found. A consequence of the lumped element approach is however that acoustic wave lengths need to be larger than geometrical lengths. The continuous coupled tubes model lacks this restriction. A row of side-resonators The numerical set-up as shown in Figure 6.3 is used to determine the impedance and reflection coefficient at x = L R. To illustrate the effect of the side resonators first the viscosity and the thermal conductivity in both the duct and the sideresonators are neglected. H resonator.8 ζ [ ] R [ ] Hz No resonators resonator resonators 43 Hz Frequency [Hz] Figure 6.33 Impedance at x = L R. L resonator = 6 mm, R resonator = 5 mm. No resonators. resonator resonators 43 Hz Frequency [Hz] Figure 6.34 Reflection (magnitude) at x = L R. L resonator = 6 mm, R resonator = 5 mm. The impedance shows a sharp minimum at 43 Hz, which coincides with the resonance frequency of the resonators. This drop to a low impedance causes

172 64 reflection of incident waves as can be seen in Figure In this simplified numerical example it is apparently possible to reflect percent of the incident energy at 43 Hz when side-resonators are used. The reference curve, i.e. when no resonators are present, represents the reflection of waves at the outlet due to the radiation condition. At 35 Hz the reflection coefficient with the side-resonators active is lower than the reference curve. This indicates that these sound waves are amplified. In Figure 6.35 the effect on the amplitude of the pressure in the duct is shown for 43 Hz when resonators are installed. Obviously, sound of 43 Hz cannot pass the acoustic mirror because a pressure minimum is enforced. For 35 Hz the downstream part of the duct can be seen as a separate tube for which 35 Hz is a resonance frequency. 3 With side resonators Without resonators At 43 Hz At 35 Hz p [Pa]..4 ^.6 Length [m] Figure 6.35 Pressure (amplitude) in the duct. The sound pressure level at the outlet (at x = L R + L end ) with and without the sideresonators is given in Figure The peaks represent the resonance frequencies in the duct. These frequencies shift somewhat when the side-resonators are activated. A clear reduction of the SPL can be seen at 43 Hz. In the calculations a prescribed pressure is used. For a comparison the SPL of a velocity source is also given. As expected the resonance frequencies of the duct differ when a different boundary condition at the entrance (x = ) is used.

173 65 4 Pressure source Velocity source SPL [db] No resonators 6 resonators Velocity source Frequency [Hz] Figure 6.36 Sound pressure level (SPL) with and without the side-resonators attached. IL [db] TL Frequency [Hz] Figure 6.37 Insertion Loss (IL) calculated with a constant pressure source and a constant velocity source. The insertion loss (IL), being the difference between the SPL with and without the side-resonators, is shown in Figure At 43 Hz a high IL can be seen. The multiple maximums and minimums at lower and higher frequencies are the result of the shifted eigenfrequencies in the duct. For a comparison also the transmission loss (TL) is shown. A high transmission loss can be seen at 43 Hz and a low TL at 35 Hz. The increasing TL for low frequencies is the result of the reflected wave pˆ due to the radiation condition at the outlet. C Design method for side-resonators The negative effect of the side-resonators on the transmitted sound for frequencies other than f resonance can be reduced when the resonators disturb the impedance of the original duct as less as possible. This can be achieved by choosing f resonance of the side-resonators at a frequency for which the impedance in the duct already shows a minimum. The results for the impedance and the reflection coefficient are illustrated in Figure 6.38 and Figure In this example the length of the resonators is now 69 mm, corresponding with f resonance = 3 Hz.

174 66 No resonators resonators resonators (s + sσ).8 ζ [ ] Frequency [Hz] Figure 6.38 Impedance at x = L R. L resonator = 69 mm, R resonator = 5 mm. With and without viscothermal effects (s + sσ). R [ ].6.4 No resonators. resonators resonators (s + sσ) Frequency [Hz] Figure 6.39 Reflection (magnitude) at x = L R. L resonator = 69 mm, R resonator = 5 mm. With and without viscothermal effects (s + sσ). The sharp minimum for the reflection coefficient is now absent (compare to Figure 6.34). This can be advantageous for a source for which the small bandwidth noise shifts somewhat during operation. It is also demonstrated here that the viscothermal effects in the resonators, represented by s and sσ, reduce the reflection coefficient. Due to the viscosity and the thermal conductivity the resonance is less pronounced so that the enforced pressure minimum at x = L R is less. The resonators have a radius of 5 mm for which the shear wave number is rather high (s VRWKH reduction is small. The accompanying IL and TL are shown in Figure 6.4 and Figure 6.4, respectively. Around f resonance there is no longer a dip in the IL nor in the TL. As a comparison the TL for the old configuration with a resonator length of 6 mm is also plotted. It shows a reduction of 5 db at f resonance but for varying frequencies of the noise source a more broadband reduction is seen for the new resonator lengths of 69 mm. An IL and TL of approximately 5 db can be realised with sideresonators.

175 67 4 Pressure source (no s + sσ) Pressure source Velocity source 4 IL [db] TL [db] Frequency [Hz] Figure 6.4 Insertion loss for a constant pressure and velocity source. No viscothermal effects Viscothermal effects Old resonator length (6 mm) Frequency [Hz] Figure 6.4 Transmission loss with and without viscothermal effects (s + sσ). Instead of adapting the resonator length it is also possible to choose an axial position in the duct in order to match f resonance with an impedance minimum of the duct without the side-resonators in place. For multiple rows with side-resonators which are designed for different frequencies (see Figure 6.3), noise containing a wider frequency band can be reflected. It is noted that for rows with similar resonant frequencies non-planar wave interaction may occur between the rows. In that case a rule of thumb describes that the rows with resonators have to be placed more than a quarter-wave length apart (Howard ). The strategy for a single row of resonators can be applied for multiple rows as follows:. The first row of resonators is placed so that it operates at a low impedance of the original duct,. The impedance in the duct is calculated with a single row of resonators upstream of the first row, 3. The second row of resonators is placed so that it operates at a low impedance, 4. Repeat step and 3 for more rows. Comparison with experiments To verify the predicted performance of the side-resonators the impedance tube as described earlier in Chapter 3 was used, see Figure 6.4. The two-microphone

176 68 technique was used to determine the reflection coefficient. In the first test set-up radial side-resonators were used. The length of the resonators could be varied via an adjustable stop. For applications such as exhaust pipes a second set-up was used with axially oriented side-resonators to reduce the radius of the device. The experimental results of the latter set-up are presented in this section. It is noted that for the one-dimensional coupled tubes model obviously the orientation of the tubes is not important. loudspeaker microphones& impedance tube side-resonators baffle open end adjustable stop side-resonator radial side-resonators axial side-resonators Figure 6.4 Impedance tube with radial or axial side-resonators. The length of the resonators as used in the numerical model is shown in Figure This length is not corrected for inlet effects. For normal incident waves the inlet effects can be accounted for by adding an incremental length with a maximum of 8R/3π (see section.4.). This value could be used as an upper limit for sideresonators. However, further information or investigation is needed to determine the correction for a number of axially and radially oriented side-resonators.