Noise reduction with coupled prismatic tubes

220
Noise reduction with coupled prismatic tubes Frits van der Eerden

Transcript of Noise reduction with coupled prismatic tubes

Noise reduction with coupled prismatic tubes

Frits van der Eerden

Steffingen

behorende bij het proefschrift

Noise reduction with coupled prismatic tubes

De lucht of de vloeistof in gekoppelde prismatische buisjes kan loodrecht invallendegeluidsgolven uit een breed frequentiespectrum grotendeels absorberen mits de juistecondities worden gekozen. (Dit proefschrift, Hoofdstuk 2)

Het 'low reduced frequency' model beschrijft nauwkeurig de stroperigheid enwarmteuitwisseling van lucht aan de wand en als gevoig daarvan de absorptie vangeluidsenergie. (Dit proefschrift, Hoofdstuk 4)

Geluid absorberend gedrag van materialen zoals schuim of glaswol kan wordenvoorspeld met een model dat bestaat uit een labyrint van kleine gekoppelde buisje. (Ditproefschrift, Hoofdstuk 5)

Bij intensief computergebn.iik dient de werkplek een raam met een aantrekkelijk ofgevarieerd vergezicht te bieden zodat de ogen voldoende periodes van rust krijgen.

De verruiming van de openingstijden van supermarkten, tot we! tien uur 's avonds,bevordert de onthaasting.

In het kader van efficiëntie is het gebruik van een grote hoeveelheid albeeldingenaantrekkelijk omdat de hoge informatiedichtheid van afbeeldingen de hoeveetheidbenodigde tekst aanzienlijk vermindert. Het eerste proefschrift in stripvorm moet helaasnog geschreven worden.

De uitdrukking 'een verhitte discussie' is misleidend, aangezien de daarbij gebruikteakoestische energie in termen van Watts weinig voorstelt.

Geluidsbeheersing is typisch menselijk aangezien het produceren van geluidsoverlastnet zo menselijk is.

Gezien de goede akoestische isolatie van thermische isolatie is te verwachten dat degeluidsoverlast in klimaten met meer extreme temperaturen minder is.

De tekstverwerker voor versiaggeving van technische aard is te classificeren in tweetypes: het programmeer-type en het WYSIWYG-type (What You See Is What YouGet). Het ugt in de lijn der verwachting dat voor de bijbehorende gebruikers dezelfdeindeling opgaat.

Frits van der EerdenNovember 2000

This research was supported by the Dutch Technology Foundation (STW). Project

TWT.3735.

De promotiecommissie is als voigt samengesteld:

Voorzitter en secretaris:Prof.dr.ir. H.J. GrootenboerPromotorProf.dr.ir. H. TijdemanLeden:

Dr.ir. W.M. BeitmanProf.dr.ir. W.F. DruyvesteynDr.ir. A. HirschbergProf.dr.ir. H.W.M. HoeijmakersProf.dr.ir. J.W. Verheij

Paranimfen:ir. T.G.H. (Tom) Bastenir. M.E. (Marten) Toxopeus

Eerden, van der, Fredericus Joseph MarieTitle: Noise reduction with coupled prismatic tubesPhD thesis, University of Twente, Enschede, The Netherlands

November 2000

ISBN: 90-36515211Subject headings: acoustics, sound absorption, resonators

Copyright © 2000 by F.J.M. van der Eerden, Enschede, the Netherlands

Printed by Ponsen & Looijen by., Wageningen

Universiteit Twente

Universiteit Twente

Intel, USAUniversiteit TwenteTechnische Universiteit EindhovenUniversiteit TwenteTechnische Universiteit EindhovenlTNO-TPD

NOISE REDUCTION WITH COUPLED PRISMATIC TUBES

PROEFSCHRIFT

ter verkrijging vande 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 Promotiesin het openbaar te verdedigen

op vrijdag 24 november 2000 te 16.45 uur.

door

Fredericus Joseph Marie van der Eerden

geboren op 13 december 1971te Tubbergen

Dit proefscrift is godgekeurd door de promotor:Prof.dr.ir.H. Tijdeman'

Summary

With so little time available nowadays for design and development, there is astrong incentive to simulate the behaviour of designs in 'virtual reality', rather thanto perform expensive and time-consuming measurements on expensive prototypes.For noise reduction, which has become an important topic in acoustics, the sametrend 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 theeffectiveness of sound absorbing materials must be known accurately. The presentinvestigation focuses on an accurate description of sound absorption. Within thisresearch a new technique to create sound absorption for a predefined frequencyband has been developed. Additionally, a simple and efficient numerical model forconventional sound absorbing materials, such as glass wool or foams, has beenformulated. It is also demonstrated that the newly gained insights are useful inapplications not directly related to sound absorption.

As a basis for the research a description of pressure waves in a single narrow tubeor pore has been used. In such a tube the viscosity and the thermal conductivity ofthe air, or any other fluid, can have a significant effect on the wave propagation.This so-called viscothermal wave propagation results in energy being dissipatedand the effective speed of sound inside the tube can be considerably reduced. Thisprinciple of energy dissipation has been applied to configurations consisting of amanifold of tubes, the so-called coupled tubes. A design strategy was developed tocreate broadband sound absorption for a wall with configurations of coupled tubes.These broadband resonators can be optimally designed so that they absorb incidentwaves for a predefmed frequency range. Experiments in an impedance tube, orKundt' s tube, proved that both the viscothermal effects on the wave propagationand the design tool are very accurate.

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 theoiy.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 twomicrophones, the use of a particle velocity sensor, the microflown, has beeninvestigated experimentally in the course of the project. With the use of acombination of a microphone and a microflown direct information on the acousticimpedance, the sound intensity and the sound energy density was obtained.

Contents

Summary

Introduction i1.1 Background i1.2 Aim of the investigation 41.3 Outline 5

Optimised sound absorption with coupled tubes 92.1 Introduction 92.2 Viscothennal wave propagation in prismatic tubes 122.3 Coupled tubes 24

2.3.1 Recursive formulation 242.3.2 Transfer matrix formulation 29

2.4 Broadband sound absorption 332.4.1 Sound absorption of a single tube 342.4.2 Sound absorption of coupled tubes 46

2.5 Conclusions 55

Impedance tube techniques to measure sound absorption 593.1 Introduction 593.2 Impedance tube techniques 65

3.2.1 The 2p method 683.2.2 The 2u method 693.2.3 The p/u method 733.2.4 The p.0 method 74

3.3 Experimental results 793.3.1 Comparison of the 2p and the 2u methods 793.3.2 Comparison of the 2p, the p/u, and the pu methods 84

3.4 Conclusions 87

Experimental verification of the coupled tubes model 894.1 Introduction 894.2 Sound absorption of single tube resonators 90

4.3 Sound absorption of coupled tubes resonators 94

4.4 Conclusions 98

Sound absorbing material represented by a network of tubes 101

5.1 Introduction 101

5.2 A random network of tubes 103

5.3 Comparison to an empirical impedance model 108

5.4 Comparison to the Limp theory 113

5.5 Comparison to the Biot theory 118

5.6 Conclusions and remarks 125

Further applications of the coupled tubes model 127

6.1 An inkjet array 128

6.2 A test set-up for a voice producing element 137

6.3 A viscothermally damped flexible plate 148

6.4 Reflection of sound in ducts with side-resonators 158

Conclusions 173

List of Symbols 175

References 179

Sound absorption mechanism of a wall with resonators 187

Propagation coefficient for different cross-sections 197

Calibration of the inicroflown 200

The Limp theory for fibrous sound absorbing materials 203

Derivation of Biot's equations of motion 207

Samenvatting (in Dutch) 211

Nawoord 213

Levensloop 215

Chapter 1

Introduction

1.1 Background 1

1.2 Aim of the investigation 4

1.3 Outline 5

1.1 Background

A typical example of an experience of noise could be the sound of a truck passingby while enjoying a quiet sleep early in the morning. The nuisance caused by noiseis seen in situations at home as well as at work. One can think of noisy domesticappliances, 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 thecomplete noise problem. For example important noise paths or vibration levels ofpanels need to be known. A well-known method for the insulation of noise is theuse of sound absorption techniques or sound absorbing materials. The acousticbehaviour of a wall with sound absorbing resonators, and to be more specific, anumerical technique to predict its behaviour, is the topic of the presentinvestigation.

A sound field

Sound can be described as a 'sensation produced at the ear by very small pressurefluctuations in the ear' (Bies 1996). Pressure and pressure fluctuations can bemeasured relatively easily in engineering. The unit for pressure is the pascal (1 Pa

= i N/rn2). It is known that the ear responds approximately logarithmically tosound energy input which is proportional to the square of the sound pressure. The

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ear can detect pressure fluctuations ranging from 20 tPa, for young persons, to 60Pa before pain is experienced. Note that the mean pressure of the atmosphere isabout i.0.i05 Pa. The high dynamic range for the square of the pressure iscompressed into logarithmic units for convenience termed decibels or dB. Thedynamic range thus varies from O to 130 dB. Sound pressure is usually measured indB 's with a sound level meter.

A sound field can be described as small perturbations of steady state variableswhich describe a medium through which the sound is transmitted. For a fluid suchas air or water the variables are: the pressure [Pa], the particle velocity [mis], thetemperature [K] and the density [kg/rn3]. The total value of these variables can beseen as a small perturbation superimposed on the mean value (the steady state).The pressure, the temperature and the density are scalar quantities whereas theparticle velocity is a vector. The word 'particle' is used here for a very small partof the medium and not for the molecules of the medium itself.

The speed with which a disturbance propagates through air or water is called thespeed of sound. The disturbance may be described as the sum of small hannonicperturbations. So in general a sound wave contains several frequencies.

o 1 2 311111 I il III

fr,

0 1 2 3 4 cm¡ J r r J ii ii i i il

Figure 1.1 A foam sample (left) and a glass wool sample (right) with a close-up above.

i

0.8

0.6

0.4

0.2 - Glass wool (50 mm thick)Glass wool (25 mm thick)

- - Foam (15 mm thick)

O125 250 500 1000 2000 4000

Center frequency of octave band [Hz]

3

The influence of the viscosity of air is normally very low and shear forces hardlyeffect the propagation of sound waves in the open air. However, the viscosity of airbecomes much more important when the air is trapped in sound absorbingmaterials. Due to the viscosity, sound energy is dissipated as heat inside thematerial. In fact the energy density of sound is small, even for sound levels ofabout 130 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 1.1. These materials are usedfor their good sound absorbing behaviour, although glass wool was originally usedfor thermal insulation.

A typical graph which depicts the sound absorption coefficient a of foam andglass wool as a function of the frequency is given in Figure 1.2. A value of a = 1.0indicates that incident waves are completely absorbed whereas a = 0.0 meanscomplete reflection of sound waves.

Figure 1.2 Absorption coefficient as a function of the frequency. Data obtained fromliterature (Bies 1996). Indication of the density: wool 60 kg/rn3, foam 27 kg/rn3.

The absorption coefficient of materials such as fibreglass, rockwool orpolyurethane foam is usually obtained empirically. However, it would be moreefficient if the effectiveness of sound absorbing materials could be predicted.Therefore it is not surprising that a variety of empirical and analytical models have

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been developed to predict the sound absorbing qualities of materials (see forinstance: Allard, Attenborough, Biot, Bolton, Cunmiings, Delaney, Ingard,Lauriks). Moreover, such methods offer the opportunity to improve the soundabsorbing qualities or to design new ones via numerical simulations instead ofperforming extensive series of experiments.

For the present investigation the acoustic behaviour of air inside a single pore ornarrow tube will be taken as a basis for studying sound absorption. In Figure 1.1small cavities and pores in the foam can be seen. This explains the assumption thatsound absorption is created inside a set of small tubes (see also Biot).

Zwikker and Kosten (1949) presented a model for wave propagation incylindrical tubes which included the effects of viscosity and thermal conductivityof the medium. This model for the so-called viscothermal wave propagation provedto be complete and accurate for both 'narrow' and 'wide' tubes (Tijdeman 1975).

It is noted that in the present investigation the effect of a mean flow through thetubes or ducts is not taken into account. However, for low flow velocities the wavenumbers of the forward and backward travelling waves can easily incorporate asmall mean flow component.

It is also noted that the use of sound absorbing material can be characterised as apassive method, as opposed to active methods. An alternative technique for noisereduction is active noise control (ANC). This technique uses active elements in aroom or a structure. Examples are loudspeakers which generate anti-noise oractuators such as piezo-materials which are attached to or imbedded in a structureto reduce radiated or transmitted noise.

1.2 Aim of the investigation

Following the considerations of the previous section the aim of the investigation isformulated as follows:

Develop and validate a numerical tool for absorptive structures, based on theviscothermal wave propagation in narrow tubes, so that the sound reduction canbe calculated accurately and efficiently.

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In order to reach this aim a number of basic aspects and questions need to beinvestigated:

. What is the sound absorbing behaviour of a single pore or narrow tube andwhat are the effects if such a tube is scaled to a larger size for moreconvenient acoustic experiments.

s What is the acoustic effect when tubes are connected or coupled to eachother.

s Can a detailed network of coupled tubes be used to represent soundabsorbing materials.

. Are there other type of acoustic problems which can benefit from thedeveloped numerical tool.

1.3 Outline

First, viscothermal wave propagation in prismatic tubes is described according tothe so-called 'low reduced frequency' model. An important parameter in this modelis the non-dimensional shear wave number which is a measure for the ratio of theinertial and viscous forces. It can be seen as an acoustic Reynolds number. Theprismatic tubes are coupled via a mass and momentum balance. Two formulationsfor coupled tubes are presented: a recursive formulation and a transfer matrixformulation.

For a single tube the model predicts the acoustic impedance, and therefore thesound absorption coefficient, at the entrance of the tube. When the other end of thetube is acoustically hard a quarter-wave resonator is obtained. It is demonstratedthat a wall with a uniform distribution of such resonators can absorb incident soundwaves 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

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frequency band. The design tool for an optimal sound absorbing wall for apredefmed frequency range is described in more detail in Chapter 2.

To validate the design tool measurements need to be performed. A relativelysimple and efficient method is the impedance tube as described in Chapter 3. It isshown how three alternative techniques make use of a new particle velocity sensor.The combination of a particle velocity sensor and a microphone provides directinformation on the acoustic impedance, the sound energy flow and the soundenergy density.

The actual validation of single tube resonators and coupled tubes resonators isdescribed in Chapter 4. A very good agreement between the experimental andpredicted results is observed. For example the effect of the number of resonatorsper unit area has been investigated. The experiments are performed for frequenciesup to 4000 Hz.

A random network of small narrow tubes is used in Chapter 5 to model theacoustic behaviour of fibrous and porous materials. With a relatively simple andcompact set of parameters different materials can be characterised. A comparisonwith empirical and analytical models validates the network approach.

The numerical tool for coupled tubes is successfully used in four applications, asdescribed in Chapter 6. In two cases sound absorption is not the issue but merelythe technique to predict the acoustic behaviour of coupled tubes. One case dealswith noise reduction in duct-systems such as air-conditioning systems. A designstrategy 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 beabsorbed, so that the sound level downstream of the position of the resonators isreduced.

The other applications are:

A test set-up consisting of a number of coupled tubes has been developed tomeasure the acoustic behaviour of a voice producing element.

In an inkjet array constantly propagating waves are created in a shortchannel with ink because at one end of the channel complete soundabsorption is realised with a broadband resonator.

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The vibrations of a flexible plate have been reduced by withdrawing extraenergy from a trapped air-layer via resonators. The flexible plate and the air-layer are strongly coupled.

Finally the conclusions of the present thesis are presented in Chapter 7.

Chapter 2

Optimised sound absorption with coupled tubes

2.1 Introduction 92.2 Viscothermal wave propagation in prismatic tubes 122.3 Coupled tubes 24

2.3.1 Recursive formulation 242.3.2 Transfer matrix formulation 29

2.4 Broadband sound absorption 332.4.1 Sound absorption of a single tube 342.4.2 Sound absorption of coupled tubes 46

2.5 Conclusions 55

2.1 Introduction

Two classes of sound absorbing structures can be distinguished in general: porousmaterials and resonance absorbers (Hecki 1995). The material of the first class hasa micro-structure of pores or fibres. Well-known examples are glass wool andsynthetic foam. These materials show a broadband sound absorbing behaviourabove a certain frequency. When the material is backed by an acoustically hardwall one can use the rule of thumb that acoustic waves are well absorbed forfrequencies for which a quarter of the corresponding wavelength is smaller than thethickness. This means that for lower frequencies the thickness of the appliedporous material has to be increased. Special porous materials that consist ofceramics or metal have been developed for applications where conventionalmaterials cannot be used (Koketsu 1980, Banhart 1994). The sound absorbingperformance is less than that of conventional porous materials but their strengthand 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 actualthickness or porosity can be a cause for large variations in the sound absorbingcharacteristics. Furthermore, a range of parameters, as given for example by the

lo

theory of Biot, have to be measured for different operational conditions. It mayalso be difficult to interpret the significance of each parameter with respect to theacoustic behaviour.

In the second class of sound absorbing structures the resonance of air or theresonance of panel-like structures is used. As an example a perforated panel and apanel absorber are illustrated in Figure 2.1.

porous liner

[i

V

Perforated panel absorbers Panel absorber

Figure 2.1 Perforated panel absorbers and a panel absorber.

A perforated panel can be seen as a row of Helmholtz resonators. A Helmholtzresonator basically can be seen as a mass-spring system with a resonance frequency

of û = 2llfo = . The spring stiffness k is represented by a volume V of airand the mass m is the mass of a small column of vibrating air in a perforation of thepanel. The vibrating air dissipates sound energy. Extra sound energy can bedissipated by placing a liner of porous material in the volume.

Perforated panels are used for the absorption of a small band of low and mediumfrequency sound. The panels use a relatively small volume. There is a considerablenumber of publications on sound absorbing materials with perforated and micro-perforated facings. For details reference is made to for instance: Ingard 1951,Heckl 1995, Kang 1999.

The same remark applies for panel absorbers. These are used for low frequencysound and also show a mass-spring resonance behaviour. A thin panel or foil isbacked by a narrow air layer that acts as a spring. The mass is represented by themass of the resonating panel. Extra porous material is used to create sufficient

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energy dissipation. Panel absorbers are generally used to absorb particular lowfrequency noise.

Besides Helmholtz resonators also so-called quarter-wavelength resonators canbe used. A wall or section with a number of these tube-like resonators is capable ofabsorbing sound of a specific frequency. Quarter-wavelength resonators, or with ashorter notation: quarter-wave resonators, are applied in general for low frequencysound. In a quarter-wave resonator the mass and spring functionality iscontinuously distributed in the tube. As a results also higher order modes can beobserved. For a Helmholtz resonator a volume with a uniform pressure distributionis required whereas a slender tube can be used for a quarter-wave resonator, whichmay be an advantage for specific applications.

In this chapter it will be shown that the effect of these type of tube resonators canbe considerably improved. The small frequency band for which the quarter-waveresonators absorb sound can be widened considerably by coupling tubes.

A simple and well-defmed sound absorbing wall with an accompanying efficientand accurate model is presented including a strategy to design such a wall. To bemore specific: the coupled tubes inside the wall are designed in such a way that atthe surface an optimal acoustic boundaiy condition is created. For a predefinedlow, 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 widefrequency band. Furthermore, the resonators can be constructed in materials thatwithstand high temperatures or aggressive environments.

The mechanism for a broadband sound absorption is the dissipation of soundenergy and the cancelling of the incident acoustic waves due to a broadbandresonance of air, or any fluid, in the coupled resonators (see Appendix A). A singletube resonator shows a significant absorption for a specific first resonancefrequency (and in general less noticeable for number of higher harmonics). Wavepropagation in a single tube or resonator forms the basis for the acoustic model. Itis shown that the viscous and thermal effects play an important role in the wavepropagation. As a next step a system of coupled tubes is described. Twotechniques, one analytical and one numerical, are developed to calculate theacoustic behaviour of a system or network of coupled tubes and ducts. The lastsection deals with broadband sound absorption of coupled tubes. The effect of thedifferent parameters on the sound absorption is shown and an efficient technique to

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compute the optimal configuration with coupled tubes for a given frequency band

is presented.

2.2 Viscothermal wave propagation in prismatic tubes

Sound propagation in prismatic tubes has been investigated thoroughly by manyauthors. A number of analytical models was presented on the basis of the followingequations: the linearised Navier-Stokes equations, the equation of continuity, theequation of state for an ideal gas and the energy equation. For an overview seeTijdeman (1975) and Beitman (1999a).

In the following sections three analytical solutions for wave propagation inprismatic tubes are presented: the Helmholtz equation which neglects viscothennaleffects, the 'low reduced frequency' solution which includes viscous and thermaleffects, and the Kirchhoff approximation which is a first order approximation ofthe low reduced frequency solution. Next, boundary conditions of a prismatic tubeand reflection and absorption coefficients in the tube are presented.

Helmholtz equation

For the most simple situation, for example the propagation of plane waves in freespace without effects of steady state temperature differences, a main flow (such as

the wind) or attenuation due to for example atmospheric absorption, the resultingwave equation for (x, y, z,t) reads:

ax2 ay2 az2 02 at2= o (2.1)

In equation (2.1) p is the pressure perturbation with respect to the mean pressure poin the air. It is a function of time t and position, where x, y and z form therectangular Cartesian co-ordinate system. The coefficient c0 is the speed of sound

in the quiescent space. For the special case of a harmonic time dependence e+iO) t

i.e. (x, y, z, t) = p(x, y, z) et (O t, the equation for the pressure perturbation reads:

a2pa2p+a2P+k20 (2.2)ax2 ay2 az2

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where k is the wave number defined as 0)/c0 with ti, the angular frequency.Equation (2.2) is the so-called Helmholtz equation. In the present investigation thepressure perturbation is assumed to be harmonic for convenience. For a longprismatic tube with rigid walls (see Figure 2.2) equation (2.2) reduces to the one-dimensional Helmholtz equation which has the solution:

p(x) = A e + PB (2.3)

The sound field of (2.3) consists of a plane wave with a complex amplitude PBtravelling in the positive x-direction with speed c0 and a plane wave with amplitudePA travelling in the negative x-direction also with speed c0.

Figure 2.2 A prismatic tube.

The amplitudes are determined by the boundary conditions on both ends of thetube. 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 frequencythe acoustic wavelength is smaller than about half the characteristic dimension ofthe tube cross-section (for a circular cross-section the cut-off frequency isdetermined byf= c0 / i .7d, with d the diameter of the tube).

With the use of the linearised momentum equation (Newton's second law and(x) a function of location and of time via e+bO)t),

ap/ax au/at (2.4)

and omitting the time dependence e+0)t the particle velocity distribution isobtained:

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i ikx -ikx'u(x)=pAe pBe )Poco

In (2.5) Po is the undisturbed or mean density. The quantity Poco is termed the'characteristic specific acoustic impedance' of the fluid and represents theimpedance of a freely travelling plane wave. The acoustic impedance is the ratio ofthe pressure perturbation and the velocity perturbation (in the case of a porousmaterial the velocity is usually directed into the surface and out of the fluid). Theterm impedance stems from the field of electrical engineering (by O. Heaviside in1886) and was later introduced in mechanics where the ratio of the force and thevelocity is referred to as impedance (i.e. "something impeding motion"). In 1914

A.G. Webster introduced impedance in acoustics. A sudden change in the acousticimpedance causes the reflection of acoustic waves. This property of acoustic wavepropagation will be used in section 2.4.

Low reduced frequency model

In small tubes or layers the wave propagation is affected by the viscosity and thethermal conductivity of the fluid. There is a variety of literature on viscothermalwave propagation of which Tijdeman (1975) and Beltman (1999) give an extensive

overview of the different analytical solutions. Of the many approaches the one of

Zwikker and Kosten (1949) has proven to be an efficient and accurate appraoch fora large number of acoustical problems. They assumed a constant pressure acrossthe tube cross-section and included the effects of inertia, compressibility, viscosityand thermal conductivity of the fluid. Their solution for the wave propagation incylindrical tubes, the 'low reduced frequency model' is written in a dimensionlessform by Tijdeman and is characterised by four dimensionless parameters:

s = i Ie-, the shear wave number

kr = i the reduced frequencyco

I/ico. = , the square root of the Prandtl number

(2.5)

(2.6)

(2.7)

(2.8)

cp7='cv

The shear wave number s is a measure for the ratio between the inertial and viscousforces and can be seen as an unsteady or acoustic (square root of the) Reynoldsnumber. It is a function of, amongst others, the dynamic viscosity p and thecharacteristic length of the cross-section i, i.e. for a layer this is half the layerthickness hi2 and for a tube with a circular cross-section this is the radius R. Forsmall values of s the viscous effects are dominant. In that case a tube is called'narrow'. For s » 1, when inertia effects are dominant, the tube is called 'wide'.The reduced frequency ¡ç represents the ratio of the characteristic length of thecross-section and the acoustic wavelength. It is noted that the viscous effectsbecome less important for a wider tube: a 10 times wider tube, bR, results in 10times lower frequencies, ¡ç /10, 50 krR remains constant whereas the resulting shearwave number s increases a factor i IO.

For a given 'ideal' fluid oand ycan be considered as constants; in the case of airthe Prandtl number equals about 0.71 (so o= 0.84) and y= 1.4. In (2.8) and (2.9)C. and Cv are the specific heats at constant pressure and volume, respectively, and2 is the thermal conductivity.

Tijdeman and Beitman show that the different models for viscothermal wavepropagation in tubes and layers, respectively, can be put into perspective by usingthe dimensionless parameters kr and s. The range of validity of the models isgoverned by the range of k,. and k,. / s. Furthermore Beitman demonstrates that formost acoustic problems the low reduced frequency model is sufficient. Comparedto more complex models, such as the simplified Navier-Stokes model, the lowreduced frequency model is a relatively simple one.

The reduced frequency model is derived from the basic equations for thepropagation of sound waves in absence of mean flow. These are: the linearisedNavier-Stokes equations, the equation of continuity, the equation of state for anideal gas and the energy equation. In general they can be written as:

p = .ii)pVx(Vxü) (2.10)

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the ratio of the specific heats (2.9)

(2.11)

16

(2.12)

pocp (2.13)

where the over-bar indicates the total quantity, i.e. the mean value plus the smallperturbation and a bolt symbol represents a vector. R0 is the gas constant (R0 = Ç, -

Cv), T is the temperature and V and ¿ are the gradient and Laplace operators,respectively. For monatomic gases applies the bulk viscosity 17 = O, for air 17 =O.641u. 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 1, i.e. k,. « 1;the acoustic wavelength is large compared to the boundary layer thickness,i.e. kriS« 1.

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 thevelocity perturbation in the axial x-direction and y the velocity perturbation in theradial y-direction. The solution for the acoustic variables p, u, y, T and p can nowbe found but in our case the solutions for the variables p and u are sufficient. Theresulting complex pressure and velocity perturbations are given as:

p(x) = PA + PB e'

G (p Fkx -rkxu(x)= A pBe )poco

These expressions are quite similar to the ones derived for the Helniholtz equation.The coefficient G is explained later. The main difference is the viscotherrnal wavepropagation coefficient T It is a complex quantity, 1= Re(r) + i Im(E), where c0I Im(E) represents the phase velocity and Re(r) accounts for the attenuation of apropagating wave. It is noticed that Eis a function of the shear wave number s andtherefore a function of the frequency. The velocity u(x) is averaged over the cross-

(2.14)

(2.15)

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 thetemperature 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 2.3 (Tijdeman 1975). 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.

Q

0.5 1 12Scaled velocity [-1

Figure 2.3 Velocity profile (magnitude) in a circular tube.

The wave propagation coefficient T depends on the geometry of the cross-section

of the tube or layer. Stinson (1992) formulated a general expression for the

propagation coefficient. For a circular, a rectangular and a triangular cross-section

Fis 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:

17

18

r_0ï

- J2(i-./is) n

G=_-'-ZTn

Jo and J2 are Bessel functions of the first kind of order O and 2, respectively. Thecoefficient n can be interpreted as a type of polytropic coefficient which relates thepressure and density, integrated over the cross-section, according to:

-- = constant (2.18)

n is a complex number and is a function of so

n=1l+ y-1 J2(isa)]lï J0(i-4fiso)

For small values of sa the polytropic coefficient n reduces to one, i.e. isothermalwave propagation, and for large values of sa the polytropic coefficient n becomesequal to ywhich corresponds to adiabatic wave propagation (see Figure 2.4).

(2.19)

1.1

lo

8

2

o

Figure 2.4 Polytropic coefficient (magnitude and phase) for a cylindrical tube.

0_1 100 101

sa [-]102 102loo loi

sa [-J

(2.16)

(2.17)

Low reduced freq.-. Kirchhoff

- - No viscothermal eff.

loo io' 102

s [-J

1

0.8

0.6

E0.4

0.2

o

- Low reduced freq.-. Kirchhoff

- - No viscothermai eff.

10' 102

Figure 2.5 Propagation coefficient for a cylindrical tube (real part and inverse ofimaginary 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 Kirchhoffapproximation approach the non-viscothermal wave propagation coefficient F= i.

19

Kirchhoff approximation

For large values of the shear wave number s, i.e. for 'wide' tubes, the propagationcoefficient r can be estimated by a first order approximation of the low reducedfrequency solution. In the present investigation large values of the shear wavenumber frequently appear. Therefore, the Kirchhoff approximation is applied,which is more efficient in terms of computing time than the low reduced frequencysolution. The propagation coefficient according to Kirchhoff can be written as:

i+l(y-1+cr (2.20)so.

The associated coefficient G is defined by:

G=_L (2.21)rBoth viscous and thermal effects are included in (2.20) via s and sci, respectively.

The propagation coefficient for a cylindrical tube according to the low reducedfrequency model, the Kirchhoff approximation, and the Helniholtz equationwithout viscothermal effects, are compared in Figure 2.5.

loo 10's [-J

T15-

'. 10 -

5- \

25

20

20

It can be seen that the Kirchhoff approximation can be used for s >10. The real partof F represents the attenuation of the propagating wave whereas the reciprocal ofthe imaginary part represents the effective phase velocity according to:

coCecc -

Im(T)

Boundary conditions

The amplitudes of the forward and backward travelling waves in a tube aredetermined by the boundary conditions at both ends of the tube. The boundaryconditions can be expressed in terms of pressure, velocity or acoustic impedance.In Figure 2.6 the possible boundary conditions are shown. For a closed end, i.e. foran acoustically hard wall, the velocity is zero. Note that in the remaining part ofthis thesis the term 'closed end' is used for an acoustically hard termination of atube. For a tube terminated with a sound absorbing material an impedancecondition is used. For pressure measurement systems, for example in a wind tunnelmodel where a thin tube connects the pressure transducer to the actual inputpressure, the bounduy conditions are the pressures at the entrance and at the end ofthe tube.

Po x Pl - pressure perturbation

U0 u1 - velocity perturbationL > - scaled acoustic impedance

Figure 2.6 One-dimensional representation of a prismatic tube withapplicable boundary conditions.

The specific acoustic impedance is usually denoted by Za and relates the pressure,at a specific position, for example at a surface, to the nonnal velocity at that sameposition. In the present investigation the impedance is scaled to the characteristicimpedance of a travelling plane wave. The characteristic impedance of a freelypropagating plane wave is usually Poco. However, in a duct where the viscothermalwave propagation is included, the characteristic impedance is poco/(-G), with G thecoefficient for the type of the cross-section of the tube considered. Therefore thescaled (dimensionless) impedance Çbecomes:

(2.22)

21

Ç(x) 9Za(X) (2.23)p0c0 u(x)

For sound absorbing materials the acoustic impedance indicates how incidentwaves are reflected at the surface. It will be shown later that the real part of theimpedance can only be larger than or equal to zero. If the real part is larger than

zero the surface absorbs energy (Pierce 1994). This principle will be used insection 2.4 'Broadband sound absorption' for the impedance at the entrance of a

tube.if the boundary conditions p(0) = po and p(L) = p' are applied for a tube of length

L, the impedance at x = O and x = L can be rewritten with equation (2.14) and(2.15) as:

siith(rkL) sinh(rkL) (2.24)

cosh(FkL)--- --cosh(rkL)PO Pi

In (2.24) the ratio p, / po can be seen as a transfer function of the tube. As anexample the case of a tube terminated by an acoustically hard wall is considered.By applying the boundary conditions po and u, = O the transfer function can be

shown to be:

i (2.25)p cosh(T'kL)

In section 2.3.1 the transfer functions of tubes are used in a recursive formulationfor coupled tubes. For more complex tube systems this is a convenient way torelate the measured pressure to the input pressure. A novel approach however is touse the coupled tubes as a resonator to absorb sound in a specific frequency range.

This will be shown in section 2.4 '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 arepresented 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 a as a function of the frequency. The soundabsorption coefficient is usually measured in an impedance tube which is

22

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 betweentwo pressure transducers is measured from which the acoustic impedance, thereflection coefficient and the absorption coefficient can be derived. The reflectioncoefficient R is the ratio of the reflected and incident wave. Using the samenotation as in (2.14) one has:

rkxR(x)= PAe

(2.26)PB e

Sometimes the amplitude reflection coefficient is used which does not include thephase information of the two acoustic waves: k = PA' PB

The acoustic energy in a plane wave is proportional to the squared amplitude ofthe wave. Therefore the squared reflection coefficient is the ratio of the reflectedenergy and the incident energy. In the case of a sound absorbing surface the energybalance at the surface where x = O gives:

a(x=O) =.2 2PB HPA

2PB

(2.27)

where the sound absorption coefficient a is the fraction of the incident energy thatis dissipated. Using (2.26) this results in:

a=1IR2 (2.28)

With known amplitudes kA and PB the impedance Ç can be derived and with theuse of the definition of the reflection coefficient R this becomes:

Ç(x) 1R(x) =

Ç(x) + 1

Three separate cases can be distinguished. First, for an impedance approachingzero the reflection coefficient R becomes -1, i.e. the acoustic waves are reflected atthe surface with a 180 degree phase shift. Such a boundary condition is called apressure-release surface and is applicable for waves propagating in water whichreflect from a water-air interface. The acoustic energy is completely reflected sothe sound absorption coefficient a = 0. Secondly, if Ç = i then R = O whichindicates that the incident plane wave is not reflected. In this case there is no jumpin the impedance so it seems that the incident wave travels to infinity. In that case

(2.29)

/

f

101

/

23

a = i which indicates that all of the incident energy is absorbed. And finally i Ç I

corresponds to R = i and represents an acoustically hard surface, i.e. the velocity

perturbation u = O at the surface. The acoustic energy is completely reflected, so

a=O.The relation between the absorption coefficient and the real and imaginary parts

of the impedance is:

4Re(Ç) (2.30)(Re(Ç) + 1)2 + ((Ç))2

It follows that the real part of Çcan only be positive because a O.

Example

The impedance at the entrance of a tube was given in (2.24). To demonstrate thedifferences between the low reduced frequency model, the Kirchhoff model and theHelmholtz model the impedance at the entrance of a cylindrical tube is shown inFigure 2.7. The tube is terminated by an acoustically hard wail.

- 0.000

- irJ4

- - it/2

= irJ4

-. No viscothermal eff.- Kirchhoff

Low reducedfreq.

No viscothermal eff.- Kirchhoff

Low reduced frequency

O 2000 4000 6000 0 2000 4000 6000Frequency [Hz] Frequency [Hz]

Figure 2.7 Acoustic impedance at the entrance of a cylindrical tube (magnitude andphase) with R=O.25 mm and L=50 mm. Tube is terminated by an acoustically hard wall.Standard air conditions are used (see List of Symbols).

When the viscous and thennal effects are neglected the first resonance frequency ofthe tube is 1730 Hz which corresponds to a minimum of the impedance. Viscousand thermal effects cause the speed of sound to be lower which explains the shift inFigure 2.7 of the first resonance frequency to 1490 Hz. The imaginary part of the

24

propagation coefficient is then 1.16, so the effective speed of sound becomes c =

c0 / 1.16. The shear wave numbers ranges from 0.5 to 12 and it can be seen that thedifferences between the low reduced frequency model and the Kirchhoff model arenot very large.

It is noted that the resonance frequency of a half open tube, i.e. one end closedand the other end open, is in fact somewhat lower because of the inlet effects. Thisis explained in more detail in section 2.4 where the effective length of a resonatoris introduced. Furthermore it is noted that the low reduced frequency model hasalso been successfully used in some recent projects (Ommen 1999, Rodarte 2000).

2.3 Coupled tubes

In this section the acoustic behaviour of a number of coupled tubes is derived. Twotechniques are presented: a recursive formulation with coupled transfer functionsand a transfer matrix formulation. Both methods include the viscothermal wavepropagation which cannot be neglected for sound absorbing resonators as will beshown.

Practical examples of the use of these techniques for networks of tubes are(provided that the air velocity is low): air conditioning ducts, gas transportationsystems and mufflers. The first technique uses a recursive formulation for thedynamic response of pressure measuring systems with cylindrical tubes aspresented by Bergh and Tijdeman (1965). It is based on a mass balance for avolume with a number of tubes connected to it. The second method is based on atransfer matrix formulation. It can also be seen as a finite element type offormulation where the linear or quadratic shape functions are replaced by theanalytical expressions for sound propagation in a tube. The degrees of freedom arethe pressure perturbation and the mass flow perturbation. Examples are describedbelow to clarify the method.

2.3.1 Recursive formulation

Expressions for the one-dimensional pressure and average velocity perturbationsincluding the viscothermal wave propagation for narrow and wide tubes werepresented in the previous section. These are used here to set up a mass balance for

a)

IXJ>

('\ p

b)

pi-1

<

Volumepi+1

Tube J+1. . pi .

Tube J.

A1

Figure 2.8 A geometric (a) and schematic (b) view ofa volume with two tubes connected to it.

For the volume 1' one assumes that the mass variation M due to the change indensity of the volume must be equal to the difference between the mass leavingtube J and that entering tube J+ 1:

dMv ¡ \ ,- / _,- kot

dkJJLJ)3)+1J+1u))e

where Q indicates mass flow. Furthermore, the pressure perturbation in the volumeis assumed to be uniform:

Pv = p(xj=Lj) = P,(Xj+1O) (2.32)

For small values of the pressure perturbation p and the density perturbation p thepolytropic relation applies:

_P Yc0 ny

So the mass M can be written as:

Mv =Vj[Po+__2_'1etny

J

25

a volume with a number of tubes connected to it. Figure 2.8 shows two tubes,labelled J and J 1, connected to volume V3. Note that at the ends of the tubes asmall index j is used and that for the tubes themselves a capital index J is used.Each tube has its own co-ordinate system Xj.

Lj+j>

(2.33)

(2.34)

26

where fl is a polytropic coefficient in the volume l/. which can be estimatedaccording to (2.19) or measured. In the paper of Bergh and Tijdeman the'instrument' volume V can also be corrected for the elastic deformation of thevolume. The mass flows entering and leaving the volume are, respectively:

Qj(xj LJ ) = A Po u(xj =L); Qj1(xj1 =0) = A1 Po u(Xj rr0) (2.35)

The complex amplitudes of the reflected and incident travelling waves in the tubes,

PA J, PBJ' PAJ+I and PBj+i are determined by the boundary conditions at bothends of the tube. Following the same approach as Bergh and Tijdeman theboundary conditions for the pressure at both ends are used to determine theseamplitudes. In the case of Figure 2.8 one has:

PAJ = (rkL)pp_1e(rkL) (rkL)

PBj = (T'kL) (rkL)p+p1_1ee e

(2.36)e

p+i pj e'C1 _pj+i +p e1'1PAJ+1 = _e_kj1 PBJi = e(rkL)J+l _e"'J+l

Finally, with (2.31), (2.35) and (2.36) the recursive formulation for the coupledtubes system of Figure 2.8 results in:

-,-1pi I

= I cosh(TkL) sinh(FkL) ( iky A1G1V.+

piiL

AG1 ny sinh(rkL)1{CoShrkLJ+1 J+1 lii

PJ J)](2.37)

Note that the transfer function p / pi of tube J depends on the transfer functionP»-' 'Pi and the ratio of the cross-sectional areas A1, / A1. In Figure 2.8a volume Vis drawn with a dotted line to indicate that a direct coupling of the two tubes is alsopossible. In that case the volume V in (2.37) is equal to zero. For the completebranch of Figure 2.8, a series of two tubes and a volume, the transfer function is:

PJ+i Pi+i P(2.38)

pi-1 PJ PJ-i

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:

pi= I cosh(rkL)

sinh(rkL)3

Pi-i LAG

27

(2.39)

The general boundary condition is an acoustic impedance condition as shown in theprevious section. The transfer function of a tube J with an impedance Ç(L) at x = Lcan be derived from (2.24):

pi i i=Icosh(rkL) +sinh(rkL)J]

°-i L C(L)

If there is a third tube connected to volume , say tube J+2 with pressures P andPj+2 at the ends of the tube, then an additional term arises in (2.37). Rewriting onlythe term between the large round brackets gives:

A1G1 {COSh(rkL)J+l !l+ A2G2 {COSh(rkL)J+2 Pj+2' sinh(rkL),. pi j sinh(rkL)2 pi j

(2.42)

In this way it is possible to create a network of tubes and volumes with multiplebranches. When each branch has a known end condition every transfer function inthe network can be calculated. This is demonstrated in the following example inwhich the acoustic impedance at a junction with two branches is calculated (seeFigure 2.9). One branch is terminated by a volume, the other branch has anacoustically hard termination. By locating the minima of the acoustic impedance atthe junction at P2 the resonances in the system, consisting of the two branches, canbe detected. A minimum in the impedance indicates that the system can be easilyexcited acoustically.

(2.41)

And for a closed tube J one has:

°=[cosh(rkL)t' /i (2.40)

Pi-1

28

P3} Branch i

} Branch 2

R10.001 mLj0.05m

R2 = 0.00025 m

2 = 0.02 m

R3 = 0.00025 m

L3=0.04mV3 = 0.0083mg

R4 0.0005 m

0.08 m

R5=0.005 mL5=0.12m

Figure 2.9 A simple example of a network of coupled tubes.

The transfer function P2/PI is calculated with (2.37) in a recursive way. Theacoustic impedance at P2 is calculated with (2.24). In Figure 2.10 the results arepresented. The propagation coefficient for air is calculated in three ways: noviscothermal effects, the Kirchhoff solution for wide tubes, and the generalsolution of Zwikker and Kosten (low reduced frequency) which is valid for narrowand wide tubes. According to the low reduced frequency solution a narrow tubecorresponds to a low shear wave number whereas a wide tube corresponds to alarge shear wave number.

The analytical model can easily be used to investigate resonance frequencies inseparate branches by switching off the successive branches. In this case the firstresonance frequency at 100 Hz belongs to the upper branch with the volume. Thisbranch can be seen as a Helmholtz resonator. The second and third resonancefrequencies at 400 and 1200 Hz belong to the lower branch and correspond to theorgan pipe resonance formula for a closed pipe. The shear wave numbers rangefrom 0.5 to 25 for the various tubes and it can be seen from Figure 2.10 that forlarge shear wave numbers, i.e. high frequencies, the viscothermal effects cannot beneglected. Furthermore, for low shear wave numbers the resonance of the upperbranch is highly damped because of viscous and thermal effects in the air.

102

101

loo

lo_I0 500 1000

Frequency [Hz]1500

- Low reduced frequency-. Kirchhoff

No viscothermal eff.

Figure 2.10 Impedance (magnitude) at junction 2 in the network of tubes asdepicted in Figure 2.9.

The recursive formulation presented here is an efficient and modular technique tocalculate the acoustic behaviour of a network of tubes and volumes, i.e. thepressure perturbation, the impedance and the velocity perturbation can becalculated directly. This technique will be used in the following sections. However,if a cross-link between branches is present the recursive formulation can no longerbe applied. In that case the system needs to be solved in a direct way rather than ina recursive way. The transfer matrix formulation is used for this purpose to set upthe equations for the unknown pressure and velocity perturbations at the ends ofthe tubes. This is shown in the following section.

2.3.2 Transfer matrix formulation

In mechanics the quantities force and displacement can be related to each other viaa transfer matrix. When the velocity is used an impedance matrix results (or amobility matrix which is the inverse of the impedance mathx). An analogueapproach can be applied in acoustics. Here the mass flow Q and the pressureperturbation p are used, where Q = Po A u. The fmal set of equations can beformulated as:

29

pi

Ui-Figure 2.11 Notation for a prismatic tube.

The pressure in the tube p(x) is given by (2.14). So for the pressures at the nodes j

and jl one has

Inverting this matrix gives the pressure amplitudes PA and PB for tube J. For themass flow one can write:

Q =Ajp0u; Qi =Ajpou+i (2.45)

and by using the velocity u(x) as shown in (2.15) the element matrix for a tuberesults:

pi

Pj+1

=e" ek1/2)j

e(TI2)j e(_T(2)j

PA

PB

(2.44)

30

[M]{p}={Q} (2.43)

In the vector {Q) the 'forcing' terms per node are ordered and in the vector {p} theunknown nodal pressure perturbations are ordered in the same way. The systemmatrix [M] contains 'mobilities' of the acoustic elements such as tubes, volumesand impedance boundary conditions. The different acoustic elements will bederived in the following sub-sections.

Element matrix for a prismatic tube

For a prismatic tube the co-ordinate system as given in Figure 2.11 is used. Notethe convention for the direction of the mass flow. As a result the element matrix for

a tube will be symmetric.

L/2>

Ix>4,F

Ah(rkL) -1 pi Q

, G= (2.46)

1 cosh(rkL) pi+i QiThe element matrix for a tube has four components which are indexed as M/1,Mj', M and M fortubef.

It is noted that when the viscothermal effects are neglected the element matrixshows a singularity if kL = ir (mod ii), i.e. when an exact multiple of half-waves ispresent in the tube.

Element for a voiwne: M

The element for a volume M connected to one or more tubes is derived byconsidering two tubes as shown earlier in Figure 2.8. The mass balance for thevolume, taking into account the directions of the mass flow, is:

0 AI J+1i" 21AJJ+1

22 pi+1

31

where M = ikyV/conv. Thus adding a volume at the junction of a number oftubes simply results in an extra term in the system matrix at the node where thevolume is present.

Element for an impedance boundary condition: Mç

An impedance condition at the end of a tube J can be written as:

A GQ+i --------p+i

O j+1(2.49)

ik yQ +--Vp =0 (2.47)

C0 v

The system matrix for the two tubes and the volume then becomes:

J --M11 M12 0 P.i-'IiQ-'

M'1 (Mj.'+M'+M) M131 = (2.48)

32

Using this relation and by eliminating Q1+ the element matrix for a single tubeterminated with an impedance becomes:

where Mç equals A G / (co C»-i). Consequently a known impedance condition atthe 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 asin the finite element method for a prescribed displacement is followed. For aprescribed pressure Pi the 1th column is subtracted from the {Q} vector, the 1th row

and column of the system matrix are set to zero and the element M is set to 1. Inthe case of two tubes connected to a volume and a prescribed pressure po at node j-

i one has:

_1 O O

O M+M«'+Mv Mj'21J+1M21 22O

To solve the nodal pressures and nodal velocities of a given network of tubes andvolumes 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 / p04J]{Q}, where the vector {u} is arranged per tube as follows:

I j j j+i J+1lUj_1 Uj U Uj+

The index J indicates that for a tube J a cross-section A has to be used.

Pj-1

Pi+1

(2.51)

Mj Mj'2

AIJ M2+Mç' 21

Pj

P»-1

Q

O

(2.50)

}T(2.52)

33

The advantage of the transfer matrix formulation described above is that theanalytical solutions are used whereas in standard finite element formulations linearor 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 tuberegardless of the number of wavelengths that fits in the tube. As a result the systemmatrix remains small. It is not surprising that with the transfer matrix technique thesaine results are obtained as with the recursive formulation technique.

Both techniques can be applied very efficiently for the acoustic behaviour of pipesystems provided that the mean velocity is low. The present investigation considerssound absorbing material as a network of holes and volumes which is presentinside the material. First the sound absorbing behaviour of a single hole isinvestigated, in this case a tube. This is done in section 2.4.1. Secondly, theabsorbing behaviour of coupled tubes is described in section 2.4.2. It is shown thatwith a combination of tubes sound absorption over a wide frequency range can beobtained. This gives the opportunity to design special purpose sound absorbingmaterial. For instance the coupled tubes can be applied in the casing of a noisydevice to absorb a range of 'annoying' frequencies.

2.4 Broadband sound absorption

Sound absorption is directly related to acoustic impedance. Hence, impedance is akey quantity in this section when studying the sound absorption with coupledtubes. Reflection of acoustic waves occurs when the impedance somewhere in thepath of the propagating wave shows a sudden change. The most simple case is arigid wall which has an infinite impedance. Reflection will also occur for changesof the cross-sectional area of a tube because the jump in the cross-section causes ajump in the acoustic impedance. The special case of an open-ended tube can becharacterised by a frequency dependent impedance boundary condition, namely theradiation impedance. For low frequencies the radiation impedance approaches zero

so that a reflection coefficient ofi results (see equation (2.29)).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 ofthe porous material acts as if it is the entrance of an infinite volume so that the

34

incident waves continue their propagation without being reflected. Thus the non-dimensional impedance has to be chosen close or equal to i to create a soundabsorbing wall.

This will be shown for a sound absorbing wall with a number of orifices or tubes,see Figure 2.12. The tubes in the wall act as resonators for specific frequencies.When the tubes are closed at one end, for example, resonance occurs when aquarter 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 ofthese orifices is known in advance so the wave propagation in these holes or tubescan be described with the theory presented in the previous section. II the wall has aregular pattern of orifices, i.e. the porosity over the wall is constant, then the soundabsorption coefficient of the wall can be calculated by considering the impedanceat the entrance of a single tube and the porosity.

2.4.1 Sound absorption of a single tube

Figure 2.12 A sound absorbing wall with quarter-wave resonators.

In the next section the sound absorbing behaviour of a single tube will bediscussed. The effects of the effective length, the radius, the viscous and thermaleffects, the cross-sectional shape and the boundary conditions at the end of the tubewill be described.

The goal is to create a sound absorbing wall with an impedance equal or close to 1.The impedance of the wall Cwall can be related to the impedance at the entrance of a

single tube by assuming that the waves are plane at a short distance from thewall, i.e. kS« i with k = ail c0 (see Figure 2.13) and the pressure perturbation andthe harmonic mass flow in a reference frame across the wall are constant.

wa1I

o

Figure 2.13 Cross-section of a wall with resonators.

The surface porosity of the wall is defined as:

N 4ube

p tube'wal1

tube

35

(2.53)A.waii

where Awaii is the total area of the wall, N is the number of identical tubes, eachwith a cross-sectional area Abe. So with the assumptions discussed earlierconcerning the reference frame the impedance of the wall can be written as:

(2.54)

This is an essential result for a sound absorbing wall with resonators. If theimpedance of the tubes and the porosity are matched in such a way that the ratio is1.0 for a specific frequency then the sound absorption is maximal for thatfrequency. It is noted that in a narrow tube Çwaii needs to be scaled according to(2.23). As an example the calculated impedance arid the sound absorptioncoefficient of a wall, with tubes which are closed on one side, is shown in Figure2.14 and Figure 2.15, respectively. The standard conditions for air are used.

36

1000 2000 3000 4000 5000Frequency [Hz]

0.8

0.6

0.4 -

0.2 -

1000 2000 3000 4000 5000Frequency [Hz]

wall

Figure 2.14 Impedance (magnitude) of a Figure 2.15 Sound absorption coefficientwall Lbe=O.O6 m, Rtube=O.005 m, of a wall with quarter-wave resonators.12=0.0135.

The geometry of the tube is given in the caption of Figure 2.14 and with thepropagation coefficient according to Kirchhoff (s > 10) the impedance at theentrance of the tube can be calculated. The accompanying optimal porosity is Q =0.0135.

The 100% sound absorption for an incident wave at 1400 Hz is physicallyinterpreted in the following way. At the first resonance frequency the pressureperturbation in the tube is amplified due to the incoming waves. For a harmonicsituation the amplitude of the pressure in the resonator is determined by the amountof damping in the tube which is described by the wave propagation coefficient.Exactly at the resonance frequency the accompanying incident sound energy iscompletely dissipated and the pressure amplitude of the reflected wave in theresonator is in anti-phase with the incident wave (see also Appendix A). In apassive way anti-sound is created for a specific frequency. The same effect couldhave been obtained with an active anti-sound source.

The reason that there is sound absorption for frequencies close to the firstresonance frequency is that due to the viscous and thermal effects a broader (andlower) 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 theentrance of the resonator.

When the wall contains tubes with different dimensions the impedance of thewall is obtained by summation of the contributions of the different tubes:

C wall =

where Q is the porosity for the tubes with impedance .

Effective length of a resonator

Due to inlet effects the effective length of the resonator is larger than the actuallength, see also Figure 2.13. Therefore an end correction is added to thegeometrical length of the tube. This effect has been studied extensively fordifferent configurations of the entrance of the tube. For example, flanged andunflanged pipes have been studied (see for instance: Levine and Schwinger). Theend correction depends on the local geometry at the entrance and termination of thetube. The effective length Leff is the geometrical length L increased by a smallincrement d. According to Rayleigh (1945) the increment d for a single tube withthe opening in an infinite baffle is equal to:

d= (2.56)3,r

where R is the radius of the tube. This can be derived by considering the acousticforce on a vibrating piston. The fluid moved by the piston with a radius R has anapparent mass which corresponds to the fluid in a cylinder of area id?2 and length d

(see also Pierce 1994). If the resonator is open at both sides then the end correctionhas to be applied for both sides. For a number of perforations in a panel equallyspaced by a distance a the end correction for each side is (see for instance Bies1996):

I

37

(2.55)

d =(i_O.441?/);3r ¡a with a> 2R (2.57)

In the next section resonators with different cross-sections are coupled. The endcorrection for a single tube centrally located in a tube of circular cross-section with

radius R2 is given by (Bies 1996):

d = 1.25 /R); with <0.6 (2.58)

38

When R2 or a tends to infinity the value of the end correction converges to thevalue for a piston in an infinite baffle. When the cross-section of the resonator isnot circular then the radius R can be approximated by:

R 2A/ (2.59)

provided that the ratio of the orthogonal dimensions is of the order of unity andwhere A is the cross-sectional area of the resonator and D its perimeter. For largerratios of the dimensions reference is made to ASHRAE 1993 (see Bies 1996) todetermine the effective radius.

It is obvious that the length is the most important geometrical parameter to tunethe resonator for a specific frequency. The effect of the end correction on the soundabsorption coefficient is shown in Figure 2.16 for a wall with equally spacedresonators with a geometrical length of 0.06 m (see also Figure 2.12).

1000 1200 1400 1600 1800 2000Frequency [Hz]

Figure 2.16 Inlet effect for a wall with resonators. Standard air conditions are used.

Evidently the inlet effects cause a significant shift of the resonance frequency. Themaximum absorption is tuned for the geometric length but the shift in thefrequency hardly affects the maximum absorption. Thus in this case the sensitivityof the maximum absorption coefficient for the length is small.

0.8

0.6

0.4

0.2

O

/

¡ II

II

I

=0.0135R 0.005mL 0.060md 0.004ma 0.076m

- Length=LLength = L + d

-

Effect of different radii

The dimensions of the cross-section of a resonator determines amongst others theratio of the viscous and inertial effects and thus affects the absorption coefficient.The effect on the absorption coefficient is shown in Figure 2.17. A comparison ismade for single cylindrical tubes with different radii. The porosity and the lengthfor the different tubes are tuned in such a way that the absorption coefficient ismaximised at the same frequency.

1

0.8

0.6

0.4

0.2

Radius = 0.Olm-. Radius = 0.005m

- Radius = 0.001m- Radius = 0.0005m

39

1200 1400 1600Frequency [Hz]

Figure 2.17 Effect of different radii for a quarter-wave resonator tuned at 1324 Hz.Standard air conditions are used.

In Table 2.1 the accompanying combinations of the radius R, the geometric lengthL and the porosity Q are given. The effective length was obtained by using thelength increase of equation (2.56). In the fifth column the effective speed of soundin the tube is shown for 1324 Hz and between brackets the corresponding shearwave number is given.

40

Radius R Length L Porosity Q Tubes per m2 Speed of sound

[ml Em] [-1 [-1 cey [rn/si (s [-1)

0.01 0.056 0.0071 23 341.8 (236)

0.005 0.060 0.0135 172 340.3 (118)

0.001 0.0612 0.069 22.10 328.8 (23.6)

0.0005 0.0591 0.138 176l0 315.5 (11.8)

Table 2.1 Data for a cylindrical resonator optimised at 1324 Hz.

For small radii the width of the absorption peak is much wider due to viscouslosses in the resonator. The required porosity is much higher in that case so thatmore resonators per unit area are necessary. Even more than due to the smallercross-sectional area of the narrow tubes. The viscosity effects in the resonatorcause the effective speed of sound to decrease considerably. This effect is directlyrelated to the propagation coefficient and therefore to the shear wave number s.

i

0.8

0.6

0.4

0.2

o

- Radius=0.lmm- - Radius = 0.2mm-. Glass wool (45mm)

Glass wool (25mm)

= 0.75

R1 = O i mm

L1 =0.060m

R2 =0.2 mmL2 =0.060m

0 2000 4000 6000Frequency [Hz]

Figure 2.18 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 absorbingfoams are approached. Figure 2.18 shows that the sound absorbing behaviour ofsmall resonators, with the porosity correctly chosen, looks much like the behaviourof conventional sound absorbing material; here glass wool. For the latter material,

0.8

0.6

0.4

0.2

01000 1200 1400

Frequency [Hz]1600

- Viscous and thermal effects- - Viscous effects only (high sa)-. Thermal effects only (high s)

No viscothermal effects

Figure 2.19 Effect of the viscosity (s) and the thermal conductivity (so). R= 1 mm.

41

with a thickness of 25 or 45 mm, measurements were performed up to 3000 Hz.The material with quarter-wave resonators has a thickness of 60 mm so that lowfrequency sound waves are more absorbed than for the glass wool.

For conventional sound absorbing materials the volume porosity is usually 0.95 <Q < 0.99. It is noted that for a homogeneous material the same values for thesurface porosity can be used. In Chapter 5 a statistical distribution of tubes in awall is used to approach the sound absorption of conventional sound absorbingmaterials.

Effect of the viscosity and thermal conductivity

The effects of the viscosity and thermal conductivity on the sound absorptioncoefficient is determined by the non-dimensional parameters s and sci, where s isthe shear wave number and a is the square root of the Prandtl number For largevalues of s and sa, i.e. » 1, the effects of the viscosity and the thermal conductivityare low. These effects are numerically shown in Figure 2.19. Just as in the previoussections the wall with resonators is optimised for a single frequency for eachconfiguration of different parameters. So the geometrical length is adjusted to getthe same frequency and the porosity is tuned for maximum absorption.

42

It can be seen that for this situation, where standard 'air' conditions are used for theparameters other than s and o the effect of the viscosity on the absorptioncoefficient is larger than the thermal effect. The absorption coefficient withoutviscothermal effects is in fact only non-zero at the resonance frequency, but forclarity s and sa are chosen relatively large so that a very sharp peak is shown inFigure 2.19. Evidently the viscothermal effects cannot be neglected for resonatorsif the sound absorption has to be calculated. The effective speed of sound for theresonator with a radius of 1 mm is shown in Table 2.2.

Table 2.2 Data for 'air-like' media in a cylindrical quarter-wave resonator.Optimised at 1324 Hz.

Effect of cross-sectional shape

The results for the absorption coefficient of resonators with a circular (with radiusR), an equilateral triangular (with sides d), and a rectangular cross-section (withsides 2a and 2b) are presented in Figure 2.20. Also the propagation coefficient for alayer is calculated. In Appendix B these propagation coefficients are given. It isnoted here that the calculation time for a rectangular cross-section is larger due tothe series solution. For a comparison the cross-sectional areas are kept constant inthe left-hand side figure. In the right-hand side figure the shear wave number iskept constant at 1324 Hz except for the layer geometry where the shear wavenumber is twice as large. To obtain the same absorption coefficient for the layergeometry it is noted that the viscous effects in one direction of the cross-section arepresent but absent in the infinite direction. Therefore the viscous effects aredoubled in the layer. Furthermore, the cross-sectional area is per unit width.

Effects (at 1324 Hz) Length L [m} Porosity Q [] Speed of sound Ceff [rn/si

s (23.6) and sa(20.0) 0.0612 0.069 328.8

large sa(20.0103) 0.0620 0.050 333.4

larges (23.610) 0.0631 0.0021 338.5

large s and sa 0.064 0.0001 343.3

0.8

0.6

0.4

0.2

/ .\\ -

\2a=20b -

- -d2a=2bR

- 2h

0.8

0.6

0.4

0.2

Figure 2.21 Resonators with different cross-sectional shapes.

Figure 2.20 Sound absorption coefficient of a wall with resonators for which the cross-section 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 thesound absorption coefficient. It can be seen in the left-hand side figure that forlarger perimeters the viscous losses are larger. The geometry and dimensions areshown in Figure 2.21, Table 2.3 and Table 2.4.

Square, Rectangular

43

o O1000 1200 1400 1600 1000 1200 1400 1600

Frequency [Hz] Frequency [Hz]

44

Table 2.3 Data for quarter wave resonators with the same cross-sectional area optimisedat 1324 Hz.

Table 2.4 Data for quarter wave resonators with the same shear wave number ofs = 23.6 at 1324 Hz.

Effect of an open tube; the radiation condition

In the previous sections the resonators had a closed end, i.e. an acoustically rigidtermination. In other words the impedance at x = L is infinite. In general theresonators may have any impedance boundary condition. A special case areresonators with an open end. These kind of resonators may be used in a wall whena 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 outsidethe resonator, is approximately zero. For a tube with the end in an infinite baffle ananalytical expression for the impedance at the end, the so-called radiationimpedance, can be obtained from the literature. It is based on the acoustic loadwhich is imposed on a vibrating circular piston in an infinite baffle (see Pierce1994 and Bies 1996). The result is given here (Ç is not scaled with the coefficient

G):

Cross-section

Em]

Length

L [m]

Porosity

Q [-1

Speed of sound

c [mis]

Cylindrical R=0.001 0.0621 0.069 328.8

Square 2a=2b=O.00177 0.0618 0.077 327.1

Rectangular 2a=20b=5.6104 0.0599 0.140 316.3

Triangular d=0.0027 0.06 14 0.088 324.9

Layer 2h=5.6.10 0.0624 0.064 330.4

Cross-section [ml Length L [m] Porosity Q []

Cylindrical R=0.00l 0.0621 0.069

Square 2a=2b=2R 0.0621 0.069

Rectangular 2a=20b=1 iR/lO 0.0621 0.071

Triangular d= Ji R 0.0621 0.069

Layer (2.$) 2h=2R 0.0634 0.035

Pierce uses rad = Rrad - Xrad as a result of the time convention e_0t.

45

0.8

0.6

0.4

0.2

0

Radius =- - Radius =-. Radius =

Radius =

001m0.005m0.001m00005m

//

/

z///

o 2000 4000 6000 8000 10000Frequency [Hz]

Figure 2.22 Sound absorption coefficient at the open end of a tuberesonator located in an infinite baffle.

Ç. trad rad rad (2.60)

The real and imaginary part of the radiation impedance are:

R(2kR)2 (2kR)4 (2kR)6

rad 2.4 2.42.6 2.42.62.8(2.61)

4[2kR (2kR)3 (2kR)5Xrad =

3 32.5 32.52.7

As an example the corresponding sound absorption coefficient of the end of thetube is plotted in Figure 2.22 for several radii. The absorption coefficient iscalculated with the use of equation (2.30).

Figure 2.22 shows that at low frequencies the sound waves propagating in theresonator are not absorbed at the end but are reflected back into the resonator dueto the mass reactance at the free end, i.e. the reflection coefficient R = -1 and a = O(note the frequency range in Figure 2.22). For higher frequencies the waves are'absorbed' due to radiation into infinity. The radiation impedance approaches theimpedance of a freely travelling plane wave, i.e. Ç = 1.

i

46

2.4.2 Sound absorption of coupled tubes

It has been shown that it is possible to design a wall with a specific number ofresonators to absorb sound energy for a narrow frequency band. These kinds ofwalls can be applied for noise problems where for example an engine motor runs ata 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, forexample, variations in engine rpm. Therefore a design tool for a more complexconfiguration of resonators is presented. By means of coupling it is possible toabsorb 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 createa maximum absorption coefficient. It will be shown that it is efficient to calculatethe absorption coefficients for a set of cross-sectional areas and lengths andsubsequently choose the cross-sectional areas and lengths for which the absorptioncoefficient matches the design requirements for the sound absorbing wall best.

The two formulations based on the basic theory for coupled tubes described insection 2.3 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 becalculated.

The most simple configuration of coupled tubes is shown in Figure 2.23. In thefollowing sections five configurations are described: 1. Two tubes coupled inseries, 2. Two single tubes in parallel, 3. Two tubes coupled in series with R2>R1,4. Three tubes coupled in series and 5. Multiple coupled tubes. In the examples thefrequency range is chosen from 1000 to 1600 Hz. In this way one is able tocompare 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> 10) so that the wavepropagation coefficient of Kirchhoff can be used.

Cylindrical tubes are used.The effective lengths as mentioned in section 2.4.1 are used in thecalculations, i.e. Leff = L + d, whereas the geometrical lengths L are given in

47

the examples. For the first tube equation (2.56) is used and for the secondtube this is equation (2.58).

Two coupled tubes in series

The geometry of two axially coupled tubes is given in Figure 2.23. Because of theassumption of one-dimensional waves the second tube does not have to beconnected concentrically to the first tube. The entrance of the first tube is located inthe surface of the sound absorbing wall.

Figure 2.23 Two axially coupled tubes.

It was seen that the impedance at the entrance of the resonator, in this case twocoupled tubes, needs to be close to one in order to achieve optimal soundabsorption. The magnitude and the phase of the impedance is shown in Figure 2.24for a closed end of tube 2. Approximately the same results can be obtained for anopen end of tube 2 if the length L2 is approximately twice as large.

10' - Coupled tubes

Tube l:openend's -

Tube 1: closed end

i02600 1000 1500 2000 600 1000 1500

Frequency [Hz] Frequency [Hz]

Figure 2.24 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: L1=O.06m, L2=O.06 m, R1=OE005 m, R2=O.001 m.

The impedance characteristic of the coupled tubes is dominated by the first tube formost frequencies (for f < 1200 Hz and f> 1500 Hz). Due to the presence of thesecond tube the first tube behaves as an open tube in the neighbourhood of theresonance frequencies of the second one (1200 Hz <f <1500 Hz). The magnitudeof the impedance at the entrance of the coupled tubes now shows two minimainstead of one. Proceeding in the same way as in the previous section the porosityof the wall is chosen so that for at least one resonance frequency the absorptioncoefficient is 1.0. The corresponding absorption coefficient is depicted in Figure2.25.

- irJ2

-

- 0.0

,T14

2000

48

102

10'

loo

0.8

0.6

0.4

0.2 - - Single tube- Coupled tubes

- - I

49

O1000 1200 1400

Frequency [Hz]

Figure 2.25 Absorption coefficient of a wall with resonators. Eachresonator consists of two axially coupled tubes. 12=0.05.

The width of the frequency band for which noise is absorbed looks promising butthe question arises from Figure 2.25 whether the absorption around 1300 Hz, whereit has a local minimum, can be increased. The five parameters to be varied in thiscase are L1, L2, R1, R2 and Q. 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 thelength of the first tube as can be seen in Figure 2.24 and Figure 2.25.Therefore the parameter L1 is set to a fixed value. The parameter R1 can be

set to a convenient value.A discrete set of parameters is chosen for f, Q, R2 and L2, i.e. f3, Q, R21 andL21. 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 accordingto the theory presented in section 2.3. A multi-dimensional array a(f3, Q3,

R21, L21) results.

To reduce the calculation time the Kirchhoff approximation for thepropagation coefficient is used.A minimum absorption coefficient a,,,, is specified so that a> a.From the array a( f3, Q, R2, L2,1) the widest frequency range and theaccompanying parameters Q, R2 and L2 are selected for which a>

1600

50

It proved to be efficient, because of the shortness of calculation times, to calculatethe absorption coefficient for a large range of the parameters and to search for a

frequency band for which the absorption coefficient exceeds the specified valuecr,,. In this way Figure 2.26 was obtained.

1

0.8

0.6

0.4

0.2

o1000

Figure 2.26 Absorption coefficient of a wall with resonators which consist oftwo wially coupled tubes.

Figure 2.26 shows that the region of high absorption for two axially coupled tubescan cover a much wider frequency band than for a single tube. Furthermore a largervalue of results in a somewhat narrower frequency band, i.e. the two peaks inFigure 2.26 lie closer together for a higher averaged sound absorption coefficient.In Table 2.5 the applied parameters are listed.

1200 1400Frequency [Hz]

1600

a>0.80a>0.90a>0.95

- a>0.99single tube

Table 2.5 Geometrical parameters for axially coupled tubes with a high level ofabsorption ( 1600 tubes per m2).

Parameter a> 0.80 «>0.90 «>0.95 «>0.99L1 [m] 0.06 - (same)

R1 [m] 0.005 -L2 [m] 0.0605 0.06 -R2 [m] 0.00085 0.00073 0.0007 0.00061

Q[-] 0.145 0.13 0.14 0.12

Two single tubes in parallel

The sound absorption coefficient of a wall with a distribution of two types of singletubes is shown in Figure 2.27. It can be seen that in this case the bandwidthcorresponding with a high sound absorption is much smaller than for the coupledsituation. Moreover there is a local minimum in the absorption coefficient whichcan only be removed by moving the peaks even closer, i.e. the resulting absorptioncoefficient is a simple linear addition of the two absorption coefficients of thesingle tubes.

= 0.014

R1 =0.005mL1 =0.057m

2 0.0135

R2 = 0.005 m

L2 = 0.063 m

0.8

0.6

0.4

0.2

o-1000

Tube 1- - Tube 2

Two tubes

-

Figure 2.27 Absorption coefficient of a wall with two types of single tube resonators.

Two tubes coupled in series with R2>R1

When the second tube has a larger radius than the first tube then the soundabsorption coefficient does not show the broadband behaviour, see Figure 2.28.Obviously there is no strong interaction between the two tubes. The first tube canbe 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 longcompared to the previous examples in order to obtain the same frequency range.

51

16001200 1400Frequency [Hz]

52

The second tube on the other hand can be seen as closed at both ends so that it alsoworks as a half-wave resonator. In this case it is only the length of the first tubethat needs to be corrected for the inlet effects, but now on both sides.

1 =0.14R1 =0.001mL1 =0.120mR2 = 0.005 m

L2 = 0.120m

Figure 2.28 Absorption coefficient of a wall with resonators which consist of axiallycoupled tubes.

Three coupled tubes in series

The results of the search procedure for three axially coupled tubes leads to thesound absorption coefficients presented in Figure 2.29. The bandwidth hasincreased drastically compared to a double tube configuration. Also in this case themaximum bandwidth depends on the specified minimum value for the soundabsorption coefficient.

1200 1400 1600Frequency [Hz]

o800 1000 1200 1400 1600 1800 2000

Frequency [Hz]

Figure 2.29 Maximum absorption coefficient of a wall with resonators which consistsof three axially coupled tubes. Parameters as given in Table 2.6.

Table 2.6 Geometrical parameters for three axially coupled tubes.Lj=0.060m, R1=0.005 m ( 3500 tubes per m2).

Multiple coupled tubes

In the search for wide band sound absorption with coupled tubes the configurationin Figure 2.30 also provides good results. An advantage of this configuration isthat, compared to the three axially coupled tubes, tube number 3 is now direcflycoupled to tube number i which results in a more efficient use of space. As a resultthe diameter of tube 3 can be chosen somewhat larger (see Table 2.7).

- triple a> 0.80-. triplea> 0.90

- triple a>0.98- double a> 0.99

single tube

53

Parameter a>0.80 ct>0.90 a>0.98

L2 [m] 0.060 0.061 0.06 1

R2 [m] 0.00155 0.00155 0.0014

L3 [m] 0.058 0.058 0.056

R3 [ml 0.00038 0.00036 0.0003

Q[-11 0.252 0.29 0.30

54

o800 1200 1600

Frequency [Hz]2000

a>0.80- cz>0.90

a>0.98

Figure 2.30 Absorption coefficient of a wall with resonators which consist of two tubesconnected to a single tube.

Table 2.7 Geometric parameters for three coupled tubes. L=0.06, R1=0.005( 3200 tubes per m2).

In general a large variety of combinations can be used to get the desired absorptionbehaviour of a wall. Figure 2.31 presents an overview and some examples ofcombinations of resonator configurations.

Parameter a> 0.80 a> 0.90 a> 0.98L2 [m] 0.054 0.054 0.056

R2 [ml 0.001 0.0008 0.0007

L3 [ml 0.068 0.066 0.065

R3[m] 0.0011 0.001 0.0008

Q [-] 0.3 0.28 0.25

mq

F_4

YA

Figure 2.31 Cross-section of various resonators and combinations of resonators.

A comparison of predicted and experimental results for a number of resonators willbe presented in Chapter 4.

2.5 Conclusions

In this chapter a strategy for the design of a sound absorbing wall with tunedresonators was presented. By applying resonators consisting of coupled tubes, awall can be designed in such a way that absorption is possible for a considerablefrequency bandwidth when compared to other resonance absorbers. The couplingprinciple and the viscothermal effects are responsible for the broadband absorption.

With the design tool presented in sections 2.4.1 and 2.4.2 an optimal distributionof resonators can be found for a predefined frequency range. The numerical basiswas presented in section 2.3 and resulted in two efficient and accurate models forcoupled tubes in general. It was seen that the viscotherrnal effects in tubes have asignificant influence on the wave propagation and that these effects are essential tocreate the wide band resonance and sound absorption. As a result of theviscothermal effects in the narrow tubes the effective speed of sound is lower. Thiswas demonstrated for different cross-sectional shapes.

Advantages of the presented sound absorbing wail with resonators are:A high sound absorption for a broad frequency band can be realised. Thefrequency band can also be located in the low frequency range.The frequency band can be fine-tuned for a specified sound spectrum.

55

Single type resonators Multiple type resonators (examples)

56

The resonators can be constructed in any material to prevent problems withfor example: ageing, extreme temperatures or aggressive environments.The wall can be constructed with perforated resonators to enable a fluid topass through or for visual inspections.The implementation of resonators with coupled tubes is not limited to axiallycoupled straight tubes. As long as the wave propagation is one-dimensionalby approximation, also flexible tubes or labyrinth-like structures can beapplied in the sound absorbing wall in order to reduce the wall thickness (seeFigure 2.32).

In the next chapter the experimental verification of the acoustic behaviour of theresonators with coupled tubes is described. An impedance tube (Kundt' s tube) is

cover plate plate with resonators

Figure 2.32 Resonator configurations to reduce the total wall thickness.

57

used in which samples of the sound absorbing wall are placed. The transferfunction of two transducer signals in the impedance tube is used to calculate thereflection coefficient, the impedance and the sound absorption coefficient. Therecursive formulation with transfer functions can be seen as the basis for thismeasurement technique, see section 2.3.1.

Chapter 3

3.1 Introduction

To verify the coupled tubes model and the predicted sound absorption behaviour ofbroadband resonators, as presented in Chapter 2, experiments were performed. Inthis chapter four techniques are described to measure the sound absorptioncoefficient in an impedance tube. Three of the presented techniques are new andmake use of a particle velocity sensor, the so-called microflown. It will be shownthat a combination of a microphone and a microflown provides direct informationon acoustic impedance, sound intensity and sound energy density. Experimentalresults of the four methods are compared to each other. To be able to repeat themeasurements 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 isreferred to Chapter 4 where the experimental results of quarter-wave and morebroadband resonators are presented.

A number of measurement techniques are available to quantify the sound absorbingbehaviour of porous materials. These techniques can also be used for a wall with

Impedance tube techniques to measure soundabsorption

3.1 Introduction 593.2 Impedance tube techniques 65

3.2.1 The 2p method 683.2.2 The 2u method 693.2.3 The p/u method 733.2.4 The pu method 74

3.3 Experimental results 793.3.1 Comparison of the 2p and the 2u methods 793.3.2 Comparison of the 2p, the p/u, and the p.0 methods 84

3.4 Conclusions 87

60

resonators. In this section a brief overview of measurement techniques for soundabsorbing materials is given. For a more extensive background the reader isreferred to the literature and standards presented in subsequent sections. It will beargued that the impedance tube method is the most convenient and efficienttechnique for experiments with coupled tubes resonators.

In general one is interested in the sound absorption coefficient a, which is thefraction of the incident sound power which is dissipated in the porous material, thereflection coefficient R, or the normal surface impedance Z. These quantities areusually measured for normal incident waves. The incident sound field can beclassified into three types: normal incidence, oblique incidence (i.e. at angle ), and

random incidence, see Figure 3.1.

Random I Diffuse

Figure 3.1 Three types of incident waves.

Typically, the absorption coefficient increases with increasing angle of incidence,

up to a certain angle of Beyond this angle, a decrease is usually observed. Theexplanation for this is the contribution of the so-called shear waves whichpropagate in the flexible porous material. This is in addition to the two dilatationalwaves in the porous material, see also Chapter 5. As a result, the absorptioncoefficient at normal incidence a,, is slightly less than the absorption coefficientmeasured at random incidence a,. for porous materials. Usually a,, is measured inan impedance tube and a,. in a room.

The acoustic measurement techniques can be divided into three categories:

Reverberant field methods

Free field methodsImpedance tube methods (Kundt' s tube)

Normal Oblique

Reverberant field method

The so-called reverberant field method is a well-known technique to measure thesound absorption coefficient for random incident waves. The experiments areperformed in a reverberation chamber in which a diffuse sound field is generated(see for example Bies 1996, Iso 354).

There is a number of standards available for the procedures as well as for thegeometry and dimensions of the test chambers. Usuafly a sound pressure field isgenerated with a uniform energy density. This is achieved with loudspeakers whichare placed in the corners of the chambers and a number of diffusers to prevent thepresence of standing waves in the chamber. A relatively large sample of the soundabsorbing material (several m2) is placed in the chamber and for a given frequencyband the reverberation time T60 is measured. T60 is the time it takes the soundpressure level to drop 60 dB after shutdown of the loudspeakers. The sameprocedure 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 thevalue of one because of extra energy loss due to edge effects and diffraction. Thiscan also be the case if the sound field is non-diffuse. Various standards state that atleast 20 modes of vibration in the chamber are required in the lowest frequencyband. As a result the room volume must be quite large. Nevertheless considerabledifferences have been observed for measurements on the same test materials indifferent reverberation chambers. It is therefore concluded that the reverberant fieldmethod 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 ofsound. The free field condition indicates that waves only propagate directly fromthe source of sound to the point of measurement. This condition can be approachedin an anechoic chamber. In practical situations there is usually reflection from theground. For these situations outdoor measurements above a reflecting plane can bemade or a semi-anechoic chamber can be used where the floor is a reflecting plane.

61

62

A number of authors have proposed methods to measure the acoustic propertiesof sound absorbing materials at free field conditions (see for example Tarnura 1995and 1990, Allard 1989a and 1989b). In general the methods are suited formeasurements with oblique incident waves. One technique is for example the pulsetechnique. A short signal is generated and the direct and reflected waves areseparated to calculate the reflection coefficient. It is noted that the sample has to beplaced outside the near field, which can pose a lower limit on the frequency bandof interest, and on the dimensions of the samples (several m2). Another techniqueuses two microphones placed close to the sound absorbing surface. With thismethod it is possible to calculate the normal impedance at the surface for obliqueincident waves. The area of the test material can be much smaller (1 m2). For lowerfrequencies (about 500 Hz) however the size of the anechoic chamber may be arestricting factor because the source should be placed outside the near field.

The possibility to measure the acoustic behaviour of sound absorbing materials atoblique incident waves is a strong advantage of the free field method. It wasalready mentioned that for oblique incident waves the shear waves whichpropagate in the sound absorbing material itself cause a different acousticbehaviour. However, for the material tested with tube resonators no shear waveswere present. Furthermore, the resonators are locally reacting. It will be explainedthat it is sufficient to use the impedance tube technique to measure the normalimpedance of a wall equipped with a number of resonators.

Impedance tube method

The most common technique used for measurements on sound absorbing materialfor normal incident waves makes use of an impedance tube. In Figure 3.2 a sketch

of two techniques is shown.

rigid tube backing plate

Figure 3.2 Schematic representation of Iwo measurement techniques in an impedancetube.

At the left-hand side a loudspeaker is placed and at the opposite side a sample ofthe test material is placed. In the tube a standing wave pattern is formed: the resultof a forward (or incident) travelling pressure wave with amplitude B' and abackward (or reflected) wave with amplitude A The frequency of the soundwaves is kept lower than the cut-off frequency (see section 2.2) to ensure thegeneration of plane propagating waves in the tube.

Earlier techniques made use of the measured standing wave ratio (SWR) for aspecific frequency in the tube. By means of a movable microphone the ratio of thepressure maximum to the pressure minimum is determined. This ratio is then usedto calculate the reflection coefficient and the acoustic impedance. An advantage ofthis method is that it is not necessary to calibrate the microphone. Drawbacks arethe complex set-up with a movable probe and the time needed to fmd the maximumand minimum pressure for each frequency of interest.

In 1980 Chung and Blazer (Chung 1980) presented a technique that is based onthe transfer function of two fixed microphones which are located at two differentpositions in the tube wall (see right-hand side of Figure 3.2). The standing wavepattern in this case is built up from a broadband stationary noise signal. With themeasured transfer function the incident and reflected waves are separatedmathematically. This leads to the reflection coefficient of the sample for the samefrequency band as the broadband signal. The impedance and absorption coefficientcan be derived as well. The method is as accurate as the SWR method andconsiderably faster. They also presented two techniques to improve the measuredtransfer function (see section 3.2.1). The transfer function method has proven to bereliable and has been standardised (ISO 10534-2, 1998).

For sound absorbing materials the impedance measured with the method asdescribed above strongly depends on the thickness of the material because sound

63

Standing wave ratio technique Two-microphone technique

movable microphone microphones

64

waves reflect at the backing plate. Therefore some authors advise the use ofacoustic properties which are independent of the test configuration such as thecharacteristic impedance and the propagation coefficient in the material (see forexample: Delany 1970, Minten 1988, Lauriks 1989, Utsumo 1989, Voronina 1998,Iwase 1998, Song 2000). One technique to derive these two coefficients is tomeasure the surface impedance of the material with two different thicknesses.

For low frequencies the impedance tube method may not give accurate resultsbecause 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 aperforated 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 ingeneral 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 2.

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 velocityperturbation u (see for instance Fahy 1995). 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

65

measured in situ instead of in a reverberation or anechoic chamber because it ispossible to perform measurements with stationary background noise. A largenumber of new techniques on sound intensity measurements is listed in theliterature and a number of standards has been developed (see for example Bies1996, Isaksson 1998).

3.2 Impedance tube techniques

In this section the impedance tube technique will be described for the two-microphone technique as well as three new techniques which make use of a particlevelocity sensor. It was explained that for the measurement of the acousticimpedance and the sound intensity the particle velocity perturbation is an importantquantity. With the use of a particle velocity sensor, the so-called microflown, theimpedance and the sound intensity can be measured directly (see also Druyvesteyn2000). Four methods are presented:

The 2p method. This is the standard technique which uses the transfer functionof two microphones.The 2u method. Instead of two microphones two microflowns are used.The p/u method. The impedance is measured directly at the sensors.The pu method. The sound intensity and the energy density is measured andfrom these quantities the reflection coefficient of the sample is calculated.

The working principle of the microflown is explained in section 3.2.2. Furthermoreit will be shown that for the p/u and the pu method a special procedure is needed tocalibrate the microflown with respect to the microphone.

To test the four methods two types of impedance tubes are used. For convenience atube with a circular cross-section and a tube with a square cross-section are used asshown in Figure 3.3.

66

transducers

speaker

Figure 3.3 Two impedance tubes, one with a circular and one with a square cross-section. Length of impedance tubes is approximately 1.0 m.

For the impedance tubes several positions for the transducers can be used. Eachseparation distance between two sensors is chosen such that for a given frequencyrange less than half a wavelength fits between the two sensors. Table 3.1 gives theseparation distances and the corresponding frequency range.

Table 3.1 Separation distances, cross-sections and frequency ranges forthe impedance tubes.

A frequency range above 4000 Hz only becomes important for sound absorbingmaterials with a small thickness of approximately 2 cm. In many applications sucha material thickness is not a problem so that the sound absorption is sufficientlyhigh above 4000 Hz, i.e. approximately 100 percent. For the present impedancetubes the cut-off frequencies are high enough to investigate the acoustic behaviourof common porous sound absorbing materials. For special material such asresonance absorbers the frequency range of interest corresponds in general to therange for porous materials. However, below that range extra sound absorption canbe gained, within a limited volume, with resonance absorbers, i.e. an interest in the

Separation distance [ml Frequency range [Hz]

x1 = 0.345 50.. .400

= 0.045 380.. .3050

x3=0.025 685.. .4000 (cut-off freq.)

Cross-section [ml Cut-off frequency [Hz]

2R = 0.05 c01 (1.7.2R) = 4000

2a = 0.04 c0/ (22a) = 4280

67

low frequency range originates from the fact that resonance absorbers can providean additional narrowband sound absorption where porous materials are usually lessefficient (Heckl 1995).

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 asample which is placed halfway in the impedance tube. The two large transducersrepresent a half-inch pressure microphone and a packaged half-inch microflownrespectively. For the 2p method '/4 inch Kulite microphones are used. For the p/uand p.0 method a ½ inch B&K microphone and a ½ inch ICP probe fromMicroflown Technologies are used. It is noted that for standing waves withfrequencies up to 4000 Hz the average pressure at a ½ inch microphone is a goodrepresentation of the actual pressure at a point (at for instance x=L) because thearea of the microphone is at most 15% of the wavelength. The niicroflowns arepositioned inside the impedance tube but the size of the sensors is much smallerthan the acoustic wavelength so that the effect on the wave propagation isnegligible.

It is noted that for accurate measurements the wall of the impedance tube has tobe rigid otherwise the signals of the microphones may be affected by vibrations ofthe wall. Therefore the impedance tubes are constructed of 5 min thick aluminiumFurthermore the distance from the sample to the sensors is kept larger than 4 timesthe width of the cross-section so that plane waves are predominant.

68

The complete measurement set-up to determine the transfer function is sketched inFigure 3.4.

Function generatorB&K 2082

.AmplifierB&K 2706

Speaker

3.2.1 The 2p method

In the 2p method the transfer function between two microphones is used. Figure3.5 shows the one-dimensional representation of the impedance tube.

samplepl P2

Ix>.

Oscilloscope

s

Sensors

Amplifier

u

To

Impedance tube

Figure 3.4 Measurement set-up.

Figure 3.5 Schematic drawing of the impedance tube.

The measured transfer function is:

pP2 5P2P1

Pi 'PiPi(3.1)

PC with FA 100Analyser card

rD-Tac sofiware

Low-pass filter

where P2 and p are the pressures in the frequency domain, S2,,1 is the cross-spectrum and is the auto-spectrum. The complex transfer function H2 can be

>x1, X2, X3 -

L

69

measured directly with a two-channel FFF analyser, see Figure 3.4. With thetheory for the one-dimensional wave propagation as presented in section 2.2 onecan derive the reflection coefficient at the surface of the sample, i.e. at x = L:

R(x-L)H2 _e''2 er

- _H2p+eX2 e_rkL

where the separation distance x2 is used. In the impedance tube the propagationcoefficient T according to the theory of Kirchhoff is used because the impedancetube can be seen as wide, i.e. the shear wave number s » 1. The absorptioncoefficient a and the impedance at the surface of the sample are:

2 l+Ra=l-IR = (3.3)lR H2 cosh(rkL)cosh(rk(Lx2))

The two microphones need to be calibrated to determine the gain and phasecharacteristics. However one can use sensor-switching to avoid this possible causeof errors (Chung 1980). Measurements are performed a second time with thesensors exchanged. Furthermore it is possible to use a third sensor to improve theaccuracy of the measurement of the transfer function. This technique makes use ofthree coherence functions between the signals of three sensors. In the experimentalresults presented in section 3.3 the transfer function improvement technique wasnot needed.

3.2.2 The 2u method

The 2u method is similar to the 2p method. However instead of two microphonestwo microflowns are used (see also Van der Eerden 1998). First the microflown isbriefly introduced and then the utilisation of the two microflowns in the impedancetube is described. Finally the governing equations for the 2u method are given.

The microflown: an acoustic particle velocity sensor

The microflown (or j.t-flown) was developed at the department of ElectricalEngineering of the University of Twente (de Bree 1996). Instead of sound pressurethe microflown measures the acoustic particle velocity, i.e. averaged over a small

(3.2)

70

fmite volume of air. The microflown consists of two cantilevers of silicon nitridewith a platinum pattern on top, see Figure 3.6. The size of the cantilevers is800x40x1 jim (lxwxh). A large number of microflowns can be created on a singlewafer.

The measurement principle of the microflown is based on the temperaturedifference between two resistive sensors which are 40 pm apart as shown in Figure3.6 and Figure 3.7.

Two sensors on top (40 m apart)

jMass Power supply Die

Signal Breaking groove

- Acoustic wave presentStationary situation

Si S2Position

Figure 3.6 Photograph of the microfiown. Figure 3.7 Photograph of the sensors ontop and estimated temperature profile.

A travelling acoustic wave causes motion of the air and as a result heat istransferred from one sensor to the other (harmonically for a single frequency). Thisresults in a temperature difference between the two sensors. This temperaturedifference causes a differential electrical resistance variation between the twosensors, which is measured. To realise a temperature difference which is highenough to determine the resistance variation the sensors are heated by a DC currentup to about 400-600 K.

The sensitivity of the microflowns shows approximately a so-called first orderlow pass behaviour. The corner frequency, above which the sensitivity drops with 6dB per octave, is between 300 Hz and 1kHz. For high frequencies more averagingand a higher signal-to-noise level may be needed.

a

I.

o

Figure 3.8 A bridge-type microflown (with sensors Si and S2). For one and threedirections.

Application in the impedance tubes

The microflowns with the two sensors on top are used for the 2u method. They areused at the same positions as the microphones. The microflowns are mounted in ahollow bolt with the same thread as the Kulite microphones (see Figure 3.9). The

71

More practical characteristics of the microflown are:The directional sensitivity varies cosine-like (as in a figure of eight). Thismeans 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 soundintensity technique with two closely spaced microphones fails.The vulnerability of the two sensors. A protective package may be used (seealso Figure 3.9 and Figure 3.10).The microflown has to be positioned outside a boundary layer, whereas amicrophone is usually flush-mounted in the impedance tube wall.

An alternative design is shown in Figure 3.8. The cantilevers of the two sensors aresupported on both sides for extra stability. Furthermore the sensors are smaller, toprovide extra sensitivity. In Figure 3.8 a microflown with three sensor pairs is alsoillustrated to measure the three orthogonal velocity components. By combining thelatter sensor with a microphone the measurement of the sound intensity vectorbecomes relatively easy.

72

sensing cantilevers of the microflowns are placed about 15 mm from the tube wall.To avoid damage of the cantilevers a protective wire frame is used.

I ½ inch

a a

Figure 3.9 Photograph of the Figure 3.10 A ½ inch microphone and a ½microflown placed in a bolt. inch packaged microflown (with enlarged

view).

For the p/u and pu method a ½ inch microphone and ½ inch microflown are used.The microflown is protected with a package which consists, among others, of twocylinders (see Figure 3.10). As a result of the amplified acoustic flow between thetwo cylinders the sensitivity of the microflown is significantly increased(approximately 15 dB).

Data processing

The measured transfer function is:

u2H2

u1 S11

With the use of H21, and the theory presented in section 2.2 the reflection coefficient

R of the sample in the impedance tube becomes:

R(x=L) =H2 _e_T2 e"'H2 _e"'2 e''

(3.4)

(3.5)

Table 3.2 Acoustic sound levels and corresponding reference levels.

3.2.3 The p/u method

When a microphone and a microflown are positioned at the same cross-section ofthe impedance tube the acoustic impedance at that position can be measureddirectly. It is noted that the shear wave number is large in the impedance tubewhich is an indication for the wave front to be plane. Therefore the particlevelocity can be measured at one point in the cross-section (outside the thinboundary layer). The position of the ½ inch probes is indicated by the large sensorsin Figure 3.3.

The transfer function is measured according to:

73

See Figure 3.5 for the parameters s and L and note the difference with equation(3.2). The absorption coefficient a and the impedance Çare given in equation (3.3).

The measurements were performed for sound pressure levels (SPL) below 130 dB(ref. 20 .tPa). This implies that the theory of linear acoustics is still valid (see forinstance Bies 1996). If only microflowns are used in the impedance tube one canuse the particle velocity level (PVL) to determine the level for which the lineartheory breaks down. In Table 3.2 the PVL, the SPL and the sound intensity level(SIL) are listed. The standard reference levels for air are chosen in such a way thatfor a plane wave with a characteristic impedance of approximately 400 kgm2s' thethree levels give the same result. For instance: 94 dB PVL (ref. 50 nm/s)corresponds to a SPL of 94 dB (ref. 20 pPa).

Sound level [dB] Reference level

Sound pressure level (SPL)

Particle velocity level(PVL)

Sound intensity level (SIL)

SPL = 20 log

PVL = 20 Iog1Eu

SiL = i0iog1L!!i.1

Pref= 20 pPa

Uref= 50 nm/s

'ref = i pW/m2

Pref

( U,f

1ref)

74

p SpuHi ----i;---

u

where and S are the cross-spectrum and auto-spectrum in the frequencydomain. Measurements of the acoustic impedance with the p/u method were firstperformed by Schurer in 1996 directly in the throat of a horn. However if thedimensions of the resonators and the dimensions of the sensors are considered thenit is more convenient to measure the impedance of the sample some distance away.If the microphone and the microflown are located at x = O then with the theory forthe one-dimensional wave propagation the impedance at the sample, i.e. at x = L,is:

cosh(rkL)p0c0 sinh(rkL)C(x=L) =

p0c0 cosh(rkL)+ sinh(rkL)

The absorption coefficient and the reflection coefficient of the sample can bederived from

The amplitude and phase characteristics of the combination of the two differenttypes of sensors need to be known. In Appendix C a calibration procedure for amicroflown and a microphone is described. The calibration is performed in thesame impedance tube with the reference microphone at the end of the tube. Thismethod 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.0method. It will be shown that besides the transfer function also the auto-spectra ofthe microphone and the microflown are needed. Therefore both the referencemicrophone and the combination needs to be calibrated.

3.2.4 The pu method

The magnitude of the reflection coefficient can be determined by measuring soundintensity and sound energy density. First the sound energy density and the sound

intensity are described.

(3.6)

(3.7)

Sound energy density

When viscothermal effects are negligible the problem of the propagation of smallperturbations in air can be treated as a conservative elastic process. In general thisis a reasonable approximation for audible frequencies in air. As a result thesummation of kinetic energy of a fluid per unit volume T and potential energy perunit volume U gives the total energy per unit volume (Fahy 1995):

E=T+U=pou(x,t)2+ p,t)2(3.8)

2 2p0c0

where p and u are time- and space-dependent perturbations. E is also called thesound energy density. The time-averaged energy density is obtained byreplacing the pressure and the velocity by the root mean square values. In thefrequency domain, and by knowing that p2 = ½ p p, we find:

=ipouu*+ pp2 (3.9)4 4pc1

where the superscript '' indicates the complex conjugate. By using the root meansquare values the energy density is independent of the position (the viscothermallosses are neglected).

Sound intensity

The instantaneous normal sound intensity I(t), where the subscript 'n' indicates thenormal direction, is defined as the work rate per unit area SS normal to that areaand can be written as:

I, (t)(dW/dt)- =p(x,t)u(x,t)

SS

For one-dimensional plane waves the relation between the instantaneous pressureand velocity perturbations is:

u(x,t) = p(x,t)/poc0 and u(x,t) =p(x,tY/poco (3.11)

75

(3.10)

76

where the superscripts '+' and '-' refer to the components propagating in thepositive and negative x-direction, respectively. So the sound intensity in the planewaves is (with the x- and t-dependence implicit):

(p+2 _(P_)2)/P0c0 (3.12)

The time-averaged sound intensity i is obtained by using the root mean squarepressures. For a pure progressive wave, i.e. when no reflections are present, onecan derive:

2jPrms =0 (3.13)Poco

while for a standing wave the time-averaged intensity 1= 0 and 1 / c0. In this

case the instantaneous sound intensity represents a purely oscillatory flow of soundenergy.

In general the sound intensity may be split into two components (Fahy 1995): anactive component of which the time-averaged value is non-zero and a reactivecomponent of which the time-averaged value is zero. The active componentindicates that there is a local net transport of sound energy and it is this value thatis usually measured. The time-averaged reactive component represents a localoscifiatory transport of energy. if the complex amplitudes for the pressure p and thevelocity u are used in the frequency domain one has:

-1 * -= = 'active + 1reactive

where the time-averaged active or mean intensity 'active and the amplitude of thereactive intensity 'reactive are:

'active =iRe{pu*} and 'reactive =iIm{pu*} (3.15)

Again it is noted that the mean intensity in a one-dimensional plane wave isindependent of position.

The amplitude reflection coefficient

(3.14)

Fahy (Fahy 1995) showed that in a tube the mean speed of the energy transport ëeis the ratio of mean intensity to the mean energy density:

- i 1R2ce = = = ° 2E 1+R

where IR is the amplitude reflection coefficient (see equation 2.26 in section 2.2)of the end of the tube, see Figure 3.11. Equation (3.16) forms the basis of the p-umethod, but now we are interested in the amplitude reflection coefficient.

microphone

microflown

Figure 3.11 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 incontrast to Chapter 2 here 1= i):

p(x) = PB {e_ikx +Re}

u(x)= PB {e__Reix} with R=--p0c0 PB

where PB is the amplitude of the incident wave. The mean intensity and the soundenergy density can now be written as:

1active=iRe{pu*}= PB2 (1 IR 12)

2p0c0

- i * PB (i+1R12)E = Pou u4p0c02 2p0c02

amplitude r- flection coefficient R

77

(3.16)

(3.17)

(3.18)

With the equations (3.16) and (3.18) the amplitude reflection coefficient and thesound absorption coefficient can be calculated:

1R12=Ec0 'active and a=1-1R12 (3.19)E c0 + 'active

78

However, in a long impedance tube the viscothermal losses during the wavepropagation become important. This is especially the case for higher frequenciesvia e'' X Therefore, if ¡ and K are measured at x = O, then the reflection at x = Lbecomes, with the help of (3.17):

R2(x=L) R2(x=O)e' L

2

(3.20)

The acoustic impedance at the surface of the sound absorbing sample cannot bederived from the sound intensity and the energy density because of their spatialindependent character. For ordinary impedance tube measurements meant todetermine a or I R I this is not a drawback. However, in finite element or boundaryelement calculations the full complex impedance, as derived by the experiments, isoften required as a boundary condition for sound absorbing surfaces. In that caseone of the other measurement methods has to be used.

The energy density is measured to eliminate the amplitude dependence ôB in themean intensity. The auto-spectra in K require a calibration of both sensors (seeAppendix C) and the sensor-switching method cannot be applied here.

Data processing

With a two-channel FFF analyser, Tacflve and K are measured as follows:

1active =Re{S}

I+

2p0c02SPP

where is the cross-spectrum and Sa,, and S,,, are the auto-spectra. For the sake of

completeness it is noted that 'reactive = - Im{S }.The 2p method is the most commonly used technique to estimate the particle

velocity. For sound intensity measurements two microphones are separated by adistance Swith a spacer. In a one-dimensional field the mean intensity is:

'active = (3.22)

(3.21)

3.3 Experimental results

This section presents the experimental results for the different measurementtechniques. First the results of the 2p and the 2u methods are compared. A samplewith 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 thatboth methods provide accurate results. In fact the latter results are already anindication that the model on viscothermal wave propagation in coupled tubes iscorrect.

In section 3.3.2 the 2p method, the p/u method and the pu method are compared.For this purpose measurements with a quarter-wave resonator are performed. Themeasured reflection coefficient of the sample serves as a comparison.

3.3.1 Comparison of the 2p and the 2u methods

An acoustically hard wall

A loudspeaker was set up to generate a random signal in frequency bands of 400Hz as shown in Table 3.3. The overlap between the frequency bands is usedbecause the FFT analyser makes use of a Hanning window. The first and last 25 Hzof each band are omitted to obtain the complete transfer function from 425 to 3925Hz.

79

80

10

101

10_1

Lower bound [Hz] Upper bound [Hz] Sensor spacing [m]

400 800 x2=0.0449

750 1150

1100 1500

1450 1850

1800 2200 x3=0.0252

2150 2550

2500 2900

2850 3250

3200 3600

3550 3950

Table 3.3 Set-up of the frequency bands and accompanying sensor spacing.

In Figure 3.12 and Figure 3.13 the measured transfer functions with the 2p and the2u methods are shown and compared to the theoretical ones.

- Microphones- - Theory (2p)

10_2 I

4000

102

10'

.10o

10- Microflowns- - Theory (2u)

10_2

400 1000 2000 3000 4000Frequency [Hz]

Figure 3.13 Transfer function for anacoustically hard wall using the 2umethod.

The standing wave pattern for different frequencies in the impedance tube has anumber of pressure nodes and velocity nodes located at the positions of the sensors.A maximum in the transfer function conesponds to a node (pressure or velocity) atthe 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

400 1000 2000 3000Frequency [Hz]

Figure 3.12 Transfer function for anacoustically hard wall using the 2pmethod.

node by a quarter of a wavelength, which explains the differences between the twotransfer functions. With the use of these peaks and corresponding frequencies thelengths in the impedance tube, L, x2 and x3, can be determined very accurately,provided that the mean speed of sound c0 is known in advance, as c0 is a functionof the temperature. A comparison of the theory and the measurements resulted inmore accurate lengths of: L = 0.2099 m, x2 = 0.0449 m and x3 = 0.0252 m. These

lengths were again used in the theory presented in Figure 3.12 and Figure 3.13.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 toprovide an accurate comparison. The peak heights are determined by theviscothermal losses which are predicted well. The viscothermal effects can also beseen in the phase angle of the transfer function, see Figure 3.14 and Figure 3.15.

90

Q 0

90

180

- Microphones (2p)- - Theoiy (no viscothermal eff.)

90

QQ

Q

i-90

180-4000 400

- Microflowns (2u)- - Theory (no viscothermal eff.)

flJ JL

81

1000 2000 3000 4000Frequency [Hz]

Figure 3.14 Phase of the transfer Figure 3.15 Phase of the transferfunction for an acoustically hard wall function for an acoustically hard wallusing the 2p method. using the 2u method.

The jump in Figure 3.12 and Figure 3.13 at 1800 Hz is caused by the use of twosensor distances x2 and x3.

The measured and theoretical reflection coefficients are derived via the transferfunctions. In Figure 3.16 and Figure 3.17 the results for the 2p and the 2u method

are presented.

400 1000 2000 3000Frequency [Hz]

82

0.8

J.0.6

0.4

0.2 - Microphones- - Theory (2p)

O400 1000 2000 3000 4000

Frequency [Hz]

Figure 3.16 Magnitude of the reflection Figure 3.17 Magnitude of the reflectioncoefficient of an acoustically hard wall coefficient of an acoustically hard wall(2p method). (2u method).

For both methods the experimental results correspond to the theoretical reflectioncoefficient of IRI = 1, except for the higher frequencies where some deviationoccurs. It was seen that the loudspeaker cannot generate the same pressure levels inthe tube for these frequencies so that the signal-to-noise ratio for the transducers isless. The oscillating variation is the result of a frequency dependence to calculateR. To be more specific: with two sensors the two unknown forward and backwardtravelling waves are determined. The condition number of the resulting matrix isfrequency dependent and shows a maximum which corresponds to the maximumdeviation of R.

Also, from equation (3.3) follows that for high values of the impedance Ç thedenominator approaches zero. This causes numerical problems and incorrect valuesof the impedance. It was seen that an inaccurate transfer function can lead to anegative real part of the impedance which is physically impossible.

It is remarked that the both the microphones and the microflowns as used in themeasurements have an almost equal sensitivity. As a result no calibration factorswere used.

A single quarter-wave resonator

The same procedure for the measurements is followed but now an aluminiumsample with a single quarter-wave resonator is placed in the impedance tube. Thelength of the resonator is 60 mm and the radius is 5 mm. Due to the inlet effects the

0.8

¿0.6

0.4

0.2

o400 1000 2000 3000

Frequency [Hz]4000

0.8

0.6

0.4

0.2

- Microphones- - Theoiy (2p)

0.8

0.6

- 0.4

0.2

Figure 3.18quarter-wave resonator.

- Microflowns- - Theory (2u)

o1000 1200 1400 1600

Frequency [Hz]

83

effective length of the resonator is 63 mm (see section 2.4.1). The porosity Q of thesample is 0.04. In Figure 3.19 and Figure 3.20 the measured and theoreticalabsorption coefficient a is given for the 2p and the2u method, respectively. It can be seen that bothmethods provide accurate results for the frequencyrange of interest, i.e. from 1000 to 1700 Hz.Furthermore the absorption coefficient can bepredicted exactly for each frequency which

indicates that the model for the viscothermal wavepropagation is accurate. For the shear wave number one has s > 10 so that theKirchhoff approximation is used.

It is noted that the repeatability of the experimental results is very good and thatboth the 2p and the 2u methods are very well suited to obtain the absorption

Figure 3.19 Absorption coefficient of a Figure 3.20 Absorption coefficient of asingle resonator (2p method). single resonator (2u method).

The quarter-wave resonator consists of a single tube which is closed at one end. Itappeared that the closed end can be approximated more accurately by animpedance condition of Ç = 1000 instead of Ç = oo. The fact that not exactly all theenergy was reflected is caused by the presence of a wall which is almostacoustically hard.

The small disturbances or peaks in the figures are the result of the in-seriesconnected frequency bands of 400 Hz.

aSample with

0-1000 1200 1400 1600

Frequency [Hz]

84

coefficient of sound absorbing samples. Also, the rnicroflowns can be seen as agood alternative to microphones in an impedance tube.

By considering the similarity of the theoretical results and the experimentalresults for the sound absorption coefficient it is concluded that the theoreticalmodel for viscothermal wave propagation in two axially coupled tubes, theimpedance tube and a resonator, is accurate for frequencies ranging from 1000 to1700 Hz.

3.3.2 Comparison of the 2p, the p/u, and the pu methods

Results of the p/u method and the p.0 method are compared to the ones of the 2pmethod in this section (see also: de Bree 2000). The experimental results are againpresented for a sample with a quarter-wave resonator because its acousticbehaviour is well known.

The impedance tube with the square cross-section as shown in Figure 3.3 is usedfor both the 2p, the p/u, and the pu method. The single quarter-wave resonator hasthe following dimensions: the radius R = 4.55 mm and the effective length L+d =72.5 mm The first resonance frequency of this resonator lies at 1200 Hz andabsorbs approximately 80% of the incident energy at this frequency.

For the measurement set-up broadband noise is generated by a DSP SigLab 20-42 box. This box is also used as a front-end for the two input signals. The SigLabbox is connected to a PC which runs the SigLab software under MatLab.

The following figures show the results for the amplitude of the reflectioncoefficient R for both the measurements and the theory. The frequency range of thegenerated noise is 50 to 4050 Hz. However in the figures a limited frequency rangeis depicted from 250 to 2000 Hz around the first resonance frequency of theresonator. It is noted that R is calculated at the surface of the sample.

T 0.6

i

0.8

T0.6

0.4

0.2

- Measurements (2p)- - Theory

500 1000 1500 2000Frequency [Hz]

Figure 3.21 Reflection coefficient (magnitude) using the 2p-method.

85

o500 1000 1500 2000

Frequency [Hz]

Figure 3.22 Reflection coefficient (magnitude) using the p/u method.

86

i

0.8

T0.6

0.4

0.2

o500 1000 1500 2000

Frequency [Hz]

Figure 3.23 Reflection coefficient (magnitude) using the p.u-merhod.

The figures show that the three methods provide identical results for the frequencyrange where the quarter-wave resonator is effective, i.e. from 1000 to 1400 Hz.Furthermore the theory predicts the acoustic behaviour of this quarter-waveresonator in a sample very well.

The new p/u and the pu measurements show some oscillating inaccuracies. Theresults may be further improved with a more accurate 'correction function' asobtained from the calibration. In Appendix C it is shown that the correctionfunction is only a third order polynomial which is fitted through the experimentalresults of the calibration. For the frequencies below 1000 Hz it can be seen that thecorrection function deviates somewhat from the calibration results. The oscillatingvariation is the result of a frequency dependence to calculate the correctionfunction (the condition numbers of the matrix for the p/u and the pu method itselfdo not depend on the frequency). In this respect it has to be remarked that thecalculation of R for an acoustically hard wall is very sensitive to measurementinaccuracies.

For the pu method the imaginary part of the reflection coefficient cannot bemeasured. However when only the absorption coefficient a (a = i -1R12) or themagnitude of the reflection coefficient R is needed this is not a drawback. For theother two methods the complex reflection coefficient is calculated from which theacoustic impedance can be derived. The complex results for R are shown in Figure

0.8

10.6

1

0.4

0.2 -

O50 1000 2000 3000

Frequency [Hz]

- Measurements (p/u)- - Theory (atx=L)-. Theory (atx= 0)

u

O

180

360

540

- Measurements (p/u)- - Theory (atx=L)

Theory (at x= 0)

7204000 50 1000 2000 3000

Frequency [Hz]

87

3.24. The frequency range of 50 to 4000 Hz is used so that also the secondresonance frequency at 3600 Hz can be seen (three-quarters of a wavelength).

4000

Figure 3.24 Reflection coefficient (magnitude and phase) of a quarter-wave resonator.

If R is calculated at x = O instead of at the surface of the sample, at x = L, then theviscothermal wave propagation in the impedance tube causes an extra energy lossover the length L. In the case of r= i the difference is negligible.

3.4 Conclusions

It was shown that the impedance tube technique is an efficient and inexpensivemethod to quantify the acoustic properties of sound absorbing material for normalincidence. Moreover, it is sufficient to measure the sound absorption with onlynormally incident waves because the resonators in the samples are locally reacting.The frequency range of interest is between 50 and 4000 Hz which is large enoughto cover most practical problems in noise control engineering.

Besides the standardised 2p method with two microphones three additionalmethods have been presented and tested. These new methods make use of a newacoustic particle velocity sensor: the microflown. The 2u method corresponds tothe 2p method but uses two microflowns instead of two microphones. It gives thesame accurate results. With the p/u method the impedance at a cross-section in theimpedance tube is directly measured. Next, one can calculate the impedance andsound absorption coefficient at the surface of the sample with the model for theone-dimensional viscothermal wave propagation. However, a more accurate

88

calibration for the combination of a microphone and a microflown, compared to theone as used in Appendix C, needs to be used. The pu method measures theacoustic energy density and the sound intensity in the impedance tube. Thecombination is a measure for the reflected energy and so the amplitude reflectioncoefficient of the sample can be calculated.

The results for the reflection coefficient of an acoustically hard wall and thesound absorption coefficient of a sample with a quarter-wave resonator werepresented in section 3.3. It was seen that the four measurement techniques give thesame results. Furthermore, the experimental results agree very well with thetheoretical results. So the viscothermal wave propagation is correctly modelled forthis 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 referredto the results presented in Chapter 4.

Chapter 4

Experimental verification of the coupled tubes model

4.1 Introduction 894.2 Sound absorption of single tube resonators 9043 Sound absorption of coupled tubes resonators 944.4 Conclusions 98

4.1 Introduction

In Chapter 2 coupled tubes in a wall were presented. In the coupled tubes abroadband resonance is present so that broadband sound absorption is created. Itwas explained that the incident waves from a source of sound are cancelled by thewaves in the resonator as well as damped by the viscothermal losses in theresonator.

The present chapter describes the experimental verification of the coupled tubesmodel. The experiments were performed in an impedance tube. Reference is madeto Chapter 3 for a description of the impedance tube technique. Some results areobtained by using particle velocity sensors, so-called microflowns, instead ofmicrophones. The microflown is also described in Chapter 3.

To verify the predictions of the coupled tubes model a number of samples withresonators were constructed. In Figure 4.1 the measurement set-up and a number ofsamples is shown. In section 4.2 the sound absorption coefficients of samples withsingle tube resonators are presented. The first sample consisted simply of a singlequarter-wave resonator. Next the number of resonators per unit area was varied toexamine the effect of the porosity parameter Q of a wall with resonators. The effect

of different lengths of the resonators was briefly investigated. The quarter-waveresonators were closed at one end and as a comparison the sound absorptioncoefficient for resonators with an open end was also measured. The latterresonators can be described as half-wave resonators.

90

Impedance tube

Filters Oscilloscope Amplifiers Speaker

Figure 4.1 Photograph of the measurement set-up and of a number of samples.

As a next step three samples with coupled tubes resonators were tested. Thesesamples show the broadband sound absorption as predicted in Chapter 2. Theresonators in the first sample simply consisted of two axially connected tubes andoperated between 1500 and 2000 Hz. The second sample was designed to absorb atleast 80 percent of the incident energy over an even broader frequency band. Theresonators consisted of a more complex configuration of coupled tubes. Finally, thethird sample was designed for a higher frequency range, i.e. from 2700 to 4000 Hz.

4.2 Sound absorption of single tube resonators

A single quarter-wave resonator

An aluminium sample with a single quarter wave resonator was placed in theimpedance tube. The length of the resonator was

50 6060 mm and the radius 5 mm. Due to inlet effectsthe effective length of the resonator was 63 mm(see section 2.4.1). The porosity Q of the samplewas 0.04. In Figure 4.3 the measured andpredicted absorption coefficients a are given for

quarter-wave resonator.the two-microphone techmque (2p method). Themeasurements and the theory agree very well. It can be seen that indeed thequarter-wave resonator absorbs the incident energy for a frequency range which is

Figure 4.2 Sample with a

0.8

0.6

0.4

0.2

O1000

Figure 4.3 Absorption coefficient of a single quarter-wave resonator(2p method). £2 = 0.04

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 isreflected from the closed end. Therefore in the application of the theory forpractical calculations the closed end is approximated with an impedance conditionof Ç= 1000 instead of Ç 00

The effect of the surface porosity Q

The sound absorption coefficient a for samples with a different number ofresonators per unit area is investigated. The length of the resonators is 50 mm andfor the effective lengths the theory of section 2.4.1 is used.

Measurements (2p)- Theory

1600

91

much wider than the quarter-wave resonance frequency of 1350 Hz. This isexplained by the viscothermal effects in the resonator, see Chapter 2. Furthermorethe measurements demonstrate that the absorption coefficient can be predicted verywell for the whole frequency range indicating that the model for viscotherxnal wave

propagation is accurate.

1200 1400Frequency [Hz]

92

0.8

0.6

0.4

0.2

o1000

Microphones- - Theory

1200 1400 1600Frequency [Hz]

1800 2000

Figure 4.4 Sound absorption coefficient of samples with a dfferent porosity Q (2pmethod).

Figure 4.4 shows that the predicted results correspond very well with theexperimental results. So the theory predicts the effect of the porosity correctly.Moreover, the peak in the absorption coefficients shifts to the right when moreresonators are used. So the effective length, which depends on the distance betweenthe resonators, is also correctly predicted by the theory, i.e. more resonators perunit area lead to higher effective resonance frequencies. Figure 4.4 also shows thatin this particular case more resonators do not lead to a higher sound absorptioncoefficient. To realise a = i at a specific frequency one needs to tune the porosity,i.e. the number of resonators per unit area. Then the amount of dissipated soundenergy is equal to the energy of the incident waves, and the harmonic mass flow atthe entrance of the resonators balances the flow of the incident waves (see section2.4.1).

The effect of different lengths

In Figure 4.5 the sound absorption coefficient of a sample with two quarter-waveresonators of different lengths is depicted. As expected the effect of the eachindividual resonator can clearly be seen. Also here the results of the theorycorrespond very well with the experimental results.

93

It was demonstrated in Chapter 2 that the combination of two separate quarter-wave 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 resonancefrequencies are too far apart.

The somewhat noisy experimental results at low absorption coefficients are aminor effect because then the sample acts approximately as an acoustically hardwall. It was demonstrated in section 3.3.1 that the impedance tube technique isrelatively sensitive to measurement errors in that case.

1000 1500 2000 2500Frequency [Hz]

Figure 4.5 Sound absorption coefficient of a sample with two resonators of differentlengths (2p method).

A half-wave resonator

The next sample contains two tubes which are open at both sides. These tubes canbe seen as half-wave resonators. An advantage of resonators which are open at bothends is that they can act as an acoustic filter and yet can let pass sound, a mass flowor light. In Figure 4.6 the sound absorption coefficient is shown with an absorptionpeak at about 1600 Hz. It is noted that the effective length of the resonatorsconsists of the geometrical length plus two small end corrections d at both ends ofthe resonators. At the open end the impedance Ç = O is prescribed and thecorrection length for a piston was used as an estimation. The total correction lengthwas 4.2 mm

i

0.8

0.6

0.4

0.2

o

94

i

0.8

0.6

0.4

0.2

Microphones- Theory

2000

i =0.022d=4.2mm

o1000 1200 1400 1600 1800

Frequency [Hz]

Figure 4.6 Absorption coefficient of a sample with two half-wave resonators (2pmethod).

The measurements prove again that the viscothermal model for coupled tubes isaccurate. Also the resonance frequency is predicted well via the application of aneffective length.

The results for the single tube resonators demonstrate that a very good agreementexists between the predicted acoustic behaviour and the measurements. Theviscosity and the thermal conductivity of the air in the resonators play an importantrole when sound absorption is considered. Evidently the treatment of viscothermaleffects was quite adequate over the complete frequency range in this case.

The saine test strategy was followed for samples with resonators consisting ofcoupled tubes.

4.3 Sound absorption of coupled tubes resonators

The measured sound absorption coefficients of three samples with coupledresonators are compared to the predicted absorption coefficients. The first samplehas a high absorption coefficient for frequencies between 1500 and 2000 Hz. Thesecond sample contains a more complex configuration of coupled tubes and iseffective in the same frequency range with an absorption coefficient of a> 0.8.The third sample is optimised for frequencies between 2700 and 4000 Hz.

Sample 1: 1500 2000 Hz

The sample has three equal resonators which consist of two coupled tubes. Thesample is designed for the frequency range of 1500 to 2000 Hz. Therefore the tubeat the entrance has a diameter of 10 mm and the coupled tube has a diameter of 1.5min. The experimental and analytical results are given in Figure 4.7.

0.8

0.6

0.4

0.2

01000

- Microphones-. Microflowns

- - Theory

50

10

- Theory: û =0.12L = 45 + 3.6 mm

I L2=49+0.Smm1500 2000 2500 = 1000

Frequency [Hz]

95

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 themeasurements for the specified frequency range. This indicates that theviscothermal effects and the coupling of tubes are correctly modelled.

The value of the sound absorption coefficient is below the maximum of 1.0. It isnoted that the optimal porosity for a sample with these resonators can be higher orlower than the arbitrary value as used for this sample. The impedance of the wallcan be used to determine the optimal porosity.

The small deviations for a in the specified frequency range are probably due toerrors in the geometry and the estimated inlet effects. The effective length L1 isobtained by using the end correction for a wall with equally spaced resonators (seeequation (2.57) with a = 16 mm) For L2 the end correction of a single tubecentrally located in a larger tube is used (equation 2.58). The end condition for thesmaller tube is estimated using Ç= 1000.

96

It is also shown that the 2p method and the 2u method provide similar results.However, there is some noise in the results at higher frequencies. The sample actsas an acoustically hard wall for these frequencies and therefore the measurementmethods provide less accurate results.

Sample 2: 1300 - 2000 Hz

The second ample as shown in Figure 4.8 was designed for a broader frequencyrange than the previous sample, i.e. a sound absorption of more than 80 percentfrom 1300 to 2000 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.

0.8

0.6

0.4

0.2

1000 1500 2000Frequency [Hz]

2500 3000

- Microphones-, Microflowns

- Theory

Theory: = 0.28

L1=47+3.6mniL2= 55 0.8 mmL3=45 +0.8mmÇ = 1000

Figure 4.8 Sound absorption coefficient of a sample with coupled tubes.

The experimental results prove that the desired more broadband sound absorptionis indeed achieved. It also shows that the coupled tubes model predicts the amountof absorption as well as the actual frequency band very accurately. Therefore it isconcluded that the theoretical model can be applied reliably for this frequencyrange to design sound absorbing walls with coupled tubes resonators for apredefined frequency range.

Sample 3: 2700 - 4000 Hz

The sample designed for a high frequency range from 2700 to 4000 Hz isillustrated in Figure 4.9. The demands on the level of sound absorption arestringent for this sample: a maximum sound absorption of 100% at two frequenciesand around these frequencies 90%. The sample was constructed from a perforatedbrass cylinder connected to a back-plate with capillary tubes. The sample had 118tubes with a diameter of 2 mm Three of the tubes were used to connect thebackplate to the sample by means of three pins. The remaining 115 capillary tubesin the back-plate were closed at one end with beeswax. As a result the length of thecapillary tubes was somewhat reduced.

In Figure 4.9 the results are presented. The word 'gap' in the legend indicatesthat in that case there is a small gap between the back-plate and the brass sample of0.2 mm The latter results indicate the effect of leakage of the coupled resonators.

I

0.8

0.6

0.4

0.2

o2000

Microphones (beeswax)- Microphones (gap)

- - Theory

2500 3000 3500Frequency [Hz]

brass sample

li

capillary tubec=0.148 -25 24

Figure 4.9 Sound absorption coefficient of a sample with coupled tubes. Di,nensions asindicated in the sketch in mm.

The experimental results indicate that the level of sound absorption meets thedesign requirements well. The sample indeed shows a high sound absorption overthe specified frequency range.

Some differences can be seen with the predicted results. These are attributed tominor deviations of the geometry of the resonators. For instance the small amount

97

98

of beeswax inside the capillary tubes causes the higher resonance peak (at 3600Hz) to shift somewhat to the right.

From the construction point of view it is noted that a small gap between theconnected tubes has little effect on the sound absorption coefficient, i.e. around theresonance frequency of the capillary tubes there is a pressure minimum at theentrance and this is precisely the location of the small gap. So fortunately a smallgap has little effect on the sound absorption behaviour.

It is noted that the repeatability of the experimental results is very good for thesamples presented in this section.

In the next chapter the coupled tubes model will be used for a comparison withmodels for porous materials. It will be shown that the acoustic behaviour in thepores can be approximated well with the coupled tubes model. The numericalresults of the models found in the literature will be compared to the results of thecoupled tubes model.

4.4 Conclusions

The experimental and the theoretical results for the sound absorption of sampleswith single tubes, i.e. quarter-wave and half-wave resonators, and samples withcoupled tubes were presented in sections 4.2 and 4.3, respectively. The differentsamples demonstrate the effects of different lengths, a varying porosity, half-waveresonators, and coupled tubes. It was also shown that the samples with coupledtubes resonators can create a more broadband sound absorption. Moreover, themeasurements provided test-cases for the theoretical results.

It was seen that the coupled tubes model predicts the sound absorption of thesamples very accurately for the complete frequency range. This indicates that themodel for the viscothermal effects in the tubes and the coupling between tubesrepresents physics well.

Finally, it is concluded that the coupled tubes model is a reliable tool for thedesign of a sound absorbing material with resonators for a specified frequencyrange. In Figure 4.10 simulated results for coupled tubes resonators are

summarised.

0.8

0.6

0.4

0.2

-. Single- - Double- Multiple- Triple

/ / \O

L._.-1 I

800 1000 1200 1400 1600Frequency [Hz]

Figure 4.10 Numerical sound absorption coefficients of a wall with dfferentconfigurations of resonators for optimal absorption (see also Chapter 2).

For a physical interpretation of the more broadband resonance and the high soundabsorption of the resonators the reader is referred to Chapter 2 and Appendix A.

1800 2000

99

Chapter 5

Sound absorbing material represented by a networkof tubes

5.1 Introduction 1015.2 A random network of tubes 1035.3 Comparison to an empirical impedance model 1085.4 Comparison to the Limp theory 1135.5 Comparison to the Biot theory 1185.6 Conclusions and remarks 125

5.1 Introduction

The coupled tubes model of Chapter 2 provides an interesting opportunity topredict the acoustic behaviour of sound absorbing materials consisting of a networkof tubes. In the present chapter this approach is described, i.e. sound absorbingmaterial is numerically represented by a labyrinth-like distribution of a largenumber of tubes. It will be shown that with a limited number of parameters thepredicted acoustic behaviour corresponds well with the results of other, moresophisticated, numerical models.

Conventional sound absorbing materials such as fibrous material (i.e. glass woolor rockwool) and porous material (i.e. with pores) such as foams, are commonlyapplied to reduce noise problems. The acoustic behaviour of these materials can bemeasured 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 soundabsorbing properties of such materials and to predict the acoustic performance inadvance.

The comparison between a numerical network of tubes, the so-called networkdescription, and models from the literature has been made for a simple case with anormal incident sound field. It is noted that, among other models, the network

102

description can also easily be used for oblique or random incident waves becausethe 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 thematerial. An advantage of a surface description is its limited calculation time in, forinstance, finite element type of calculations. Drawbacks are its restriction to locallyreacting materials and normal incident waves. Furthermore, the model is limited toa class of materials, for instance fibrous rockwool materials, due to its empiricalcharacter.

The volume description actually takes into account the interaction between theframe of the material and the fluid trapped in the frame. The frame can be seen asthe skeleton of the sound absorbing material and often physical parameters aregiven for the frame instead of for the solid material of which the frame isconstructed. In Table 5.1 a survey of volume descriptions as used in the presentinvestigation is given. A more extensive description of the Limp and the Biottheory, including some appropriate references, is given in sections 5.4 and 5.5.

Rigid Limp Biot Network

Volume Volume Volume Volume

Infinitely stiff No stiffness Poisson' s ratio Infinitely stiff(no mass) (with mass) Young's mod.

MechanismsInertiaoffluid jInertia of frameViscous effects JThermal effects

Wave types lx dilatational lx dilatational 2x dilatational lx dilatationallx shear

Table 5.1 Comparison of models for sound absorbing materials.

For the theories as described in the literature a subdivision can be made inincreasing order of complexity: the so-called 'Rigid' theory treats the frame asinfinitely stiff, the 'Limp' theory treats the frame without stiffness but includes themass, and the 'Biot' theory includes both the stiffness, mass and the acousto-elasticcoupling between the frame and the fluid. For the volume description of Biot threetypes of waves with different wave numbers are present inside the material. This isthe result of the extra degrees of freedom for the frame. Measurements have

ApproachFrame

103

indicated that indeed three types of waves propagate in materials with an elasticframe and a fluid phase (Geerits 1993). Although the Biot theory is the mostcomprehensive one, the large number of parameters needed to describe the bulkmaterial and some difficult techniques to obtain these parameters are drawbacks.Furthermore the computational effort in finite element calculation is high becauseof the number of degrees of freedom per node. In the Rigid, Limp and Networkdescriptions 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 touse volume description of sound absorbing material with a limited number ofparameters. The description is based on the coupled tubes model as presented inChapter 2. Furthermore, the results can easily be translated to an efficient surfacedescription (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 complexBiot theory for normal incident waves.

5.2 A random network of tubes

Two examples of a random network of tubes as applied in the present calculationsare depicted in Figure 5.1. Each tube is identical, has length L and radius R and isrepresented by a short line. The length and radius of the tubes can be seen asequivalent ones for the pores inside the material. The tubes are connected to eachother (see Figure 5.1) to form a type of network. Tubes which end inside thematerial are considered as being closed with an acoustically hard termination.

At the left-hand side of the network incident waves are assumed via a constantpressure for normally incident waves or via a complex pressure distribution foroblique incident waves. At the right-hand side an acoustically hard wall isprescribed to simulate a sample of material in an impedance tube. Other boundaryconditions, such as an open end, can also be prescribed.

Both networks are constructed from a so-called 'full' network of tubes (here:30x19 tubes on a square grid). To create a labyrinth-like network tubes arerandomly deleted from the full network. In this way the two networks as shown inFigure 5.1 were obtained. Only 85 percent of the full network is shown on the left-

104

hand side and on the right-hand side this is 70 percent. To eliminate tubes from thefull network a uniform distribution of random numbers was used.t

....__.u. ._............I......u_.._. uuuu__..__._.........._......_._.._..._-_..u.........._

......................u........._...u..................._......_.._....-........._.u........_......0..u....u .._...._....._u___.uuuu._.uuuuuuuuu...................u....._.........................._.._....u. uuuuuuuuu... ... ....uu_.Length [m]

u.. ...uu . LINH IIII mu III UI I

''U!IÎIII!IIIMIUiii'i..._.._._.

u u

..

Length [ml

Figure 5.1 Examples of a random network of coupled tubes (85% and 70% of fullnetwork, 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 thenumber of horizontal tubes. The height of the material is not specified. Instead theporosity Q of the sample is used so that the acoustic behaviour per unit area iscalculated. Summarising: the four main parameters for the network to describeacoustic materials are: L, R, Q and the percentage % of the full network. Theseparameters describe the bulk behaviour of a sound absorbing material. In generalthe porosity of a material is known. The equivalent length, radius and percentagecan 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 beenused for the drainage dominated flow in porous media (Aker 1998). There a full network isused with a random variation of the radius of the pores.

O 0.038o 0.038

0.8

0.6

0.4

0.2

oO

- - Full network85 % of full network

- 70 % of full network55 % of full network

105

percentage and the radius are the main bulk parameter to describe a soundabsorbing material.

The computed sound absorption coefficients for various samples of the type asdepicted in Figure 5.1 are shown in Figure 5.2 and Figure 5.3. The total networkhad a thickness of 38 mm. The length and radius of each tube were chosen as L =2.0 mm and R = 0.1 mm The surface porosity had a value of Q = 0.95. For soundabsorbing materials a volume porosity of 0.95 is common and it is assumed herethat the surface porosity equals the volume porosity.

The results show a striking resemblance with the measured absorption coefficientof two samples of glass wool (25 min and 45 mm thick) as shown earlier in Figure2.17. In Figure 5.2 the effect of the density of the distribution of tubes is shown. Acombination of the viscotherrnal losses in the tubes and the effect of the labyrinth-like structure is the cause for this sound absorbing behaviour (see also section 5.3for the effect of the so-called flow resistivity).

- Basic configuration- - Thickness = 76 mm

1.0 mmQ=0.80

Figure 5.2 Sound absorption coefficient Figure 5.3 Sound absorption coefficients.for various networks of tubes. L=2.0 mm, R Basic configuration: L=2.0 mm, R = 0.1= 0.) mm, Q=0.95, thickness = 38 mm. mm, Q=0.95, 70%, thickness = 38 mm.

M interesting example to apply the network description to is aluminium foam (Banhart1994). The frame is rigid and the porosity Q is known in advance. The main bulkparameters % and R need to be determined. See also the paper of Lu, Chen and He (2000).

1000 2000 3000 4000Frequency [Hz]

40001000 2000 3000Frequency [Hz]

106

In Figure 5.3 the three other parameters are varied:The thickness of the sample is doubled (30x38 tubes, thickness = 76 mm).As a result also lower frequency sound waves are more absorbed.The length of each tube is set to L = 1.0 mm. This change has only a smalleffect on the sound absorption.The porosity Q= 0.8. 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 networkwith R = 0.1 mm and Q= 0.95 is again the basic configuration (thickness = 38 mm,30x20 tubes).

0.8

0.6

0.4

0.2

R=0.4mm- - R=0.2 mm- R=0.lmm- R=0.05 mm

Oo

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 viscouseffects are governed by the shear wave number s which is proportional to the radiusR. As follows from Figure 5.4 the slope in the low frequency range is determinedby the radius. At higher frequencies the viscosity effects for a radius of R = 0.05mm are still too high and acoustic waves are not able to penetrate sufficiently in thenetwork, i.e. the jump in the impedance at the interface with the numerical soundabsorbing material is too large.

The absorption coefficient of the samples is calculated with the transfer matrixprocedure as described in section 2.3.2. The nodal pressures p at each junction 'i'are solved via the inverse of the system matrix. The nodal velocities u are solved

40001000 2000 3000Frequency [Hz]

Q p0c0 fly

u(x=O)j=1

where p, is the prescribed pressure at the surface of the sample (at x = 0) and n isthe number of tubes in the y-direction (vertical direction in Figure 5.1). With theuse of Ç the reflection and absorption coefficients are calculated. On an ordinaryPC the calculation of the sample shown in Figure 5.1 takes one or two minutes for40 frequencies. It is noted that a new calculation with a different randomdistribution but the same percentage of the full network gives slightly differentresults for a This was done by starting the random generator at a different state sothat a different labyrinth-like structure results with approximately the same numberof tubes. As will be shown later on a finer network with more tubes reduces thiseffect (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 witha uniform random distribution of identical tubes. Obviously a random parametercan also be used for the radius R and the length L of the pores. Some first resultsare presented in Figure 5.5 and Figure 5.6 for a nonnal distribution of the radiusand length. A full network of 30x20 tubes is used. In the figures the 95% densityregion of the parameter to be varied is indicated. The other parameters are:thickness = 38 mm and Q= 0.95.

107

subsequently. These quantities provide the information for the non-dimensionalacoustic impedance via:

Ç(x=O) px=o (5.1)

108

0.8

0.6

0.4

0.2

uL

- R=0.lmm- - 0.05<R<0.l5mm

0.01 <R< 0.20 mm

0.8

0.6

0.4

0.2

-0.731

19 IC = Re(Ç) + Im(Ç) i = [ + 9.08[ ,f)] +

/ 1000 j j

It can be seen that for these random distributions of the radius the effect on thesound absorption coefficient is small. For a variation of the lengths it was alreadyseen that this parameter is of less importance.

5.3 Comparison to an empirical impedance model

In 1970 Delany and Barley (Delany 1970) presented a simple power-law for theacoustic impedance of fibrous sound absorbing materials. Based on measurementson a range of these materials they showed that the impedance can be given as afunction of the frequency f divided by the flow resistivity q):

i (5.2)

The quantity q) in (5.2) has units [Nsm4] which differ by a factor of 1000 from theunits as originally used by Delany and Bazley. The flow resistivity is frequencydependent but for fibrous materials usually the static flow resistivity is used. Thestatic value is measured relatively easily from:

(5.3)

0 1000 2000 3000 4000Frequency [Hz]

Figure 5.5 Sound absorption coefficientof a full network with a normaldistribution for the radius R of the tubes.

O0 1000 2000 3000 4000

Frequency [Hz]

Figure 5.6 Sound absorption coefficientof a full network with a normaldistribution for the length L of the tubes.

109

where zip is the static pressure drop across the sample. L is the thickness of thesample and u the flow velocity through it. According to Biot the resistivity isrelated to the dynamic viscosity of the fluid 4u, for a Poiseuille type of flow, as:

1/1 (5.4)

with being the so-called Darcy coefficient of penneability. The flow resistivity isalso referred to by other authors as the static flow resistance per unit width Øo.

Picard describes a method to measure q) dynamically (Picard 1998) and it is notedthat various authors such as Zwikker & Kosten and Biot use a cylindrical pore forfibrous (!) materials to approximate the dynamic flow resistance per unit width as afunction of the shear wave number s.

In the present investigation the static flow resistivity is used for a morestraightforward comparison. In Table 5.2 the range for q) for fibrous materials asmeasured by different authors is indicated. The range for the porosity Q isindicated as well.

Table 5.2 Range of possible values for various fibrous sound absorbing materials.

The absorption coefficients as calculated with the empirical impedance model ofDelany and Barley are given in Figure 5.7 for different values of q).

Author Flow resistivity Ø [Nsm] Porosity Q [-]

Delany & Barley (1970) 20,000 - 800,000

Picard (1998) 9,000 135,000 0.924 - 0.98

Allard (1986) 1,000 88,000 0.97 - 0.983

Attenborough (1983) 5,000 100,000 0.85 0.97Launks (1989) 4,800 - 47,000 0.9 - 0.975

110

1

0.8

0.6

0.4

0.2

oO

- 4) = 10,000 [Nsm4]4) = 20,000 [Nsm4]

4) = 40,000 [Nsm4]

-. 4) = 100,000 [Nsm4]

Figure 5.7 Sound absorption coefficient according to Delany and Bazley.

Clearly a higher flow resistivity causes a lower sound absorption. The reason forthis behaviour is that for a high flow resistivity the fibres and the fluid inside theframe are vibrating more in phase than for a small value of Ø. Therefore thedissipating mechanism, i.e. the viscous drag force, is less effective. This behaviourcan be shown in detail with the theory of Biot which uses both degrees of freedomfor the frame and the fluid (see section 5.5).

To compare the results of our network description with the prediction of theempirical impedance description, the flow resistivity of the network needs to beknown. The network description provides an easy way to calculate the flowresistivity as a function of the frequency. With the help of equation (5.3) is

calculated. The length L of the sample is known. A pressure drop of 1.0 Pa isprescribed and the resulting flow velocity is determined. The static flow resistivityis determined for a frequency approaching 0.0 Hz. It was observed that the flow atboth ends of the sample is approximately the same for frequencies close to 0.0 Hzso that it is concluded that the static flow resistivity can be determined accuratelyby performing the calculation for a very low frequency. In Figure 5.8 the staticflow resistivity is given for a number of samples. Results are given for twodifferent thicknesses. Extra numerical samples were used with a different randomdistribution of tubes, i.e. by starting the random generator at a different state, toinvestigate the sensitivity on Ø.

40001000 2000 3000Frequency [Hz]

300000

100000

'E50000

-e-

20000

-O- Width=38mm-A- Width=76mm

111

10000100 90 80 70 60 50

Percentage of full network [%]

Figure 5.8 Static flow resistivizy as afirnction of the network configuration.

It can be seen from Figure 5.8 that the range for the predicted ç) lies within theranges as shown in Table 5.2. Furthermore, the thickness of the sample hardlyaffects ç) so that the flow resistivity is indeed per unit width. For a low densenetwork ç) is rather high. For these densities there is a lot of 'dead' material insidethe network, i.e. tubes without any connection to the entrance of the sample. As aresult the differences between these networks for random distributions with thesame densities are large.

The sound absorption coefficient as measured indirectly by Delany and Bazley arecompared to the results of a network with a flow resistivity of 22,000 Nsm (thedensity of the network is then 86%), a thickness of 76 mm (30x39 tubes), aporosity of 0.95 and pores of L = 2.0 mm and R = 0.1 mm.

The results for the network description for a agree quite well with theexperimental results of Delany and Bazley for the complete frequency range.Evidently, a network of tubes is well suited as a volume description for a soundabsorbing material.

112

0.8

0.6

0.4

0.2 - Random network (22,200)- - Delany & Ba.zley (22,000)-. Delany & Bazley (10,000)

1000 2000 3000 4000Frequency [Hz]

Figure 5.9 Sound absorption coefficient of a networkcompared 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 offersalso the possibility to study the effects of the frequency on the flow resistivity. Aharmonic pressure perturbation is prescribed at both sides of the network to realisea pressure difference, i.e. pj(x=0) > p2(x=L). The flow resistivity is determined withequation (5.3) and the flow velocity u is determined on both sides of the numericalsample, i.e at x = O and at x = L. The numerical dynamic flow resistivity of aspecific network of tubes is given in Figure 5.10.

The dynamic flow resistivity is clearly frequency dependent and decreases to alower value for higher frequencies. This behaviour for a network of tubes can beascribed 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 theresult of the storage and dissipation of pressure and velocity perturbations insidethe network. As a result the dynamic flow velocity at the low pressure side issomewhat lower. Furthermore at the low pressure side the effect of resonances inthe sample is much smaller and therefore the velocity is lower. The high flowresistivity at 2000 Hz is a result of a distinct anti-resonance in the sample.

1.5

E

0.5

4X 10

(withuatx=0)(with uatx=L)

113

0 1000 2000 3000 4000Frequency [Hz]

Figure 5.10 Flow resistivity Ø as a function of the frequency. Density of the network:86%, thickness = 76 mm, Q = 0.95, Tubes: L = 2.0 mm and R = 0.1 mm (30x39 tubes).

Measured data on fibrous rockwool samples as reported by Picard (1998) show aslight increase of the flow resistivity as a function of the frequency. Obviously theacoustic behaviour of fibrous material and labyrinth-like material with tubesdiffers. Further study is needed to give more insight into the behaviour of the lattermaterials but it appears that considerable difference between the static and thedynamic flow resistivity may occur. It is noted that the network descriptionprovides a quick way to determine the dynamic flow resistivity.

In the following two sections the results for the network description are comparedwith 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 andfluid. The fibres have no stiffness, i.e. they are limp, but the mass of the fibres andthe drag force due to the presence of the fibres is accounted for. In Appendix D theLimp model is described in more detail. In this section the basic results of the Limptheory are presented.

The Limp theory has been developed by Ingard (1981) and Göransson (1993). Asa consequence of the assumption of limp fibres the application of the model is

114

restricted to materials for which the thickness is less than the shortest wavelengthin the frequency range of interest. This imposes an upper limit on the frequencyrange so that resonance of the fibres, of which the elasticity is neglected, isavoided. The main parameters of the Limp model are:

the volume porosity Q (also used as surface porosity),the static flow resistivity Ø,the structure factor K, which approaches unity for fibrous materials,the density p5 of the solid material (not the frame).

The wave equation for Limp material is given as:

V2p+kL2p 0 (5.5)

where kL is a complex wave number which can be written as:

iØQ

kL=--.JF(o) ; F(w)=K5(o/jo

(5.6)co

1

a)(1-Q)p5

In (5.6) is the isothermal speed of sound because it is assumed that for lowfrequencies the thermal fluctuations in the pores are absorbed by the surroundingsolid material.

It is noted that the 'Rigid' description (Morse & Ingard, 1968) can easily bederived from the Limp description by assuming the mass of the fibres to beinfinitely large. The fibres are unmovable (rigid). The interaction between theframe and the fluid is represented by the viscous flow across the fibres. As aconsequence the Rigid description is valid for high frequencies or for materialswith heavy fibres.

The relation between the displacement U of the fluid, with U being a vector andaveraged over a fmite elemental volume, and the displacement in the pores U' isgiven by:

U=QU' (5.7)

The displacement U is not explicitly used in the Limp description so that thenumber of degrees of freedom is decreased. As a result a single dilatational wavepropagates inside the Limp material. For the present investigation however it willbe shown that it is more convenient to use the average velocity of the fluid u(u = io U) to determine the acoustic impedance of a sample.

Air region

I x1> I Rigidback

Figure 5.11 Set-up for the calculation of the absorption coefficient ofa sample of Limp material.

At the left-hand side of the air region a harmonic pressure perturbation Po isprescribed and as a result waves are travelling back and forth. In the air region thepressure and particle velocity perturbations are assumed to be:

p(x1 , t) = frA eTkx1 + PB eX1 }e0

u(x1, t) =G

{rkx1 _Fkxi}eiøtPAe PBepoco

Pl PLimp material 2

115

To compare the acoustic behaviour of a network of tubes to a sample of Limpmaterial a one-dimensional numerical set-up as shown in Figure 5.11 is used.

(5.8)

where PB and kA 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 ofbrevity. In the Limp material applies:

p(x2 , t) = {kAL eikL X2 + PBL e kLx2

Hu(x2 , t) = {kAL ekL X2 - PBL e_i kLx2 }e»t

poco

The coefficient H is given as:

iØ/F(w) KH= K(W) SPS (z)

= Q iiØQ

p

where F(w) was already introduced in equation (5.6).

(5.9)

(5.10)

116

In this set of equations the unknowns are the four amplitudes of the waves

PA' PB PAL and bBL. The four boundary conditions to solve the unknownamplitudes of the harmonic pressure and velocity perturbations are:

The functionfta) in boundary condition 3 relates the free fluid in the air region andthe fluid in the pores (see Appendix D):

1iw(1Q)p

f(o)= (5.11)

Q zw(lQ)p50

For low frequenciesftw) approaches l/Q. This indicates that the fluid displacementin the pores U' equals the displacement of the free fluid (see equation (5.7)). Forhigh frequencies J(co) approaches 1.0. Now the averaged fluid displacement in thepores U is equal to the free fluid displacement in the air region.

By solving the system of equations and application of the boundary conditions thepressure perturbation and the velocity perturbation as a function of the position andthe frequency is known. This provides the infonnation to calculate the soundabsorption coefficient a of the sample of Limp material. Computations areperformed for the following values of the main parameters: Q = 0.95, K. = 1.0, p =2000 kgm3 These values are representative for glass wool.

1. Prescribed pressure: p(xj=O) = po,

2. Continuity of pressure: p(xj=Lj) = p(x2=O),

3. Continuity of mass flow: u(xj= L1) = u(x2= O).ftco),

4. Acoustically hard back: u(x2=L2) = O.

0 1000 2000 3000Frequency [Hz]

Figure 5.12 Sound absorption of asample of Limp material for different q)(thickness = 76 mm).

Network (22,200)Limp (22,000)

117

In Figure 5.12 the effect of the flow resistivity q) is shown. Again a high value of q)results in less sound absorption in the material (for extreme low values of q) alsoless sound absorption can be seen). The resemblance between the results with theLimp material and as predicted with a sample of a network of tubes is quite good,both in Figure 5.12 and Figure 5.13. The network description predicts a broadbandsound absorption which is comparable to Limp material although it neglects theinteraction with a frame. For the present material the energy dissipation because ofthe inertial forces acting on the fluid in the Limp material is small (and the Rigidtheory may be applied).

Although in the impedance description the effect of the thickness of the sample isimplicitly taken into account via the flow resistivity per unit length, it can be seenthat for the Limp theory the same kind of results as for the network description arepredicted if the thickness is reduced. As expected low frequency waves are lessabsorbed for thin samples.

As for the effective speed of sound for both descriptions it is noted that the Limptheory uses the isothermal speed of sound (here: 290 mIs) whereas the networkdescription uses the propagation coefficient r of the 'low reduced frequency'model. ris frequency dependent via the shear wave number s. For the pore radiusas used in Figure 5.12 and Figure 5.13 (R = 0 1 mm) the shear wave number at1000 Hz is s = 2.0 and the effective speed of sound is only Ceff = 225 mIs. A larger

pore radius (R= 0.2 mm) gives: s = 4.0 and Ceff = 280 mIs. It can be seen from

0.8

0.6

0.4

0.2

O

,1

0.8

0.6

0.4

0.2

- Network (22,200)- - Limp (10,000)

Limp (22,000)- Limp (40,000)

Limp (100,000)

4000 0 1000 2000 3000 4000Frequency [Hz]

Figure 5.13 Sound absorption of a sampleof Limp material (thickness = 38 mm).

118

Figure 5.4 that in that case the absorption coefficient more closely resembles theresults of the Limp theory, as expected.

The pressure distribution in the one-dimensional Limp material is given in Figure5.14 for a frequency of 1000 Hz. The Limp material has a thickness of 76 mm anda flow resistivity of 22,000 Nsm4.

o

-ltf2

-lt

312it

2it

Figure 5.14 Pressure perturbation (magnitude and phase) along the horizontal axis(at 1000 Hz).

For this situation 85 percent of the incident energy is absorbed inside the Limpmaterial. It can be seen that the magnitude of the pressure perturbation dropsrapidly inside the material due to the interaction of the fluid with the fibres. This isin contrast to a standing wave pattern as seen in resonators. There the pressureperturbation 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 areobtained.

5.5 Comparison to the Biot theory

Biot (1962) formulated his equations for sound absorbing material as a function ofthe displacements of the elastic frame Uframe and the displacements of the fluidinside the frame Ufld. The current section briefly introduces Biot's volume

0.250.1 0.15 " 0.2Length [m]

0 0.05

dz

zy

'X

Fluid

Elastic frame

119

description. For more detail the reader is referred to the literature or the summaryin Appendix E.

In the second part of this section the sound absorption coefficient of Biot materialis 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 willbe described. It is recalled here that the network description can potentially be usedalso in three-dimensional problems.

Introduction to the theory of Biot

Figure 5.15 presents a schematic view of a piece of Biot material with pores and anelastic frame. It suggests anisotropic behaviour. However, Biot's theory isisotropic.

Figure 5.15 Simplified representation of a piece ofporous material.

Via stress-strain relations and Lagrange' s equations of motion Biot derives thefollowing equations for the elastic frame and the fluid, respectively:

ÏVV2U frame + (A + N)V(V Ufranw ) + QS7(V Ufi) 1U frame + P12Ufluid) +b_(Uframe Uflj1j)

QV(V Ufrmee) + RV(V Ufld) = (Pi 2j frame + P22t' fluid ) - b(Uframe - Uflid)

(5.12)

In (5.12) N, A, Q and R are the elastic coefficients. They can be seen as theequivalent elastic constants when the frame consists of a homogeneous medium. N

U ame

U uid

120

is the shear modulus (one of the constants of Lamé) of the elastic frame which ismeasured in vacuum. The other constants are defined as follows:

A=Kb--N+ K2 (1 Q)23 Q

Q=(1Q)Kj (5.13)

R = QK1

where Q is the porosity, K the bulk modulus of the fluid (Kf = poco2) and Kb thebulk modulus of the elastic frame:

Kb2(l+Vb)

N3(l-2Vb)

(5.14)

Poisson' s ratio of the frame Vb 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 stresscaused 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:

PII (1Q)Ps+Pa

P12 = Pa(5.15)

P22 =Qpf+p0

where Pa is a mass coupling factor due to extra mass of the fluid and the frame thathas to be accelerated because of the so-called tortuosity of the material. It isdefmed as:

PaP0('s1) or p=p0(q2-1) (5.16)

with K the structure factor and q the tortuosity factor. These factors account fortwisted pores or random directions of fibres.

The viscous effects are accounted for in the viscous factor b:

b=ØQ2 or b=Q2F(w) (5.17)

121

It is a function of the flow resistivity Ø and the porosity Q. For higher frequenciesthe factor b is multiplied by the function F(w) which accounts for the assumptionof a Poiseuille flow in the pores breaking down for higher frequencies. In Chapter2 it was seen that the 'low reduced frequency' model, originally derived byZwikker and Kosten, predicts a Poiseuille type of flow for the velocity profile forlow shear wave numbers and a plane wave front for high shear wave numbers (seeFigure 2.2 for the amplitude of the wave front, which is complex in general). It willnot be surprising that Biot used the velocity profile according to Zwikker andKosten (Biot 1956). As for the thermal effects Biot also used the fluid density Pofor higher frequencies with the polytropic coefficient n(so) as given in Chapter 2.

For the present investigation, for a frequency range between 0 and 4000 Hz,Biot' s theory without the high frequency adjustment is used in order to have a morestraight-forward comparison.

Compared to the previous models Biot' s theory is rather complex. However, it isthe most complete one. It is capable of describing three types of waves inside thematerial: a so-called fast and slow dilatational (compression) wave and a shearwave. Bolton, Shiau and Kang (Bolton 1996) presented compact expressions forthese wave numbers.

A drawback of the theory is the use of a large number of parameters which aredifficult to measure in practice: Q, K1, N, Vb, Ø, q or K and p. Some need to bedetermined for the frame while others represent the bulk material. When oneconsiders an uncertainty range for each parameter it may be expected that thepredicted results for Biot material are rather inaccurate. A second drawback is thatthe finite element formulation of B jot'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 whenthe pressure and the fluid displacement potential are used (Göransson 1998)). Thisincreases the computational times dramatically in acoustic problems whichnormally 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 5.11 is used again, but the Limpmaterial is replaced with Biot material. For this specific case the curl-free version

122

of the equations (5.12) is used, i.e. rotations in the material are not described(VxU =0):

V2 (PU frame + QUflUjd) = 2(Pi 1U frame + p12U fluid) + b- (U frame - U fluid)

(5.18)

y2 (QU + RU fluid) = -- (p12U frame + p22U fluid) - b(Uframe - U fluid)

where P = A + 2N.

Proceeding in the same way as for the Limp material one-dimensional planewaves in the Biot material are assumed.

As indicated in the previous section there are two dilatational waves propagating inthe material. The corresponding wave numbers are k1 and k,,. The backward andforward travelling waves become:

Ufranw(X2,t){ÛAÍ e1X2 +ÛBI e tkjx2 +ÛAJJ eih1X2 +ÛBJJ e_h1X2}ek0t

U fluid (x2 , t) = {mJÛAJ e'(1 x2 + mJÛBJ e_I1 x2 + mI!ÛAIJ e k,1 x2 + mIJÛBJI e' k11 x2 } e°(5.19)

In (5.19) m, and m11 are the ratios between the amplitudes of the two dilatationalwaves of the frame and the fluid. The ratios can be calculated from the eigenvalueproblem of Biot material. When in equation (5.18) a harmonic time dependence issubstituted one obtains:

[_k2P+w2p1, ioib k2Q+a2p12 +iwbl {Ûframel 1

[_k2Q+W2Pl2+kOb _k2Ro2p22_iwb]tUfluid

A solution of (5.20) is possible if the determinant of the symmetric matrix M iszero. The condition Det[M] = O leads to a quadratic equation for k2 which provides

the wave numbers k, and k,1 for the positive and negative x-directions. Substitutingthese wave numbers in (5.20) gives the ratios m1 and m11.

(5.20)

Next the boundary conditions to solve the 6 unknown amplitudes in the air regionand the Biot material are concerned. The boundary conditions are:

Pfluid

= m2)Pfree field with: axr = .

U frame+

aUfldax ax

(5.23)

Finally the set of complex amplitudes in the air region and the Biot material can besolved as a function of the frequency. The pressure and velocity perturbations inthe air region are used to calculate the acoustic impedance and the absorptioncoefficient at the surface of the Biot material.

The parameters as used in the numerical example are listed in Table 5.3. In thetable also some characteristic values are given. Note that both Allard and Dhainautuse an extra 10 percent structural thmping for the frame.

For the air:Prescribed pressure: p(xj=C)) = Po

For the 'Free air - Biot material' interface:Continuity of mass flow,Continuity of fluid pressure,Continuity of frame stress,

For the acoustically hard wall:Suppressed frame displacement: U(x2=L2) = O,Suppressed fluid displacement: UflUjd(x2=L2) = O.

For the continuity of mass flow one has:

U free fluid = (1 Q) U frame + Q U fluid

The average pressure of the fluid in the Biot material is given as:

aUírame aUfldwith: = Q e + R , e = , e = (5.22)

ax ax

where ' is the x-component of the fluid stress tensor s. e, and e, are theaccompanying components from the solid and fluid strain tensor.

For the continuity of stress for the 'frame - free fluid' interface one has:

123

(5.21)

Table 5.3 Standard values and reference values as used for the Biot material.

In Figure 5.16 the absorption coefficients of Biot material for different values ofthe flow resistivity are depicted. In Figure 5.17 the thickness is reduced to 38 mm.

0.8

0.6

0.4

0.2

- Network (22,200)- - Biot (10,000)- Biot (22,000)- Biot (40,000)

Biot (100,000)- Network (22,200)- Biot (22,000)

0.8

0.6

0.4

0.2

4000

124

Parameter Example Allard (1993) Dhainaut (1996)

glass wool light glass wool heavy glass wool

Q E-] 0.95 0.94 0.98 0.90

0 [Nsm] 20,000 40,000 4,000 100,000

q [-1 1.0 1.06 3.2 3.2

Po [kgm3] 1.19 1.2 1.25 1.25

(1 Ps [kgm3] 100 130 15 130

N [Nm2] 2.0 iO5 2.2(1+0.li)106 1.85(1+0.li) io 1.85(1+0.li) iO5

Vb [-1 0.2 O 0.4 0.4

1000 2000 3000 4000 0 1000 2000 3000Frequency [Hz] Frequency [Hz]

Figure 5.16 Sound absorption of a sample Figure 5.17 Sound absorption of a sampleof Biot mate rial for different Ø (thickness = of Biot material (thickness = 38 mm).76 mm).

A comparison of the results of a sample of Biot material with results of the networkdescription, although both techniques are different, gives a satisfactory agreement.The Biot material behaves just like the Limp material because the same glass woolmaterial is used and only normal incident waves are applied. The main difference isthe effect of the elastic frame, via N, which is rather stiff for glass wool. In thenumerical model of the Biot material it was seen that the fluid and framedisplacements, Ufld and Uframe, have a phase difference of 90 degrees for lowfrequencies which apparently causes the low frequency sound absorption.

5.6 Conclusions and remarks

In the present chapter the coupled tubes model was used as an alternative techniqueto predict the acoustic behaviour of conventional sound absorbing material, such asglass wool or foams. This so-called network description uses a random distributionof a large number of coupled tubes (or pores) which can be seen as a representationof the interior of the material. It makes use of a small set of basic physicalparameters to characterise the material: the porosity Q, the uniform randomdistribution to approach the flow resistivity Ø, and the pore length L and pore radiusR. In general the porosity Q of a material is known or easily measured and it wasseen that the parameter L is of minor importance. Therefore only two parametersneed to be established to represent the bulk material.

A comparison of results with well-known models was used to validate theacoustic behaviour of the network description for a simple one-dimensional case. Itproved that the network description is a simple and efficient technique. Anadvantage of the network description compared to the Biot theory is that it uses arelatively small set of parameters. It is also possible to estimate a dynamic flowresistivity 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 thehelp of an impedance tube by matching the measured absorption coefficient withthe numerical frequency dependent absorption coefficient. As a next step one canuse the set of parameters obtained to select processing parameters to produce, forexample, metallic foam with better sound absorbing capabilities. Also, one canpredict the acoustic behaviour in different cases. For example the effect of obliqueor random incident waves on the sound pressure level can be simulated for aparticular design. A pressure gradient inside the material can be generated with aprescribed complex pressure across the numerical sample to simulate the angle ofincidence. The effect of random incident waves is estimated with an average ofincident waves at different angles. Obviously, this method can be carned out forvarious 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 2 the direct relation between the pressure perturbation and velocityperturbation in a tube was shown. So with the pressure known at both ends of the

125

126

tubes the velocity at only the surface of the sample, and thus the impedance, isknown.

A volume description such as the Biot theory is used in general in extensive finiteelement simulations with a complex sound field. It was seen that a more efficienttechnique is to use a surface description of sound absorbing material. Therefore itmay be useful for some applications to calculate the acoustic impedance ofmaterials as an average of incident waves at different angles. Such an average canbe calculated in an easy way for the network description by prescribing complexpressures across the numerical sample. This may be a problem for commercialfinite element codes.

A maj or difference of a volume description with a network of tubes compared tothe 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 foroblique and random incident waves.

Chapter 6

Further applications of the coupled tubes model

6.1 An inkjet array 1286.2 A test set-up for a voice producing element 1376.3 A viscothermally damped flexible plate 1486.4 Reflection of sound in ducts with side-resonators 158

In the present chapter the coupled tubes model is used in a number of differentcases. In two cases sound absorption is not the issue but merely the technique topredict the acoustic behaviour of coupled tubes. The first case is described insection 6.1 and uses coupled tubes to model the acoustic behaviour of ink in theinkjet array of a print head. The problem to be tackled is to reduce the differencesin performance of the jets from the inkjet arrays. A comparison of numerical andexperimental results proves that the coupled tubes model is well capable ofpredicting the acoustic behaviour of an inkjet array. Furthermore a new design witha broadband resonator shows that the performance of the arrays will be much lesssensitive to small deviations in, for example, the geometry.

The second case concerns the design of an experimental test set-up for measuringthe acoustic behaviour of a voice producing element. This element seeks to restorespeech in cases where the vocal folds have been removed due to cancer. Theacoustic loads on the element are represented by the vocal tract, i.e. the mouthcavities, as well as the subgiottal tract, i.e. the lungs. Both tracts are modelled withthe coupled tubes model and closely reflect the physical acoustic behaviour. Thedesigned tracts were applied in the experiments.

The third case describes the application of resonators to reduce the vibration of aflexible plate. The plate is backed by a small air cavity. It is shown that byextracting energy from the air layer via the resonators the flexible plate is dampedbecause of the acousto-elastic coupling of the plate and the air layer.

Finally, a design strategy is presented to place resonators perpendicular to theaxis of a duct. The resonators create a so-called 'acoustic mirror' for acoustic

128

waves in the duct. Waves within a small particular frequency band are reflected. Asa result the amount of noise leaving a duct can be significantly reduced.

6.1 An inkjet array

The newly developed technique to optimise sound absorption with coupled tubeswas used in a numerical study of the inkjet array in a print head (see Figure 6.1). Incooperation with Stork Digital Imaging B.V. improvements in the performance ofthe array were sought. The deviations in performance of the jets from the inkjetarrays were mainly caused by the variable effective speed of sound in ink withinthe array. The speed of sound is reduced in the array because of both viscothermaleffects and the presence of a flexible cover plate. The flexible cover plate withnozzles 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 betweenthe arrays differs too much.

This section shortly describes the basic principle of the inkjet array and itsperformance when a broadband resonator is applied. A comparison withexperimental results shows that the numerical model predicts the acousticbehaviour of the array very well. In contrast to the previous sections, where theresonance of air in tubes is treated, here the acoustic behaviour of ink in a smallchannel is described. It is shown that the low reduced frequency model forviscothermal wave propagation can also be applied for ink Furthermore, it isshown that the coupled tubes model is a suitable design tool for the presentapplication.

Working principle of the inkjet array

Stork Digital Imaging produces inkjet printers for the professional market. One oftheir products is used for printing on textiles. Textile widths of 1.5 meters can beprinted on. The print head is very small and has 16 nozzles (in Figure 6.1 the inkjetarray 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 thenozzles under high pressure. Ink is supplied via a relatively large reservoir.

Nozzles

Channel

Cover plate

Housing

Piezo

Reservoir

129

Figure 6.1 Schematic drawing of an inkjet array.

After a short length, LB, the ink jet breaks up into drops (see Figure 6.2). The dropsare selectively charged so they can be directed in a magnetic field to create therequired textile print. Non-charged drops are captured and re-used in the printingprocess.

Nozzle Inkjet Drop

Breaking length LB

Figure 6.2 Breaking length for an ink jet.

The drops are charged and directed within a limited distance. Hence the breakinglength LB needs to be within a certain range. For the present investigation it isimportant to know that the length LB is controlled with a pressure perturbationwhich is superimposed on the average pressure in the ink. By means of apiezoelectric element this perturbation is generated for a single, optimally chosenfrequency. A standing wave pattern is created in the ink channel because theopposite side of the channel is acoustically hard. At the location of each pressuremaximum a relatively small nozzle is situated so that for each nozzle

130

approximately the same breaking length is created (see Figure 6.3). The actualdimensions and frequencies are company confidential.

Piezo Reservoir Channel Nozzles

I

6

e,,,e,,.e..ses1Supply

Pressure amplitude

i 15 14 13 12 1110 9 8 7 6 5 4 3 2 1

Acousticallyhard wall

Figure 6.3 Working principle of the inkjet array.

The first arrays produced, however, showed a large variation in breaking lengthsLB. Possibly this was caused by a lower effective speed of sound in the ink in thechannel due to the glue and the flexible cover plate. Therefore an approximationfor 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

infmitesimal volume dV are (Lap 1998):

A(x, t) = A0 + A(x, t), the cross-sectional area of the channel,

(x, t) = + p(x, t), the density of the ink,(x,t) = p + p(x,t) , the pressure in the ink,

ii(x, t) = u(x, t), the velocity perturbation of the ink. The mean speed u0 is

much smaller than u.

131

The difference in the mass flow Q entering and leaving the infmitesimal volume dVequals:

dQ = (Aii)txax

and the change in the mass flow within the volume dV due to the changes indensity and the cross-sectional area amounts to:

dQ = -CÄ)dr (6.2)

So the mass flow balance for the volume dV is:

(6.1)

Neglecting the higher order terms gives:

I òA 1 ap au

A0at p0at ax

Next, with the linearised momentum equation ap/ax = Po au/at the velocityperturbation u can be eliminated:

i 1 a2p iA0 at2 po at2 - Po ax2

The pressure perturbation p is related to the adiabatic bulk modulus K0 of the inkaccording to:

(6.6)Po

where K0 = c02Po.

Furthermore it is assumed that, because of the flexible cover plate, a pressure riseleads to a larger cross-sectional area:

AP=KA-

(6.3)

(6.4)

(6.5)

(6.7)

where the modulus KA relates the pressure p to the relative change of the cross-sectional area.

132

Substituting equation (6.6) and (6.7) into equation (6.5) gives the wave equationin the channel with a flexible cover plate:

Evidently, the effective speed of sound in the channel can be written as:

2 l(Ceff

l/Ko+l/KA]po

Thus the limited stiffness of the channel causes the effective speed of sound in inkto be lower than the undisturbed speed of sound c0. As a result the actual standingwave pattern in the channel differs from the predicted pattern so that the location ofthe nozzles is not optimal. In other words: the amplitude of the pressureperturbation at the entrance of a nozzle depends on its specific location. Hence thebreaking length LB 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 pressureperturbations at the various nozzle positions, can be demonstrated by varying the

(i i i

(\E;- KAJat2 ax2

a2p 2Pat2

Ceff

' .\ e JetSo- Jet6e Jet 7o Jet8

e Jet 9-c- Jet 10e- Jet 11o Jet 12

-k- Jet 13- Jet 14

Jet 15

AJetl60.2 0.4 0.6 0.8 1

Normalised frequency [-]

Figure 6.4 Measured breaking length LB as afunction of the frequency.

-4 Jet 1O- Jet24- Jet 3

-O Jet4

(6.8)

(6.9)

/

133

excitation frequency of the piezo-electric element. For a range of frequencies thebreaking length LB for each nozzle is determined A characteristic result isillustrated 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 achievedLB, for the different nozzles. At the frequency of 0.8 this particular inkjet arrayperforms best because LB is almost equal for each nozzle. In practice the piezo-element of all inkjet arrays operates at a normalised frequency of 0.5.

The experimental results serve as a test-case for the numerical model. It will beshown 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 soughtin a channel without a standing wave pattern. By leaving the standing waveprinciple in the inkjet array the problem with different pressure perturbations at thedifferent nozzles is absent. So the design of the array should be in such a way thatat the end of the channel, opposite to the location of the piezo element, a non-reflective boundary condition is created.

'excitation 16 15 2 1 - Nozzles¿IiiPI6 Pi PI

>< >

lnkjet array ,Coupled resonator

134

With the theory of Chapter 2 a broadband resonator which consists of two coupledtubes was designed. In Figure 6.5 the numerical model for the inkjet array and theresonator is depicted.

The resonator is shown on the right-hand side and the excitation pressure Pexcitationon the left-hand side. The amplitudes of the pressure perturbation at the entrance ofthe nozzles is represented by i. The pressure perturbation at the exit of the nozzlesis set to zero. Given the dimensions of the closed end of the channel an optimalconfiguration of smaller channels can be determined so that the absorptioncoefficient a of the closed end is almost 1.0 for a wide frequency band. A singlechannel resonator could be used for a specific frequency but because of theexpected variations of Ceff a broadband resonator is preferred. For the frequencyrange of interest a resonator which consists of two coupled tubes is sufficient. Thepredicted sound absorption coefficient of the wall on the right-hand side of thearray channel is given in Figure 6.6.

i

0.8

0.6

0.4

0.2

O0

- a at end of channel

0.2 0.4 0.6 0.8Normalised 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 100 kHzwhich is quite large. Such a broadband sound absorption is possible because thenormalised frequency of 0.5 represents a rather high frequency.

The calculated amplitudes of the pressure perturbation at the entrance of thenozzles without the resonator are shown in Figure 6.7.

1

10'

Ef

loo

a

101

135

0 0.2 0.4 0.6 0.8 iNormalised frequency E-]

Figure 6.7 Dynamic pressure transfer function at the position of the nozzles without abroadband resonator.

Along the vertical axis the transfer function H11 is shown which represents the ratiobetween the pressure Pi at nozzle 'i' and the excitation pressure Pexcization.

It seems reasonable to assume a direct relation between the magnitude of thepressure perturbation at the entrance of a nozzle and the breaking length LB of thejet. This means that a high value of Jl corresponds with a short breaking length.Another observation is that at the operational normailsed excitation frequency of0.5 the breaking lengths for all nozzles is equal (see Figure 6.7). For that situationthe spacing of the nozzles and the effective speed of sound in the ink is optimal. Adifferent excitation frequency (or a different Ceff) leads to different values of LB.Furthermore there is a striking resemblance between the results in Figure 6.7 andthe measured breaking lengths of Figure 6.4. Therefore it appears justified that thenumerical model of the inkjet array can be used for the channel with a resonatorpresent at the end of the channel.

In Figure 6.8 the numerical results are given for the transfer functions at thenozzles with the resonator present.

136

10'

Ef

loo

aI-

lo-I0 0.2 0.4 0.6

Normalised frequency [-]

Figure 6.8 Dynamic pressure transfer function at the position of the nozzles with abroadband resonator.

The reduction in the variations of H,3 is about an order of magnitude compared tothe previous results and the effect of the resonator is clearly present, i.e. theamplitudes H,,, which remain at the level of the original arrays, are almostindependent of the frequency. At the boundaries of the frequency domain the effectof 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 ofmicrometers squared. The cross-sectional area of the smallest channel of theresonator is about 15 times smaller than that of the array channel. With currentmanufacturing techniques this is possible to produce.

It can also be shown that the low reduced frequency model is valid for the presentfrequency range and dimensions, i.e. k,. < 1.0 and k,. /5< 1.0.

Concluding remarks

The numerical study of the inkjet array with a broadband resonator was a newapplication for optimised sound absorption. Moreover the coupled tubes modelshowed to be an effective design tool. It was also demonstrated that instead of airany other fluid can be used in the coupled tubes model. Alternative techniques

0.8 i

137

were used by Stork Digital Imaging to improve the performance of the inkjet arrayso that the technique based on the broadband resonator was not implemented inpractice.

6.2 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 avoice producing element is described. This element restores the ability to speak forpatients who have undergone a removal of the vocal folds due to cancer. The studywas performed in cooperation with the department of BioMedical Engineering ofthe University of Gromngen, 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 subgiottal tract for the simulation of the acoustic properties of the lungairways.

In brief: with the coupled tubes model a system of tubes was designed to create thedesired acoustic impedance of the section below the voice producing element, thesubgiottal 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 treatmentpossible is the operative removal of the larynx. In medical terms this is called alaryngectomy. 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 themto vibrate. The sound generated in this way passes the vocal tract up to the lips toform the desired sound of speech. The vocal tract can be seen as an acousticconverter. It will be clear that the lack of speech is a severe handicap in the humancommunication process and therefore alternative ways of voice production aresought.

138

On the left-hand side in Figure 6.9 the situation after laiyngectomy is sketched.The neck is reconstructed after the removal of the larynx. It can be seen that thetrachea, the pipe from the lungs, is led outside (via a so-called stoma).

Nose

TongueLips

Esophagus

Pseudoglottis

Shunt valve

Trachea

Lungs

Flexible lip

} Vocal tract

Voice producing element

Subglottal tract

Figure 6.9 Schematic drawing of the vocal tract, the voice producing element, and thesubglottal tract.

The conventional solution to restore the possibility of voice production is to putsoft tissue from the esophageal entrance into vibration. This tissue is called thepseudoglottis. In earlier days the pseudoglottis was set into vibration by air injectedfrom the mouth into the esophagus. Since about 1980 a 'simple' shunt valve isplaced into the tracheo-esophageal wall as indicated in Figure 6.9. By closing thetracheostoma in the neck the air flow is directed into the esophagus. The flowcauses vibration of the pseudoglottis and the resulting vibrations of the air can bearticulated into speech. Unfortunately the speech quality is poor. The soundproduced 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 atypically monotone voice results.

To overcome these drawbacks a project supported by the Dutch TechnologyFoundation (STW) was started in 1996 (GGN.3712) to develop a voice producingelement which can be placed in a shunt valve (the Groningen button LR 11 miri by

-

Air flow

Shunt valve

Voice producing element

139

Medin Instruments, Groningen). The four participating groups are: the Departmentof BioMedical Engineering of the University of Groningen, the EarNoseThroat andHead & Neck Department of the Vrije Universiteit, Medical Centre Amsterdam,the Department of Biomedical Engineering of the University of Twente and theFluid Dynamics Laboratory of the Eindhoven University of Technology.

It was chosen to place a prosthesis, the voice producing element, directly in theshunt valve (de Vnes 2000). The prosthesis consists of a metal tube with an innerrectangular cross-section in which a flexible silicon lip is glued (see Figure 6.10).

Flexible lip

One cycle of the vibrating flexible lip

Figure 6.10 A prototype of the voice-producing element, placed inside aGronin gen button shunt valve.

The air pressure, built-up in the trachea, causes a vibration of the flexible lip via afluctuating air flow so that a tone is generated. The prosthesis is different for menand women: for men an average frequency of 110 Hz is desirable and for womenthis frequency is 210 Hz. Furthermore a higher air pressure causes a higherfrequency of the silicon lip. This corresponds to the effect of normal speech wherea 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 thetest set-up (see Figure 6.11) three sections are distinguished: the vocal tract whichrepresents the speech articulation section, the voice producing element, and thesubgiottal tract which represents the trachea and the lungs. The subglottal and the

140

vocal tract were introduced in the test set-up because it was known that the effectof the voice producing element changes in different acoustic circumstances.Therefore the physical acoustic behaviour of the vocal tract and the subgiottal tractwere simulated with a system of coupled tubes. Experimental results of theimportant harmonic frequencies of vowels (Peterson 1952) and the acousticimpedance of physical lungs (Ishizaka 1976) were used as an objective for thedesign of the tracts.

An in vitro test set-up

The acoustic behaviour of the subglottal tract and the vocal tract on the voiceproducing element is measured with the set-up as shown in Figure 6.11.

Microphone

Vocal tract model

roducing element

act modelFlow transducer

Pressure transducer

Sound absorbing material

Figure 6.11 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 aflow 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 airflow can escape the vessel via the system of tubes representing the lungs, the voiceproducing element and the model of the vocal tract. The produced sound ismeasured with a microphone, as indicated in Figure 6.11.

4000-

3000-

2O

J

1000-

500o 4Ó0 8Ó0 100

First formant frequency, F1 [Hz]

141

The design strategy to create the desired acoustic behaviour of the vocal tractmodel and the subgiottal tract model with a number of coupled tubes is describedin 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 beformed with the mouth of which the vowels can be seen as the basic sources ofsounds. Peterson and Barney (1952) investigated the acoustics of these vowels byplotting the first so-called formant frequency against the second formant frequency,see Figure 6.12. The first and second formant frequencies are the first and secondmain frequency components of the auto-spectrum of the vowel.

Figure 6.12 Vowel triangle of Peterson and Barneyrepresenting the frequency areas of the different vowels.

Three extreme vowels were selected for the simulation of the vocal tract in the testset-up: the Ial, the lii and the lui. The vocal tract can be considered as acombination of resonance tubes (Mol 1970). With the help of the theory on coupled

tubes presented in Chapter 2 the dimensions of the tubes were determined in such away that the first and second resonance frequency of the coupled tubes system fall

142

in the regions as given by Peterson and Barney. The final geometry of the coupledtubes for the three vowels is shown in Figure 6.13.

p2

Ial

L1 =80 mmR1=3Omm

L2=98mmR2 = 11 5mm

61 mmR1= 5 mm

L271 mmR2= 13.5mm

'YVoice producing element

L1=26mmR1 =5mm

L2= 60 mmR 17mm

L3=55 mmR3=75mm

L4= 64mmR4= 17 mm

Figure 6.13 Configuration of tubes for the vowels Ial, li! and 1w'.

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 pressureperturbations at both ends of the coupled tubes system, for example for the vowelIal, is derived as:

_P2 Pi_P2Iltot--.---Pi PO PO

In Figure 6.14 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 Figure6.13, the effective lengths of the tubes are used (see Chapter 2 equation (2.58)).

(6.10)

102

101

10o

101

i020

- - Vowel k/ IE

Vowelif/ r

Vowel 44/

/

500 1000 1500 2000Frequency [Hz]

2500

Figure 6.14 Transfer function of coupled tubes which represents different vowels.

The peaks at the resonance frequencies clearly show up in Figure 6.14. Theseresonance frequencies could be distinguished by the ear when the experimentalmodels for the vowels were used in the test set-up.

143

Table 6.1 Comparison of the measured and predicted resonance frequencies ofcoupled tubes which represent the vowels Ial, lii or 1w'.

The performance of these 'acoustic filters' was also measured. At the position ofthe 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 theoutlet of the vocal tract model the sound was measured with a microphone in the

Vowel Resonance

frequencyMeasured [Hz] Predicted [Hz]

Ial ist 600 700

Ii'

2

ist

1000

250

1180

280

lui

2'"

ist

2100

190

2240

210

2' 560 590

144

same way as sketched in Figure 6.11. The auto-spectrum of the latter microphonewas measured. In Table 6.1 the measured and simulated results are given for thecoupled tubes for the vowels Ial, Iii and lui.

The measured resonance frequencies show a reasonable agreement with thepredicted ones. The differences can be explained via the assumed boundaryconditions at both ends of the coupled tubes. At one end the excitation is amembrane instead of the assumed acoustically hard wall and on the other side aradiation boundary condition would be more appropriate. However, at this stagethe results of the measured resonance frequencies could be used very well andthere 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 (1970) already used a lumped element technique for the coupledtubes and volumes. The lumped elements are described with equivalent circuitryformulas, where the inductance (or mass) L = p01/A, the capacitance (or spring) C =1A/p0c02, and resistance (or energy loss) R are used. To simulate the differentvowels Fant used a double Helmholtz resonator and gives the correspondinglengths, cross-sections and volumes. The resonating mass in a tube, here the so-called neck of a Helmholtz resonator, is estimated as a lumped mass with a certainlength. The volumes of the Helmholtz resonators behave as springs. So the maindifference is that the present model is a continuous one. This offers the freedom touse also a combination of tubes because the mass and spring functionality arecontinuously distributed in the tubes. Furthermore higher order modes can becalculated as well whereas this is impossible with the lumped element approach.

The subgiottal tract for the simulation of the lungs

The acoustic load on the voice producing element on the side of the lungs is givenby the acoustic impedance of the trachea and the lungs, i.e. the subgiottal tract.Measurements of the impedance of the subgiottal tract of Japanese laryngectomeeshave been reported by Ishizaka (1976). The peaks in the impedance spectrum lie at640, 1400, 2100 and 3000 Hz. Furthermore the geometry, i.e. lengths and cross-sections, of each successive generation in the lungs is reported in the literature sothat a system of tubes can be constructed. This approach is used in the Groningenproject. Two models have been constructed, see Figure 6.15.

Generation 1

Generation 2

Generation 3

Generation 4

Air flow

L1= 125mmR1 9mm

L2= 51.6mmR2= 6.1 mm

L3 = 19 mmR3=42mm

L4= 7.6 mmR4= 2.8 mm

Closed ends Impedance Ç

Figure 6.15 Schematic representation of the subgloual tract: lung models 1 and 2.

The first model resembles the geometry of the lungs reasonably well. Forconvenience the last generation of tubes is terminated with acoustically hard walls.The air flow is provided via a capillary tube and numerical simulations showed thatthis tube has no influence on the acoustic behaviour of the model.

The second model consists only of the generations i and 2 and is terminated witha thin layer of glass wool. This second model was introduced because the resultsfor 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 inChapter 2. The transfer function for a tube with a symmetric Y-junction at one endyields:

-1pi I

= I cosh(rkL),sinhi(rkL)

'2 A+1G+1{cosh(rkL)J+i (6.11)

LAG sinh(T'kL)1

1J]

The tube is labelled J, pj is the pressure perturbation at the entrance of the tubeand p, the perturbation at the Y-junction. ris the wave propagation coefficient, k isthe wave number, L is the length of a tube, A is the cross-sectional area and G is thecorresponding coefficient for the type of the cross-section (see Appendix B).

The recursive formulation of transfer functions is used to calculate the transferfunction of the tube of generation 1. With the latter transfer function the acousticimpedance at the entrance can be calculated (see section 2.3.1).

The acoustic impedance of both models was measured in the impedance tube asdescribed in Chapter 3. The numerical and experimental results for the first model

145

Model i Model 2

Voice producing element

146

are given in Figure 6.16 for i and 2 generations and in Figure 6.17 for 3 and 4generations.

300010_2 I I I

500 1000 1500 2000 2500Frequency [Hz]

Figure 6.16 Acoustic impedance of lung model ¡ (sections 1 and 1+2).

102

IÇ I for 1 generation- - IÇ I for 2 generations

Measurements (1 gen.)- Measurements (2 gen.)

- Ç I for 3 generations- - Ç I for 4 generations

Measurements (3 gen.)

Figure 6.17 Acoustic impedance of lung model i (sections 1+2+3 and 1+2+3+4).

The correspondence of the results of the numerical and the physical models isreasonable. The reason for the noisy results for higher values of the impedance isthat the model consisted of a number of separate sections with tubes so thatpressure leakage can occur between these sections. Also, the last section consisted

300025001500 2000Frequency [Hz]

1000

147

of tubes which were closed on one end with a rather flexible aluminium tripwhereas the theory uses Ç= 1000 at these ends.

The peaks in Figure 6.17 in the case of 4 generations lie close to the frequenciesas measured by Ishizaka; however, the peaks are too high. Therefore, moredamping was needed in the subgiottal tract. This was solved by numericallyapplying an impedance boundary condition at the ends of the tubes which can beseen as the use of damping material at the tubes' end. Consequently the coupledtubes model showed that the resonance and anti-resonance frequencies change. Thefmal model, model 2 in Figure 6.15, was designed with only 2 generations and anacoustic impedance at the ends of approximately Ç = 0.2. This value for Ç wasrealised in practise with a thin layer of glass wool. The results, depicted in Figure6.18, show a reasonable agreement between the measured and simulated acousticbehaviour.

Simulation- - Measurements

500 1000 1500 2000Frequency [Hz]

Figure 6.18 Acoustic impedance of lung model 2.

The damping caused by the glass wool is less in the measurements than in thesimulations. The acoustic behaviour of the glass wool was not actually measuredbecause the measurements proved that model 2 was suited for use in the test set-up.

148

Concluding remarks

It was demonstrated that the coupled tubes model was successful in predicting theacoustic behaviour of the vocal and subgiottal tracts. For the vocal tract theacoustics of different vowels was simulated in an efficient and flexible way. Thiswas also the case for the design of the system of tubes to simulate the acousticimpedance of the trachea and the lungs.Measurements proved that the numerical predictions were sufficientiy accurate.The designs were successfully used in the in vitro test set-up for the voiceproducing element.

6.3 A viscothermally damped flexible plate

In the present section the viscothermal effects in air are used to reduce thevibrations of a flexible plate which is backed by an air filled cavity. The dampingof the plate is achieved in two ways. Firstly, the viscothermal effects in a thin airlayer trapped under the plate cause dissipation of energy via the so-called acousto-elastic coupling of the plate and the air layer. Secondly, a number of tunedresonators is used to create extra damping in the air layer. The extra damping isadded in a small frequency range where the damping of the plate by the air layer is

low.

245

49050

30 Rigid frame

Flexible

loo 4(1' Iclamped plate

2h0 Airtight volume

Movable rigidbottom

Figure 6.19 An airtight box with a flexible clamped plate and a movable bottom.Dimensions in mm.

Flexible plate

Air layer

Fixed surface

149

The vibration of plates is a major topic in noise problems and the goal in general isto reduce the radiated soundt. An example where air, or any fluid, can be used infor instance sound transmission problems are double wall panels (Beltman 1998b,Basten 1998). For demonstration and feasibility purposes the problem is restrictedto a rectangular clamped plate with a trapped air layer as shown in Figure 6.19.

The set-up was previously used by Beitman (1 998a) to validate a computationalmethod with new viscothermal acoustic finite elements in acousto-elastic problems.

In the following section the basic equations for both the analytical and finiteelement method are reviewed briefly. Next, the effects of the trapped air layer onthe dynamical behaviour of the flexible plate are shown. Finally the numerical andexperimental results are given for a viscothermally damped flexible plate whenresonators are also present.

Basic equations

Consider the squeeze film damping problem as depicted in Figure 6.20.

Figure 6.20 Squeeze film damping problem.

An air layer with a mean thickness of 2h0 is trapped between the flexible plate andthe fixed surface. The plate performs a small harmonic oscillation around theequilibrium position 2h0 t:

h(x,y)= 2h +h(x,y)eU»t (6.12)

Sound radiation is not necessarily directly coupled to the vibration level of the plate. Byusing the so-called 'radiation modes' of a plate configuration one knows the relativeimportance on the radiation. This topic is beyond the scope of the present investigation andreference is made to for instance Currey 1995, Chen 1997 and Gibbs 2000.

Note that the width of the air layer is 2h0 which results is a shear wave number ofs = h0 'J(oplp).

150

For the air layer the so-called 'low reduced frequency model' is used. It isemphasized that in this model there inherently are no pressure gradients across thelayer 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:

-2- -2- 2

_PL_k2r2p=_P0C0 'k2r2i (6.13)ax2 a2 2h0 y

where represents the total pressure (po + p e i0)t) and T is the propagationcoefficient in the air layer (see Appendix B). In the propagation coefficient theviscothermal effects are accounted for. Fis a function of the shear wave number s,which represents the ratio between the viscous and inertial effects in the fluid. Alow value of s indicates that the viscous effects are dominant.

For the deformation of the plate a standard plate equation can be used (forinstance Kirchhoff). Also here, a right-hand side term is present where the forcingterm consists of the pressure perturbation.

The acousto-elastic coupled system can be solved analytically in some cases, seeBeltman (1999b, a one-dimensional case with an infinite flexible plate) and Basten(2000, a two-dimensional case with a double wall panel). However, for most casesan analytical solution cannot be found. Therefore the finite element formulation is

used. The basic equations are reformulated and the acousto-elastic coupling isestablished by demanding continuity of the normal velocity across the interface.This leads to the following coupled system of equations, see Bellman 1998a for an

extensive derivation:

I [Msi [01 11{u}1 kKsI [Kd]1[{U}1 [Fext

L - -

w21 - - -- k k

[M C(s)] [M a (s)] L{P}J [0] [Ka] I(p}J {0}

(6.14)

The mass matrix consists of a standard structural part [M5], a frequency dependentacoustic part [Ma(s)] and a frequency dependent coupling matrix [Ma(s)]. Thestiffness matrix has a conventional structural part [K5], an acoustic part [KS] and acoupling matrix [Kc]. The system matrices in (6.14) are frequency dependent,asymmetric and complex.

The finite elements for the viscothermal air layer were implemented by Beitmanin the finite element code of B2000 (Merazzi 1994). He also implemented one-dimensional elements for tubes which include the viscothermal effects. The latter

300D Ilmodeo 21 mode

3lmode- Calculations

oloo 101 102

Gap width 2h [mm]

25

20Cap

=a)

15

a)oQ

u.E35

O

D llmodeo 21 mode

- Calculations

151

elements can be coupled to other (standard, volume) acoustic elemeñts and areused to model the resonators. The coupling elements, to accomplish the acousto-elastic interaction between acoustic elements and structural elements, wereimplemented by Grooteman (1994).

A flexible plate coupled to a viscothermal air layer

A finite element model in B2000 was used to calculate the effects of theviscothennal air layer on the flexible plate. This was done for different widths ofthe air layer. h Figure 6.21 the eigenfrequencies of only the first three structuralmodes for the box as shown in Figure 6.19 are depicted versus the gap width 2h0. Itis noted that for these acousto-elastic coupled modes the structural part dominatesthe modes.

Figure 6.21 Eigenfrequencies of the Figure 6.22 Damping coefficient of theflexible plate (Belt,nan 1998a). flexible plate (Beitman 1998a).

For large gap widths the eigenfrequencies and mode shapes approach the ones invacuum. When the gap width is reduced the influence of the air layer becomesapparent. The eigenfrequencies of the acousto-elastic coupled system differconsiderably from the eigenfrequencies in vacuum. The eigenfrequency of the firstmode increases initially but decreases for even smaller gap widths (the modeshapes for different gap widths are depicted in Figure 6.24). Similar results can be

104 6 8Gap width 2h0 [mm]

152

seen for the 31mode*. The 21-mode on the other hand shows a continuouslydecreasing 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 6.21 show that the low reduced frequency model and theacousto-elastic coupling are accurate.

The amount of damping of the plate for each mode shape i can be calculated fromthe complex eigenfrequencies according to:

Im(to)100% (6.15)

where is the viscous damping coefficient and a the angular eigenfrequency. InFigure 6.22 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 21-mode is more damped than the 11-mode. This is explained by theadded stiffness and added mass effects on the dynamical behaviour of the plate, see

Figure 6.23.

11 mode: Added stiffness effect

t tttMrt21 mode: Added mass effect

Figure 6.23 Added stiffness and mass effect due to the change in the cavity volume.

For the 11-mode the air in the cavity is compressed and decompressed. Because ofthe acousto-elastic coupling the flexible plate behaves as if it possesses a higherstiffness compared to the situation in vacuum. The 21-mode on the other handcauses 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 addedmass effect, which in turn causes a large amount of damping due to viscous effects

* The term '31-mode' indicates that there are 3 half-wavelengths in the x-direction and ihalf-wavelength in the y-direction.

11

21

31

Figure 6.24 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 thesymmetrical and the asymmetrical modes. Obviously the symmetrical modes of aplate with a small air layer are less damped by the squeeze film effect than theasymmetrical ones. Therefore extra damping is needed for the frequenciescorresponding with the symmetrical modes.

153

in the air layer. It was shown that the thermal effects are of lesser importance forthis kind of systems (Fox 1980).

The structural mode shapes in Figure 6.24 show that for a smaller gap width thesymmetric modes, i.e. the 11- and the 31-mode, try to escape the change of volumeunder the plate by changing their mode shape. By doing so the added stiffnesseffect is reduced and the added mass effect is amplified. This explains the increaseand decrease of the eigenfrequencies as shown in Figure 6.21. For the asymmetric21-mode the mass effect is the most important effect so that a continuous decreaseof the eigenfrequency can be seen for a decreasing gap width.

= 10mm 2h0=SOmm 2h0=300mmMode

154

As a first attempt quarter-wave resonators are connected to the air layer to realiseextra viscous damping for the frequency of the 11-mode. The set-up is shown inFigure 6.25. The plate is excited by a harmonic force F close to the centre of theplate. The response is measured with an accelerometer. In this way the frequencyresponse H of the plate can be determined, where X is the amplitude of thedisplacement (i.e. Ï = w2 X).

Resonators at the left side

Flexible clamped plate

Movable rigid botto

Resonators almost left

Excitation force F

490/2 Accelerometer

490/2

Resonators in the middle

Figure 6.25 Possible locations of the resonators to create extra damping in the narrowair gap.

To determine the effect of the position of the resonator three locations have beeninvestigated.

The FEM model without the resonators as well as the experimental results showedthat with a gap width of 2 mm the 21-mode of the flexible plate is sufficientlydamped. The resonance frequency of the 11-mode is approximately 133 Hz so thequarter-wave resonators were tuned to absorb maximally at 133 Hz (i.e. the length,

the radius and the surface porosity are: L=O.586m, R=1.5mm, .0=0.144). Thepredicted effect of the resonators on the frequency response as well as the effect ofthe position of the resonators is given in Figure 6.26.

-4 -4xlO xlO

- With resonators- Without resonators

Structural 11mode -

o60 80 100 120 140

o120 125 130 135 140 145

Frequency [Hz] Frequency [Hz]

Figure 6.26 Calculated frequency response (magnitude) of the flexible plate (gapwidth = 2 mm, frequency = 133 Hz).

The amplitude of the response at the resonance frequency of the 11-mode has beenreduced with more than 50 percent. Furthermore, the right-hand side shows that, asexpected, the best position of the resonators is at one side of the air layer. For the11-mode of the plate the corresponding pressure in the air layer shows a maximumat both ends of the gap, due to the closed ends, so that the effect of the resonators ismore pronounced at that location. This can also be observed in Figure 6.27. Alongthe horizontal axis the length of the plate is shown and along the vertical axis thepressure perturbation. The pressure is normalised between 1.0 and 0.0 and theexcitation force is maintained constant for both situations. The resonators arelocated 30 mm from the plate edge.

The left-hand side of Figure 6.27 shows that the amplitude of the pressureperturbation in the air layer is reduced with more than 50 percent. So bywithdrawing energy from the air layer, and thus from the complete coupled system,the plate is damped.

155

Structural 11mode..

- / -.../ 'ç.

-

no resonators- - r. inthe middle- r. almost left

r. at left side

156

T 0.6

n.- 0.4

- With resonators- - Without resonators

100 200 300 400 490Length [mm]

O

Figure 6.27 Calculated amplitude and phase of the pressure distribution in thenarrow gap (gap width = 2 mm, frequency = 133 Hz). Reference = Fexcjta,on.

The right-hand side indicates that the resonators also introduce a phase shift.Therefore the now somewhat asymmetric pressure distribution results in a morepumping-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 Figure6.28). Therefore it is concluded that the energy dissipation in the resonators mainlycontributes to the damping of the plate.

X

60 80 100 120Frequency [Hz]

Figure 6.28 Calculated frequency response (magnitude) of the flexible plate.

140

- With resonators- Without resonators

7t

O

100 200 300 400 490Length [mm]

- Resonators at left side- Resonators at both sides- - Without resonators

it 3/2

lt 5/4

lt 3/4

irJ4

2

-4X 10

21-mode

o55 lOO

/ /

41-mode

- With resonators- - Without resonators

I I

150 200 250 300Frequency [Hz]

157

For the experiments the resonators were placed at one side in the rigid bottom plate30 mm from the edge. The reason for this choice is that for a gap width of 2 mmthe resonators with a diameter of 3 mm simply cannot be placed at the far left sideas indicated in Figure 6.25. Furthermore, the experimental results withoutresonators showed that the 11-mode lies at a somewhat higher frequency of 150Hz. It was necessary to use a new flexible plate, apparently slightly thicker, whichexplains this higher frequency. The gap width is again 2 mm. As a result theparameters for the quarter-wave resonators are: L=0.526 m, R=1.5 mm, Q=0.144.

In Figure 6.29 the measured frequency response of the plate is given.

Figure 6.29 Measured frequency response (magnitude) of the flexible plate (gapwidth = 2 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 canbe concluded that the effect o the extra resonators is conectly modelled, apartfrom the use of a new flexible aluminium plate. The asymmetric 41-mode is alsoshown in Figure 6.29 since the symmetric 31-mode lies in a higher frequencyregion due to the added stiffness effects. For this particular case the resonators havea small negative effect on the frequency response of the 41-mode.

158

Concluding remarks

It was demonstrated in this section that resonators, connected to a viscothermal airlayer, reduce the vibrations of the first symmetric mode of a flexible plateconsiderably. This principle can be used in applications where air-damped platesare 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 ofdamping can be increased in particular for frequencies for which the effect of thesqueeze 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 techniquecan be applied as long as the one-dimensional wave propagation in the resonatorsis preserved. It is also interesting to investigate the effect of Hehnholtz resonators.

6.4 Reflection of sound in ducts with side-resonators

This section deals with noise reduction in duct (or tube) systems such as airventilation systems, gas transportation systems or exhaust pipes. This is achievedby placing tubes perpendicular to the direction of the sound propagation in the duct(see Figure 6.30). The tubes are referred to as quarter-wave side-resonators. A setof side-resonators can be considered as a reflection damper because noise isreflected in the duct whereas an absorption damper dissipates acoustic energy.

Closed end Transmitted sound

Duct

Source of sound Sound wave Side-resonator

Figure 6.30 Main duct with side-resonators.

159

It will be demonstrated that the coupled tubes model, including viscothermaleffects, 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, basedon the recursive transfer function formulation, are given. The equations are used todetermine, respectively, the acoustic impedance, the reflection coefficient, theinsertion and the transmission loss in a duct when a row of side-resonators isapplied. 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 theexperiments described in Chapter 3 and 4. As a result the frequency range (from1000 to 2500 Hz) is rather high for normal exhaust pipes or air-conditioning ducts.For high frequencies it is common to use absorption dampers. Furthermore it isnoted that the viscous effects become less important in wider duct systems: a 10times wider duct, bR, decreases the frequency with a factor 10, kilO, so kRremains constant whereas the resulting shear wave number s increases a factor l 0.

Performance of mufflers

The general term for a device that reduces noise in a duct and at the same timeallows the passage of a fluid is a 'muffler'. There are many publications onmuffling devices, see for example Munjal (1987) or Bies (1996). A study onmufflers is beyond the scope of the present investigation. The main goal of thisstudy 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 theTransmission Loss (li). The insertion loss is defmed as the reduction (in dB) ofsound power transmitted through a duct with and without the muffler in place. Thetransmission loss of a muffler is defined as the difference (in dB) of transmittedpower by the muffler and incident power at the entry of the muffler. Mufflingdevices which are based upon reflection are called reactive while sound absorbingmufflers are called dissipative.

The performance of reactive mufflers, such as devices with quarter-wave side-resonators, depends on the impedance of the source and the termination (outlet) ofthe duct. In the case of low impedance sources the energy reflected by the side-resonators has little effect on the generated incident energy. Typical examples of

160

low impedance sources, also called acoustic pressure sources, are for instancecentrifugal and axial fans and impeller type compressors and pumps. However,when high impedance sources are used the reflected energy is built up in the ductand even more noise can be radiated at the outlet of the duct. High impedancesources are characterised by a fixed cyclic volume displacement and can be seen asconstant 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 6.31).It is assumed that frequencies of the propagating waves in the duct are below thecut-off frequency so that the one-dimensional coupled tubes model can be applied.The side-resonator is closed at one side, at p, so that a quarter-wave resonatorresults. At p' a noise source is present and at p the duct has an open end. For anopen end the radiation impedance ad can be applied (see Section 2.4.1).

Geometric

p'

1 2

P2

Schematic

p3ad

Figure 6.3] Two coupled ducts and a side-resonator, geometric and schematic view.

Following the notation as used in Chapter 2 the transfer function for the side-resonator, with an acoustically hard wall at one side, becomes:

Ri = [cosh(FkL)3]' (6.16)P2

where Nres is the number of identical resonators connected to the duct at the sameaxial location (see Figure 6.32). From equation (6.18) can be learned that a largenumber of resonators has the same effect as one resonator with a larger cross-sectional area.

With the transfer function P2 / p the acoustic impedance and the reflectioncoefficient R at P2 can be calculated. The reflection coefficient is a measure for theamount of incident energy that is reflected. R and Ç can also be calculated at x = LR

with the two-microphone method, at p' and P2 as shown in Figure 6.32. Thedimensions of the duct and the resonators are of the same order of magnitude asused in the experimental set-up with an impedance tube.

Pi

R resonator = 5 mm

)fresonance = 1430 Hz

erad

LRLendL1

Rduct

= 500 mm= 120 mm= 50mm=25mm

Figure 6.32 Duct with a single row of side-resonators (second row = not active).

161

Tube 2 has a prescribed impedance boundary condition which gives the following

transfer function:

1f1=Ícosh(rkL)2 i (6.17)+____sinh(rkL)2]P2 L Crad

Coupling these tubes gives the recursive transfer function for tube 1:

P2cosh(rkL)1

sinh(rkL)1 AG _--l...(6.18)

{cosh(rkL)2sinh(r k L)2 P2 JA1 Gi

Pi NA3G3 {cosh(rkL)3 R.

P2 Jres . . IsmhrkL)3

1-

LR Lend

162

In the duct four plane travelling pressure waves can be distinguished withamplitudes PA' PB' PC and ßD. The so und energy of a plane wave in a duct can beestimated with:

2 2¡Prms Pp0c0 2p0c0

where p is the amplitude of the plane wave. With the use of (6.19) an energybalance can be derived for a narrow control volume located at x = LR. It is notedthat due to the viscothermal effects a small amount of the incident energy, a j í 12,

is dissipated in the resonators.

'dissipated = 'in 'out PBI2 = (IBI2 - 1PAl2)_ (IßDI2 - kC12) (6.20)

Rewriting (6.20) gives the reflection coefficient R and transmission coefficient r ina duct with side-resonators and a reflecting boundary condition (non-anechoic).

a = 1- 'PA2

PD 12 - PC' -1 1Rl2 - r'PB'2 IPBI2

So in addition to the energy balance for a sound absorbing wall, where only a andR play a role, the transmission coefficient r emerges. Obviously the transmissioncoefficient r represents the fraction of the incident power that is transmitteddownstream of the side-resonators. The coefficient r is related to the transmissionloss TL according to:

TL=101ogr [dB] (6.22)

The insertion loss IL is defmed as the reduction of transmitted power when amuffler is installed. Usually the same reduction is measured when the soundpressure level is used so that the IL can be written as:

IL = SPLthØut SPLWjth [dB] (6.23)

where SPL represents the sound pressure level downstream of the mufflerwhile the muffler is inactive and SPLth the level when the muffler is active. In thenumerical and experimental results presented in the following sections, thepressure was determined inside the duct so that the effect of the outlet on the ILwas not taken into account.

(6.19)

(6.21)

102

loi

lo

lo-I

-210

o

-. No resonators- 1 resonator- 12 resonators

500 1000 1500 2000Frequency [Hz]

Figure 6.33 Impedance at x = LR.Lreso,,ator = 60 mm, Rresonator = 5 mm.

0.8

0.6

- 0.4

0.2-. No resonators

- I resonator- 12 resonators

163

Insertion loss can be seen as the quantity directly related to noise reduction of thesource-duct-muffler system. The transmission loss is used in general to measure theeffect of a muffler with an anechoic outlet, i.e. in a laboratory. In the special caseof a pressure source and an anechoic termination of the duct (J = O) IL TL

applies.

It is noted that in the literature the performance of mufflers is often calculated withan electrical circuit to model the acoustic system. So-called lumped elements are

used and acoustic analogies for electrical inductances, capacities and resistancesare 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 6.32 is used to determine the impedanceand reflection coefficient at x = LR. To illustrate the effect of the side resonatorsfirst the viscosity and the thermal conductivity in both the duct and the side-resonators are neglected.

o2500 0 500 1000 1500 2000 2500

Frequency [Hz]

Figure 6.34 Reflection (magnitude) at x = LR.Lreso,,,,jor = 60 mm, Rresopg.,jor = 5 mm.

The impedance shows a sharp minimum at 1430 Hz, which coincides with theresonance frequency of the resonators. This drop to a low impedance causes

164

reflection of incident waves as can be seen in Figure 6.34. In this simplifiednumerical example it is apparently possible to reflect loo percent of the incidentenergy at 1430 Hz when side-resonators are used. The reference curve, i.e. whenno resonators are present, represents the reflection of waves at the outlet due to theradiation condition. At 1325 Hz the reflection coefficient with the side-resonatorsactive is lower than the reference curve. This indicates that these sound waves areamplified.

In Figure 6.35 the effect on the amplitude of the pressure in the duct is shown for1430 Hz when 12 resonators are installed. Obviously, sound of 1430 Hz cannotpass the 'acoustic mirror' because a pressure minimum is enforced. For 1325 Hzthe downstream part of the duct can be seen as a separate tube for which 1325 Hzis a resonance frequency.

- With 12 sideresonators- - Without resonators

0 0.2 0.4 0.6Length [ml

Figure 6.35 Pressure (amplitude) in the duct.

The sound pressure level at the outlet (at x = LR + Lend) with and without the side-resonators is given in Figure 6.36. 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 1430 Hz. In the calculations aprescribed pressure is used. For a comparison the SPL of a velocity source is alsogiven. As expected the resonance frequencies of the duct differ when a differentboundary condition at the entrance (x = 0) is used.

A

110

100

90

80

70

60

50

- No resonators- 12 resonators

Velocity source

500 1000 1500 2000 2500Frequency [Hz]

40

20s.'

20o

- Pressure sourceVelocity source

165

500 1000 500 2000 2500Frequency [Hz]

Figure 6.36 Sound pressure level (SPL) Figure 6.37 Insertion Loss (IL) calculatedwith and without the side-resonators with a constant pressure source and aattached. constant velocily source.

The insertion loss (IL), being the difference between the SPL with and without theside-resonators, is shown in Figure 6.37. At 1430 Hz a high IL can be seen. Themultiple maximums and minimums at lower and higher frequencies are the resultof the shifted eigenfrequencies in the duct. For a comparison also the transmissionloss (TL) is shown. A high transmission loss can be seen at 1430 Hz and a low TLat 1325 Hz. The increasing TL for low frequencies is the result of the reflectedwave frc due to the radiation condition at the outlet.

Design method for side-resonators

The negative effect of the side-resonators on the transmitted sound for frequenciesother than fresonce can be reduced when the resonators disturb the impedance of theoriginal duct as less as possible. This can be achieved by choosing freso,,nce of theside-resonators at a frequency for which the impedance in the duct already shows aminimum. The results for the impedance and the reflection coefficient areillustrated in Figure 6.38 and Figure 6.39. In this example the length of theresonators is now 69 mm, corresponding with freso,c = 1230 Hz.

166

- No resonators12 resonators

- 12 resonators (s + sc5)

500 1000 1500 2000 2500Frequency [Hz]

0.8

0.6

500 1000 1500 2000 2500Frequency [Hz]

Figure 6.38 Impedance at x = LR. Lresotor Figure 6.39 Reflection (magnitude) at x == 69 mm, Rreso,,aior = 5 mm. With and LR. L resonator = 69 mm, R resonator = 5 mm. Withwithout viscotherinal effects (s + s a). and without viscotherinal effects (s + sa).

The sharp minimum for the reflection coefficient is now absent (compare to Figure6.34). This can be advantageous for a source for which the small bandwidth noiseshifts somewhat during operation. It is also demonstrated here that the viscothermaleffects in the resonators, represented by s and scr, reduce the reflection coefficient.Due to the viscosity and the thermal conductivity the resonance is less pronouncedso that the enforced pressure minimum at x = LR is less. The resonators have aradius of 5 mm for which the shear wave number is rather high (s 100) so thereduction is small.

The accompanying IL and TL are shown in Figure 6.40 and Figure 6.41,respectively. ArOufldfresonance there is no longer a dip in the IL nor in the TL. As acomparison the TL for the 'old' configuration with a resonator length of 60 mm isalso plotted. It shows a reduction of 25 dB atfresonce but for varying frequencies ofthe noise source a more broadband reduction is seen for the new resonator lengthsof 69 mm An IL and TL of approximately 15 dB can be realised with 12 side-resonators.

- 0.4

- No resonators0.2 - 12 resonators

- 12 resonators (s + sa)

20O

- - Pressure source (no s + scT)- Pressure source

Velocity source

500 1000 1500 2000 2500

40

20

1

H

o

20O

- - No viscothermal effectsViscothermal effects'Old' resonator length (60 mm)

167

500 1000 1500 2000 2500Frequency [Hz] Frequency [Hz]

Figure 6.40 Insertion loss for a constant Figure 6.41 Transmission loss with andpressure and velocity source. without viscothermal effects (s + s0).

Instead of adapting the resonator length it is also possible to choose an axialposition in the duct in order to match freso,'nce with an impedance minimum of theduct without the side-resonators in place.

For multiple rows with side-resonators which are designed for differentfrequencies (see Figure 6.32), noise containing a wider frequency band can bereflected. It is noted that for rows with similar resonant frequencies non-planarwave interaction may occur between the rows. In that case a rule of thumbdescribes that the rows with resonators have to be placed more than a quarter-wave

length apart (Howard 2000).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 impedanceof the original duct,The impedance in the duct is calculated with a single row of resonatorsupstream of the first row,The second row of resonators is placed so that it operates at a lowimpedance,Repeat step 2 and 3 for more rows.

Comparison with experiments

To verify the predicted performance of the side-resonators the impedance tube asdescribed earlier in Chapter 3 was used, see Figure 6.42. The two-microphone

40

20

-1

168

technique was used to determine the reflection coefficient. In the first test set-upradial side-resonators were used. The length of the resonators could be varied viaan adjustable stop. For applications such as exhaust pipes a second set-up was usedwith axially oriented side-resonators to reduce the radius of the device. Theexperimental results of the latter set-up are presented in this section.

It is noted that for the one-dimensional coupled tubes model obviously theorientation of the tubes is not important.

loudspeaker

microphones 1 & 2

impedance tub

12 side-resonators

adjustable stop'I

side-resonator

baffle open end

Figure 6.42 Impedance tube with radial or axial side-resonators.

The length of the resonators as used in the numerical model is shown in Figure6.43. This length is not corrected for inlet effects. For normal incident waves theinlet effects can be accounted for by adding an incremental length with a maximumof 8R/37t (see section 2.4.1). This value could be used as an upper limit for side-resonators. However, further information or investigation is needed to determinethe correction for a number of axially and radially oriented side-resonators.

radial side-resonators axial side-resonators

Figure 6.43 Definition Of Lresonator

In Figure 6.44 and Figure 6.45 the experimental and numerical reflectioncoefficients are depicted for a single row of 12 side-resonators with a length of 40mm and an optimal length of 49 mm, respectively.

0.8

0.6

0.4

0.2 - Measurements- - Theory

Reference

Adjustable stop

o500 1000 1500 2000

Frequency [Hz]

Figure 6.44 Reflection coefficient at x =LR. L resonator = 40 mm, R resonator = 5 mm.

Lresonator

I

0.8

0.6

0.4

0.2

o2500 500 1000 1500 2000

Frequency [Hz]

Figure 6.45 Reflection coefficient aix =LR. Lresonajor = 49 mm, Rresonator = 5 mm.

169

2500

The numerical results agree quite well with the experimental ones. Clearly, for aresonator with a length of Lresonator = 49 mm, a more broadband reflection can beseen. The small difference between numerical and experimental results in thehorizontal direction in Figure 6.44 can be attributed to the estimated effectivelength of the resonators. The experimental values of the reflection coefficientsagree very well with the predicted ones, which indicates that the viscothermal wavepropagation is correctly modelled.

Another measurement set-up used 4 microphones in order to determine thetransmission loss. The set-up is shown in Figure 6.46.

170

loudspeaker

impedance tube

0.8

0.6

0.4

microphones I & 2

open end

Figure 6.46 Impedance tube with axial side-resonators and four microphones.

A comparison of the experimental and numerical results is shown in Figure 6.47for the reflection coefficient, in Figure 6.48 for the insertion loss and in Figure 6.49for the transmission loss.

baffle

172

45mm95mm

1000 1500 2000 2500Frequency [Hz]

Figure 6.47 Reflection coefficient due to 12 side-resonatorswith arbitrary chosen resonator lengths (Lresonor = 60 mm).

0.2 - Measurements- Theory

Reference

45 mm 12 side-resonators

413mm microphones 3 & 4

40- Measurements- - Theory

30- J

20 -

-o-

10-

-10

J-

Frequency [Hz] Frequency [liz]

Figure 6.48 Insertion loss in a duct due to12 side-resonators (Lresonator = 60 mm,Rreso,,jo,- = 5 mm).

40

-to

- Measurements- - Theory

30 -

fJ\ -

Figure 6.49 Transmission loss in a ductdue to 12 side-resonators (Lresonaror = 60mm, Rreso,,,,or = 5 mm).

171

hi general the experimental results correspond well with the numerical results. It isnoted that the length of the resonators was defined in the way as shown in Figure6.43.

Concluding remarks

A short study on noise reduction with quarter-wave side-resonators connected to aduct was presented. A comparison with experiments, with a source that can be seenas a constant pressure source, demonstrated that the coupled tubes modelaccurately predicts the effect of these resonators. It is noted that further study isneeded to determine the effective length of the side-resonators. It was alsodemonstrated that by using a design method a more broadband insertion loss isobtained. The design method prescribes that a row of side-resonators is placed insuch a way that they operate at a low impedance of the original duct.

Note that with the coupled tubes model as presented in Chapter 2 it is possible aswell to predict the effect of side branches which consist of Helmholtz resonators.

1000 1500 2000 2500 1000 1500 2(100 2500

Chapter 7

Conclusions

A design tool was presented to create sound absorption for a wall with adistribution of resonators for a predefined, broadband frequency range. Theresonators consist of coupled tubes. By tuning the radius and the length of thecoupled tubes and by optimising the number of tubes per unit area, the energy ofthe incident pressure waves can be almost completely absorbed for the specifiedfrequency range.

The dissipation of energy in the narrow tubes of the resonators is accounted forvery accurately in the 'low reduced frequency' model. This model describes theviscothermal wave propagation in both narrow and wide tubes.

The effectiveness of the design tool is validated for frequencies between 500 and4000 Hz in an impedance tube. The agreement between the predictions of thedesign tool and the results of experiments is very good. In an impedance tube onlynormal incident pressure waves are generated. For oblique or random incidentwaves the resonators were not tested. However, the sound absorption of resonatorsfor oblique incident waves, with a known angle of incidence, can be simplycalculated because the resonators are locally reacting. The saine broadband soundabsorption can be obtained.

The acoustic behaviour of conventional sound absorbing materials, such as glasswool or foams, can be predicted with a network of small coupled tubes. It is shownthat for a one-dimensional case the sound absorption as predicted by the network ofcoupled tubes corresponds well with empirical and theoretical models known from

174

the literature. The network description is relatively simple and computationallyefficient.

A design strategy, based on the description with coupled tubes, was presented foroptimal reflection in ducts with side-resonators. The resonators cause propagatingnoise in a duct to be reflected, not to be absorbed, so that the sound level beyondthe position of the resonators is reduced.

As a spin-off the description with coupled tubes, including the viscothermaleffects, was also successfully used for: an improvement of the performance of aninkjet array, a test set-up for a voice producing element, and a reduction of thevibrations of a viscothermally damped flexible plate.

List of Symbols

Roman

a Half the width of a rectangular cross-section [mlA Cross-sectional area [m2]

A Pressure amplitude of an incident wave [Pa]A Elastic coefficient of Biot [N/rn2]

b Half the height of a rectangular cross-section [m]c0 Speed of sound in quiescent space [mis]Ceff Effective speed of sound [mis]Ee Mean speed of energy transport [mis]C,, Specific heat at constant pressure [JIkgK]Cv Specific heat at constant volume [J/kgK]d Diameter of a tube Em]

d Length increment due to inlet effects [rn]E Instantaneous sound energy density per unit volume [Jim3]k Time averaged sound energy density per unit volume [Jim3]

f Frequency [Hz]G Parameter which depends on the cross-sectional shape [-]h0 Half the width of an air layer Em]

FI2,, Transfer function of two microphones [-]

i = IiI Imaginary number [-]¡ Instantaneous sound intensity [J/(m2s)]Ï Time averaged or mean sound intensity [Jl(m2s)]Ja(s) Bessel function of the first kind and of order n [-]k = w/co Wave number [m']kr = lw/co Reduced frequency [-]K, Structure factor for porous materials [-JKf = Poco2 Bulk modulus of fluid [Nim2]K,, Bulk modulus of elastic frame [Nlm2]l=R, l=h0 Half the characteristic length scale of the cross-section [m]L Length of a tube [mlM Component of the element matrix of a tube [msi

176

Mç Element for an impedance boundary condition [nis]

M Element for a volume [msI

M Mass variation in a volume [kg]

n Polytropic coefficient [-I

ny Polytropic coefficient for a volume [.-]

N Elastic coefficient of Biot: shear modulus of frame [N/rn2]

p Pressure perturbation [Pa]

po Mean pressure [Pa]Undisturbed pressure plus pressure perturbation [Pa]

PA Complex amplitude of plane wave [Pa]

P = A + 2N Elastic coefficient of Biot [N/rn2]

q Tortuosity of porous material [-]

Q Mass flow perturbation [kg/s]

Q Elastic coefficient of Biot [N/rn2]

R Reflection coefficient [_]

R Radius [ml

R Elastic coefficient of Biot [N/rn2]

s = i ,.jû)po Iii Shear wave number [-1

S2,j Cross-spectrum [dB]

SPIP1Auto-spectrum [dB]

t Time [s]

T Kinetic energy per unit volume [Jim3]

Mean temperature plus temperature perturbation [K]

u Particle velocity perturbation [rn/si

i Undisturbed velocity plus velocity perturbation [rn/si

U Potential energy per unit volume [J/m3]

U Displacement [m]

V Volume [m3]

x1, x2 or x3 Distance between two microphones [ml

x Cartesian co-ordinate [m]

y Cartesian co-ordinate [ml

z Cartesian co-ordinate [m]

Za Acoustic impedance [kg/(m2s)]

Greek

a Sound absorption coefficient [-I

Flow resistivity [Ns/m4]

177

y = Gp/cv Ratio of specific heats [-]

712 Coherence function [-]

r Viscotherrnal wave propagation coefficient [-1

A Thermal conductivity [J/msK]

Dynamic viscosity [Pa s]

Poisson's ratio of elastic frame [-]

Mean density plus density perturbation [kg/rn3]

Po Mean density [kg/rn3]

Ps Density of solid material [kg/rn3]

p» Dynamic coefficient of Biot [kg/rn3]

Pa Mass coupling factor of Biot [kg/rn3]

= .,JAU Gp/A Square root of the Prandtl number [-]

w Angular frequency [radis]

Q Porosity of a surface or a volume [-]

Viscous damping coefficient [-Il

Darcy' s coefficient of penneability [m2]

Ç Acoustic dimensionless impedance [-1

Miscellaneous

[M] System matrix of number of tubes and volumes

{p } Vector with nodal pressures

(Q) Vector with nodal mass flows{u) Vector with velocities for each tube's endU Vector with displacements

J Index

i Index

Reo RealpartIm{ } Imaginary part* Complex conjugate

Standard air conditions: c0 = 343.3 m/s, Po = 1.22 kg/rn3, /1 = 18.2.1OE6 Ns/m2,

7= 1.4, 0= 0.845

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Appendix A

Sound absorption mechanism of a wall with resonators

The pressure distribution, the sound energy flow and the sound energy density areconsidered in an impedance tube as well as in the resonators. The resonators areplaced in a wall at one end of the impedance tube. With the use of numericalresults the sound absorbing mechanism of the resonators is explained. A singlequarter-wave resonator and a coupled tubes resonator respectively are used.

A single tube quarter-wave resonator

In Chapter 2 the theoretical model was presented for a wall with quarter-waveresonators. The wall is capable of absorbing 100 percent of the incident energy atthe resonance frequency of the resonator. In that case the porosity of the wallequals the impedance at the entrance of the resonator:

Cresonator d Qwall - an CresOnatOr - wall' wall

An impedance of = 1.0 indicates that acoustic waves in the medium propagatewithout a disturbance. In other words: the wall seems to be located at infinity sothat no waves are reflected.

To demonstrate the acoustic behaviour of the resonator an impedance tube and aresonator are used as indicated in Figure A. 1. At one end of the impedance tube, atx1 = L1, the wall with the resonator is present. It is noted that the geometricallengths are given whereas the effective lengths are used in the calculations, see alsosection 2.4.1.

Cwaiil.0 (A.1)

188

Tube 1: Impedance tube

Tube 2: Quarter-wave resonator

Wall

2O.136L2=O.06mR2=O.005m

L1=O.18mR1=O.043m

Figure A.1 A single quarter-wave resonator connected to an impedance tube.Standard air conditions are used.

The distribution of the pressure perturbation in each tube (i = 1,2) is given by:

(rkx). -(rkx).p(x)=pAe '+pBe

G ( (Fkx). -(Tkx)u(x) = 'FAi e - PB e

poco

where the pressure amplitudes of the forward and backward propagating waves,indicated with an 'A', are unknown. These are derived by imposing boundaryconditions at both ends of the tubes. Here the pressure perturbations are used asboundary conditions.

The tubes are coupled as described in Chapter 2. For the resonator the transferfunction is the following:

P2_ i

Pi - cosh(rkL)2

and for the impedance tube, which is coupled to the resonator, applies:

_1

_=Icosh(rkL)i+5h1(Tk1 ( A2G2

IA1G1 sinh(rkL)2

{cosh(rkL)2 _RiflI (A.4)pli JI

Finally the pressure amplitudes can be derived by use of the transfer functions(A.3) and (A.4):

PO

PiPi--PoPO

P2P2P1Pi

189

PAl' PBI=1' (A.5)

PA2' PB2

The complete pressure distribution and the velocity distribution are known as afunction of position and frequency. Therefore the impedance, and also the soundabsorption coefficient, at the end of the impedance tube can be calculated (seesection 2.2). For the arbitrarily chosen, yet optimised combination of theparameters Q2, L2 and R2 in Figure A. 1, the sound absorption is given in FigureA.2.

0.8

0.6

0.4

1600o1000 1200 1400

Frequency [Hz]

Figure A.2 Sound absorption coefficient of a wall with quarter-wave resonators.

In the following sections the acoustic variables in the impedance tube and in theresonator are considered when a= 1.0 and when a= 0.03.

Complete sound absorption (a = 1.0)

The pressure along the impedance tube and the resonator for a frequency of 1337Hz is depicted in Figure A.3. At x1 = O a hannonic pressure PO with amplitude 1.0

Pa is generated.

- 1337Hz.0.2

190

0.05 0.1 0.15 A 0.2 0.244 0Length [ml

NN

NN

NN

NN

N

2lt0.05 0.1 0.15 A 0.2 0.244

Length [ml

Figure A.3 Pressure perturbation (magnitude and phase) in the impedance tube andthe quarter-wave resonator (at 1337 Hz).

On one hand, the constant amplitude and the constant slope of the phase indicatethat there is a forward propagating wave in the impedance tube. So no reflectionsare present. On the other hand the pressure amplitude and the constant phase in theresonator demonstrate that there is a standing wave pattern in the resonator. Theamplitude at the end of the resonator is rather high: 75 Pa = 53 Pa (rms) = 128 dBSPL, whereas the incident sound power level is: 1.0 Pa = 0.71 Pa (rms) = 91 dBSPL.

The sound power P in the tubes, with the flow of sound energy in the positive x-direction, is given in Figure A.4 and is calculated with the use of the soundintensity J and the cross-sectional area A:

P(x) = 1(x). A = .Re{p(x) u(x)*} (A.6)

The energy density E in each tube is shown in Figure A.5, where E is given by:

E(x) !pOu(x).u(x)*+ 4p0c02

p(x) p(x)* (A.7)

NN

N

- Impedance tubeResonator

x 10_6

0.05 0.1 0.15 " 0.2 0.244Length Em]

- 10 r60

10 r

10_o

o

191

0.05 0.1 0.15 A 0.2 0.244Length [ml

Figure A.4 Sound power in the tubes Figure A.5 Sound energy density in the(at 1337 Hz). tubes (at 1337 Hz).

Figure A.4 shows that there is a constant flow of energy in the impedance tube as aresult of a forward propagating wave. The energy flow at the acoustically hard endof the resonator is zero since the velocity equals zero at this position. The energydensity however in the resonator is much higher than in the impedance tube. So alarge amount of potential and kinetic energy is stored in the resonator. The energyratio between the two tubes is 2600 which is an indication of the strength of theresonance. It is important to note that the strength of the resonance is controlled bythe amount of viscothermal losses in the resonator. When there is more damping,i.e. the shear wave number is lower, then the pressure amplitude as well as theenergy density in the resonator are lower.

In the case of 100 percent sound absorption the energy of the incident wave iscompletely dissipated in the resonator. In the resonator itself both the forward andthe backward propagating waves are attenuated due to viscothermal effects. InFigure A.7 the amplitudes of both waves are shown as a function of the position inthe resonator. A summation of both complex valued waves leads to the pressureperturbation as given in Figure A.3.

10 - - Impedance tubeResonator

192

36

- Forward travelling wave B- Backward travelling waveA

0.18 0.2 0.22 0.24Length [ml

Figure A.6 Amplitude of the forward and backward propagatingwave in the resonator (at 1337 Hz).

The ratio of wave A and wave B at the entrance of the resonator is given by:

-_=eRe{1 2L2(A.8)

PB2

where the real part of the propagation coefficient Frepresents the attenuation and2L2 is the total length which an incident wave travels in the quarter-wave resonator.For the present situation the ratio is 0.973 so that only a small fraction of theenergy is dissipated per unit length. But because of the large amplitudes in theresonator the dissipated energy equals the incident energy.

It is noted that the waves A and B are in anti-phase so that the incident wave Bcancels out the backward wave A.

The reason for 100 percent absorption is that the number of resonators (or theporosity) is chosen in such a way that the incident energy is exactly absorbed in theresonator. The amount of dissipated energy per resonator is given by the strengthof the resonance in the resonator and the viscothermal effects. Furthermore,because the system is linear an increase of the incident energy results in the sameincrease of dissipated energy in the resonators.

Little sound absorption (a = 0.03)

As a comparison the results for a frequency for which the sound absorption is loware also presented. At 1202 Hz the sound absorption coefficient is only 0.03.

Figure A.7 shows the amplitude and the phase of the pressure in the twoconnected tubes.

- - Impedance tube- Resonator

193

- itI2

- lt0.05 0.1 0.15 ' 0.2 0.244 0 0.05 0.1 0.15 " 0.2 0.244

Length [ml Length [ml

Figure A. 7 Pressure perturbation (magnitude and phase) in the impedance tube andquarter-wave resonator (at 1202 Hz).

In the impedance tube a standing wave pattern can be seen which indicates thatreflection occurs at the wall with the quarter-wave resonator. The resonance in theresonator is much smaller than at the frequency where a = 1.0, see also Figure A.3.As a result less energy is dissipated.

The sound power and the energy density are shown in Figure A.8 and Figure A.9.The energy flow is also much lower in the impedance tube. However due to theviscothermal wave propagation and the presence of the resonator there is a smallflow of energy. When the viscothermal effects are neglected no energy flow ispresent for a standing wave pattern.

A comparison of the energy densities in the resonator and the impedance tubesgives a ratio of 40 so that indeed the resonance is much less pronounced comparedto the situation where a= 1.0.

194

Tube 1: Impedance tube

Wall

- - Impedance tube- Resonator

0.14L2=0.06mR2=0.005m

L3=0.06mR3=0.0007m

L1 = 0.18 mR1=0.0134m

xl

Figure A.10 A 'two coupled tubes' resonator connected to an impedance tube.

For a specific combination of the parameters the resonator shows a broadbandsound absorption as shown in Figure A. 11, see also section 2.4.2.

X lo10-2

' 1.5

n, l0q) =

ano

q)0io-5

- - Impedance tube- Resonator

o6lo

o 0.05 0.1 0.15 " 0.2 0.244 o 0.05 0.1 0.15 A 0.2 0.244Length [ml Length Em]

Figure A.8 Sound power in the tubes (at Figure A.9 Sound energy density in the1202 Hz). tubes (at 1202 Hz).

A 'two coupled tubes' resonator

A similar explanation as for the quarter-wave resonator can be given for a coupledtubes resonator. Here two coupled tubes form the resonator which is connected toan impedance tube as depicted in Figure A.1O.

Tube 2 + 3: Coupled tubes resonatorp3

0.8

0.6

0.4

0.2

awall

1277Hz» 1418Hzo H

1000 1200 1400 1600Frequency [Hz]

Figure A. 11 Sound absorption coefficient of a wall with coupled tubes resonators.

The pressure distribution in the tubes is calculated at two frequencies: at 1277 Hza= 1.0 and at 1418 Hz a= 0.96, see Figure A.12.

Impedance tube- Resonator (First tube). Resonator (Second tube)

1277 Hz.

1418 Hz.-

O

it/2

7t

3/2,t

2z

5/21t

195

60 L- Impedance tube

- Resonator (First tube). Resonator (Second tube) / ./

40 Ica

1277 Hz._3/.

n. be

20 1418 Hz. I

0o 0.1 A02 A 0.3 0 0.1 A02 A 0.3

Length [m] Length Em]

Figure A. 12 Pressure perturbation (magnitude and phase) in the impedance tubeand quarter-wave resonator.

The pressure distribution in the impedance tube and the coupled tubes resonator issimilar to the situation as shown earlier for the quarter-wave resonator. So thereader is referred to the interpretation given there. However, there are now tworesonance frequencies present. At 1277 Hz the pressure in the first and second tubeof the resonator can be considered as in phase, whereas at 1418 Hz the pressure inthe second tube can be considered as out-of-phase. Also the frequency region

196

between these frequencies shows a highly resonant behaviour which explains thebroadband sound absorption.

The results for the sound power and the sound energy density are given in FigureA.13 and Figure A.14.

6C,,

t:o

o

X 10

At 1277 Hz

At 1418 Hz

- - Impedance tube- Resonator (First tube)-. Resonator (Second tube)

0.1 A02 A

Length [ml0.3

I 0

E 10

E -510

10_6

o

- - Impedance tube- Resonator (First tube)-. Resonator (Second tube)

At 1277 Hz

At 1418 Hz

0.1 A02 A

Length [mJ0.3

Figure A. 13 Sound power in the tubes. Figure A. 14 Sound energy densizy in thetubes (upper lines at 1277 Hz).

It can be seen that the major resonance takes place in the second tube of theresonator. However, the first tube is essential for the 'two degree of freedom'system to produce resonance at two frequencies in the frequency range of interest.The energy ratios at 1277 Hz are: E2 / E1 = 26 and E3 / E1 = 1700. At 1418 Hz theyare 30 and 1500, respectively.

Appendix B

Propagation coefficient for different cross-sections

In the case of one-dimensional viscothermal wave propagation in prismatic tubesthe pressure and velocity perturbations are given by:

p(x)= frA +PB e_Tu (B.1)

G ( Fkx rkxu(x)=\pAe pBePoco

The propagation coefficient Fand the coefficient G were solved for circular cross-sections by Zwikker and Kosten. Based on this solution the formulation for ageneral cross-section was derived by Stinson (1991, 1992) (see also Roh 1991,Attenborough 1983):

It can be seen that in the function F successively the thermal conductivity % and theviscosity dU are used. Note that if the formulation is in a non-dimensional form, seeTijdeman (1975) and Beitman (1999a), then G is divided by The corresponding

shear wave number is defined as:

s = R, (B.5)

where Rh is the so-called hydraulic radius which equals Rh = 2A/D.

197

(B.2)

F=iy(yl)F1

(B.3)pocp ]

FIJIPo

G=irF1--- (B.4)Po

198

Circular cross-section with radius R

The function Fin (B.3) is a ratio of Bessel functions:

F J2(i..fiR..fTh)(77) -

J0 (i RJJ/)

Writing Fand G as a function of s and sogives:

r_0T.

- J2(i-.Jis) n

G=_2'5)= i T2' Jo(i'.Jis) F n

where

s=Rfwpo

v/iThe velocity profile for a cross-section (as shown in Figure 2.2) is:

U(X,Y)=0(Y) i) u(x)J0(s)

n =+ y-1 J2(isa)]'

T J0(i-/iso)

where y is the radial direction in the cross-section of the tube and u(x) is theaveraged velocity perturbation in the cross-section. For low and high values of s(s « 1, s » 1) the velocity profile can be estimated with:

- 'z

U(X,y) ow =_52O Y ).u(x) U(Y,X)ghs =l.u(x) (B. 11)

Rectangular cross-section with sides 2a2b

The function F for a rectangular cross-section with the sides 2a and 2h is:

.4w 222 2ttI1

F(i) = ¡2 2 an ßm (an + ßm + - I

¡7a b n=Om=O77)

lita =(n+)-2alizO

ßm =(m+)

(B.6)

(B.7)

(B. 10)

n,m = 0,1,2,...

(B. 12)

s---a+blJ p

Equilateral triangle with side d

The function F in this case is as follows:

e2(i7)e(17)

F(i7)tanh(e(17)) d 13io

-And the shear wave number is given by:

d /wp0

p

An infinitely wide layer with spacing distance 2h

The function F is as follows:

tanh(Jiwh2/77)F(17)=l

Jioh2 /,The accompanying shear wave number is given by:

s = hv/I

if the parameters s and soare used then r becomes:

i =11+i±1t1*15) 1TÌ1tanh(/s) 'z 2' ( -,/isc

I's

199

A large number of terms are needed for the function F(7). Up to 200 terms wereused in the present investigation. The shear wave number is given by:

(B.l3)

(B. 16)

(B.17)

(B. 18)

200

Appendix C

Calibration of the microflown

Before the sound absorption coefficient a of a sample is measured in an impedancetube the two sensors must be calibrated. The procedure and the calibration resullsfor the combination of a microphone and a microflown in the impedance tube aredescribed in this appendix The combination of the two sensors is used for the p/umethod and the pu method, see Chapter 3.

For the pu method both the auto-spectra and the cross-spectrum of themicrophone and the microflown are also required. It is shown that these spectra arecalculated with the use of the calibrated transfer function and the calibrated auto-spectrum of the microphone.

Combination of a microphone and a microflown

For the calibration of the combination of a microphone and a microflown theposition of both sensors is as indicated in Figure C. 1.

KL>1

microphonemicroflown

Figure C.1 Set-up for the calibration measurements.

The theoretical transfer function between the microflown and the microphone

can be shown to be:

Hu(x=O) G 1' cosh(FkL)

p(x=L) poco Ç(x=L) j__Is11(TkI)+

where the right-hand side of the impedance tube is approximately acoustically hard(Ç(L) = 1000) so that there is always a pressure maximum The amplitude and thephase of the measured and theoretical transfer function H,11 are given in Figure C.2

(C.!)

40

50

60

70- Measured

- Interpolated

1000 2000 3000Frequency [Hz]

4000

300

Ï 250

t: 150

100o

200

..-' '1000 2000 3000 4000

Frequency [Hz]

Transfer function (phase).

From a comparison of the measured and theoretical values a correction function isobtained as a function of the frequency, i.e. the gain and the phase lag aredetermined. The djfference for the amplitude and the phase is given in Figure C.4

and Figure C.5.

1000 2000 3000Frequency [Hz]

201

and Figure C.3. A minimum occurs when the velocity has a minimum value at the

location of the microflown.

4000

Figure C.4 Amplitude difference in dB. Figure C.5 Phase difference in degrees.

In the figures a simple third order polynomial is used for the interpolation to obtainthe correction function. Here: from 250 to 4000 Hz. A better interpolation shouldbe possible when the measured data is weighted with for instance the transferfunction. Also, because the correction function is smooth, it is possible to neglect

i- I ii'iHi'i \\f/180

lO0 1000 2000 3000 4000 0

Frequency [Hz]

Figure C.2 Transfer function Hí (magn.). Figure C.3

-I- - Measured- Interpolated

lo'- Measurements

lo- - Theory

T l0'

10_2

180 - Measurements- Theor

¡90-

-90-

202

the measured data at the frequencies where the transfer function shows a minimum(and where the coherence is low).

The correction function is used in the impedance tube measurements for the p/umethod and the p.0 method. The measured transfer function H is corrected in thefollowing way:

201ogfl, - 201ogH,+Correction [dB] (C.2)

arg(fl,) - arg(H,)+ Correction [deg] (C.3)

where H,

is the corrected transfer function.

The correction procedure as described above was applied in section 3.3.3.

Auto-spectra and cross-spectrum

The cross-spectrum between the signals of the microphone and the microflown is:

Spu = "pIu 5pp = "u/p Spp (C.4)

The auto-spectrum of the signal of the microflown is simply:

S, (C.5)

Appendix D

The Limp theory for fibrous sound absorbingmaterials

In addition to the information on the Limp theory as presented in section 5.4 thisappendix presents the basic equations for the forces inside fibrous porous material.The interaction between the fibres and the fluid in the so-called pores isrepresented by an inertial force which acts on the fluid and by a viscous drag whichis the basic mechanism for the absorption of sound energy. In Figure D. i anelementary volume with a cross-section of a fibre and the fluid in the pores isdepicted.

Cross-section of a fibre

Fy''-'fibre

U,fluid

203

Elementary volume dV of fluid

Figure D. i Fluid and fibre displacements.

The porous material is regarded as homogeneous. The pressure in the fluid and thedisplacements of the fluid and the fibres perform small harmonic fluctuationsaround an average value. As a consequence the theory is linear.

The porosity Q of the porous material is defined as the ratio of the volume of thefluid inside the pores and the total volume. Due to the porosity the average fluiddisplacement Uflja, with Ufld being a vector, is less than the fluid displacement inthe pores U'fld:

Ufld = Q U'U (D.l)

204

Another important parameter in the Limp theory is the mass M of the frame Theframe is the set of fibres in vacuum. M is defined via the bulk density of the porousmaterial ph:

Pb=2PO+M ; M=(lQ).p3

where p is the density of the fibres itself.

Inertial force acting on the fluid

The inertial force which acts on the fluid of a porous material is:

i'1inert - K5p0

2dV

where K5 is a structure factor which accounts for inaccessible pores or pores thatare difficult to access. The factor Q is the result of the use of the averagedisplacement of the fibres Ufibres (see equation (D. 1)).

Viscous drag force acting on the fluid

The viscous drag force which acts on the fluid is assumed to be proportional to theflow resistivity Ø. For a rigid porous material this viscous force is:

dFdrag =ØdV (D.4)

When the fibres are free to move harmonically, so that the time dependence isei)t, the force is proportional to the difference between the displacement of thefibres Ufibre5 and the displacement of the fluid in the pores U'Jjd:

ø'O)(U'fluidUfibres) = 0)2MUfibres (D.5)

where Ø' is the resistivity in a pore (Ø' = Q q)). Equation (D.5) provides the relation

between the displacements of the fibres and the fluid in the pores:

Ufibres

U'fijd o)M

So the viscous drag force for the Limp material becomes:

(D.6)

cJFdrag = io)ø'(U'flUjd Ufibres) = ' 1UflUddV

Wave equation for Limp material

With the use of the equation of continuity and the momentum equation the waveequation for Limp material with a complex wave number kL can be written as:

V2p+kL2p 0 (D.8)

The wave number is given in section 5.3.

Boundary conditions

The accompanying boundary conditions for the wave equation can be one of the

following ones:A rigid wall: p/i O , where is the normal direction,A flexible wall: U,aii = Ubres. In this case the degrees of freedom for thefibre displacements are required.A free fluid: continuity of the mass flow and continuity of the pressure.

For the latter boundary condition the continuity of mass flow, divided by iW, iswritten as:

IITfl TT - flPo 'j-' free fluid - fibres) - ' Po (U fluid -1flbres)

where the interface is moving with the displacements of the fibres. With the help ofequatìon (D.6) one can derive the ratio between the displacement of the free fluidand the average fluid displacement:

iwM1+

YT,i 0Ufree fluid ufluid iwM

0

(D.9)

(D.10)

It is noted that the Limp model (as described by Göransson, 1993) has beenimplemented in the finite element program B 2000 by Blom (1995). The boundaryconditions for a rigid and flexible wall have been taken into account via interface

205

(D.7)

206

elements. The irterface elements between Limp materïal and a free fluid have notyet bleen imp1thented.

Appendix E

Derivation of Biot's equations of motion

In section 5.4 Biot' s equations of motions were presented. This appendix brieflydescribes the stress-strain relations and Lagrange's equations of motion for aelemental volume of porous material. Together these equations form Biot'sequations of motion. The elastic and dynamic coefficients used in these equationswere introduced in section 5.4.

Biot considers an elemental volume of porous material (recall Figure 5.13) whichconsists of an elastic frame and cylindrical pores with a fluid. The volume isinitially assumed to be isotropic for simplicity. For anisotropic material, such astransversely isotropic glass wool, reference is made to Biot (1955). Furtherassumptions are:

Linear stress-strain relationsSmall strainsNo gravitational forcesA Poiseuille type of flow in the pores for low shear wave numbers

The averaged displacements of the elastic frame Uframe and the fluid in the poresUfld are the degrees of freedom for the elemental volume, where U is a vector:

207

The average displacement of the fluid is defmed in such a way that the product of

Ú fluid and the fluid part of the cross-sectional area ( = Q dydz ) represents thevolume flow. For the frame displacements the solid part of the cross-sectional area

is used ( = (l-a dydz ).

Ux

Uy (E.l)

Uz

208

Stress - strain relations

For the elemental volume the strain tensor is separated into a frame strain tensorand a fluid strain tensor e

e

e

The components of these tensors are given by:

eaUx frame

aU

aUYframeeay

aUYflUdE =) ay

eEc+e+e2 =V.Uflujd

where V is the nabla operator (a/ax, a/ay, a/az).

The stresses acting on the porous material are also averaged over the cross-sectional area of an elemental volume. The total stress tensor is divided into thestress tensor of the frame oand the stress tensor of the fluid s:

The dilatation of the frame and the fluid are defined as:

ee+e+e =VUframe

x Exy E

x yyyz

X Zy ZZ

(E.2)

[auam au frameetc.etc. eXY=2

ay+

ax j[auXfluid auY fluid

etc.etc.= 2 ay + ax j

(E.3)

(E.4)

with Q being the porosity and p the pressure perturbation in the pores.

a O.Xz 00totaI

0= oO.yz

s= O O (E.5)

O0

where the scalar is defined as:

= Qp (E.6)

[auxframe auflUd auYflUd aUZflUdP12

+auyframe a11zframe

/1 auX fluid i f

auYflUd f auflUdP22 at

Jat

Jat

In (E.l0) Pii, P12 and P22 are Biot's dynamic coefficients.

209

Biot formulated the stress- strain relations in the same way as Hooke' s law in terms

of the so-called Lamé moduli:

a=2Ne+Ae+Qe =2Ne

a, =2Ne+Ae+Qe a =2Ne ¡=Qe+Re (E.7)

a =2Ne+Ae+Qe o =2Ne3,

or

= N(VUframe + UfraV) SAV U frame + ÖQV Ufluid(E.8)

QVUfra +RV-Ufl

where Sis the Kronecker delta. N, A, Q and R are Biot' s elastic coefficients.

Equations of motion

Lagrange' s equations of motion in the x-direction, including a dissipation functionD, can be written as:

a àT aD a- a7at frame aUx frame

= + a ax(E.9)

aT a a

at ÒÚXfluid J aufld axz

The kinetic energy T of an elemental volume of isotropic material is derived as:

21I auxframe (auyframe(auzframeIT=

atJ

atJ

atJ

(E.l0)

atL

at at at at at

210

The dissipation function D, also known as Rayleigh' s dissipation function,depends on the relative velocity between frame and fluid:

D bkUx frame Úfld)2 +(Ú frame -y fluid)2 +(UzfroJ,w _Uzfiujd)21(E.11)

where the coefficient b represents the viscous losses in the pores.For the y- and z-directions similar equations can be used.

Finally, with the use of the equations above, the equations of motion as derived byBiot can be written as:

NV2Ufr + (A+ N)V(V .Ufr0,) + QV(V Ufl) = l[Tframe + P12Ufld )+b(U frame -flu1d)

QV(V U frame) + RV(V U fluid) j(P12Uframe + P2211 fluid )b(Ufr Uflu)

(E.l2)

Samenvatting

De korte tijd die vandaag de dag beschikbaar is voor onderzoek en ontwikkelingleidt tot een grotere behoefte orn het gedrag van ontwerpen te simuleren, zoals ineen 'virtual reality'. Een simulatie is goedkoper dan het uitvoeren van dureexperimenten met kostbare prototypes. Deze ontwikkeling is ook zichtbaar in deakoestiek waar geluidsreductie meer en meer in de belangstelling komt te staan.0m het geluidsniveau in een open of gesloten ruimte te kunnen voorspellen moetende belangrijke invloeden van flexibele panelen en geluid absorberende materialenbekend zijn. Het huidige onderzoek is daarom gericht op een nauwkeurigebeschrijving van geluidabsorptie. Het onderzoek heeft geleid tot een nieuwetechniek orn voor een brede frequentieband geluid te absorberen. Daarbij is eeneenvoudig en efficient numeriek model verkregen orn conventionele geluidabsorberende materialen, zoals glaswol of schuim, te beschrij ven. De nieuwemodellen en inzichten zijn ook gebruikt voor toepassingen die met directgerelateerd zijn aan geluidabsorptie.

De basis voor bet onderzoek is de beschrij ving van geluidsgolven in een enkel smalbuisje, ook wel pone genoemd. In zo'n buisje kunnen de viscositeit en dewarmtegeleiding van de lucht, of in het algemeen, van een gas of vloeistof, een niette verwaarlozen effect hebben op de geluidsvoortplanting. Het gevoig van dezogenoemde visco-thermische geluidsvoortplanting is dat er geluidsenergieverloren gant en dat de effectieve geluidssnelheid in het buisje lager is. Dit verliesaan energie is gebruikt voor configuraties die bestaan uit gekoppelde buisjes. Eenontwerpstrategie is ontwikkeld orn voor een brede frequentieband een hogegeluidabsorptie te creëren met deze gekoppelde buisjes. Deze zogenoemdebreedbandige resonatoren kunnen zodanig worden ontworpen dat ze invallendegeluidsgolven absorberen voor een van te voren bepaald frequentiespectrum.Experimenten in een impedantiebuis (Kundt's tube) bewijzen dat het model voorvisco-thermische geluidsvoortplanting en het numerieke ontwerpgereedschapnauwkeurig zijn.

211

212

Verder is aangetoond dat op microschaal een netwerk van gekoppelde buisjesgebruikt kan worden orn conventionele geluid absorberende materialen te

beschrij ven. Deze beschrijving is eenvoudig en efficient vergeleken met bestaandemodellen voor geluid absorberende materialen, zoals een empirische beschrijvingen de zogeheten Limp en Biot theorie. Daarnaast is het model van gekoppeldebuisjes succesvol toegepast bij: de verbetering van een inktjet array uit eenprintkop, een nieuw ontworpen experirnentele opstelling orn een spraakvormende

prothese te testen, het reduceren van het trillingsniveau van een flexibeleluchtgedempte plaat, en een ontwerpstrategie orn optimale geluidreflectie tecreëren in een buis met behuip van zogenoemde zij -resonatoren. Bij deze laatstetoepassing wordt geluid gereflecteerd , en met geabsorbeerd, orn het geluidsniveau

dat de buis verlaat te reduceren.

Voor de bekende meetmethode in de impedantiebuis met twee microfoons is in deloop van het onderzoek een alternatieve meetmethode, met een rnicrofoon en eendeeltjes snelheidsmicrofoon, de zogeheten microflown, experimenteel onderzocht.De combinatie van een microfoon en een microflown geeft direct inforrnatie overgrootheden als: de akoestische impedantie, de geluidsintensiteit en de geluidenergiedichtheid.

Nawoord

Met veel plezier heb ik ruim vier jaar binnen de vakgroep Technische Mechanicaen Kuststoffen gewerkt ann het onderzoek waarover je je binnen twee minuten eenbeeld kan vormen als je twee pagina' s terugbiadert. Deze vakgroep, en in hetbijzonder natuurlijk de Dynamica groep, bestaat uit een levendige en gezelligeverzameling mensen. Graag wil ik als eerste mijn promotor Henk Tijdemanbedanken voor het creëren van die open en gezellige sfeer. Als voormaligebuurman op de gang hoorde ik menigmaal het lachen van Henk (ook van dieandere Henk) door de muur heen. Verder is de Dynamica groep haast ondenkbaarzonder het 'duo' Ruud Spiering & Peter van der Hoogt die elkair perfect aanvullenen uitspelen tij dens onze vaste vrijdagmiddag vergaderingen. Gelukkig mocht ikdie laatste vergadenngen ook nog meemaken met Andre de Boer erbij. Verderwordt hèt gezicht van de vakgroep natuurlijk bepaald door Debbie Zimmerman &Annemarie Teunissen. Ook een 'smart' duo waar ik veel mee gelachen heb.

Tegenover mijn kamer, op vijf passen afstand, ligt het befaamde dynamicalab.Veel lichaamsbeweging heb ik dus niet gehad tijdens mijn onderzoek. Naast mijnonderzoek gelukkig we!, maar daar korn ik nog op terug. Dankzij Bert Wolbertkunnen de promovendi en de studenten zich in jeder geval uit!even bij deexperimenten. En dat deden (en doen) we: Marco Beitman, Tom Basten, MarcoOude Nijhuis, René Visser (leuk hè) en Clemens Beijers, als onderzoeker-in-opleiding dan ook. Heren, bedankt voor de gezellige tijd.

Bij de Dynaimca-studentenkamer, slechts tien passen verderop in de gang, stondde deur ook altijd uitnodigend open. En die uitnodiging werd dus menigmaalmisbruikt. Al was het vaak alleen maar voor de koelkast. Bij deze wil ik devolgende ingenieurs bedanken voor hun bijdrage aan het onderzoek: MarcelMolenaar, Arnoud Breedijk, Sander Groten, Theo de Gruiter, Martijn Kamphuis,Hedzer Tillema, Martin Bnnkhuis en Sebastiaan Wolbers.

Onze deur stond ook akijd open. Daar waren we best strikt in. Graag bedank ikJoeri Lof voor de vele en vele uurtjes die we samen doorbrachten. Dat is toch lang

213

214

niet zoveel als met Hans-Elias de Bree (werk) of Marten Toxopeus (privé), terwiji¡k met hun heel wat boompjes heb opgezet.

Het onderzoek valt binnen het STW project 'Modellering van akoesto-elastischeinteractie'. De halfjaarlijkse vergaderingen en de contacten daarnaast waren openen inspirerend. Met name dank ik de secretaris van S1'W: Jean Pierre Veen, en laterLeo Korstanje. Graag bedank ik ook de comrnissie, en met name Jan Verheij, voorhet kritisch doornemen van de versiaggeving.

Als je mij vraagt: 'Wat heb je nu aiiemaal gedaan in die vier jaar?', dan kan iknatuurlijk heel simpel, en heel flauw, naar dit boekje wijzen. Maar gelukkig staatniet alles op papier dus er valt nog een hoop te verteilen. Wat ik wel graag wilvermelden is dat ik een hele leuke tijd bij de Stretchers heb gehad. Als jebijvoorbeeld nog nooit unihockey of mertebagbal hebt gespeeld dan heb je tochwat geniist. En dat geldt ook voor het paardrij den.

De gezellige weekendjes bij de familie en bij vrienden gaan gewoon door. Toch?Huize Hellendoom was en is in ieder geval een ideale thuisbasis en zeer belangrijkals informatiecentrum voor en door de rest van de familie. En waar zouden we zijnzonder de trein? Neen, waar zou de NS zijn zonder mij. Pijnacker stond wel ergvaak op mijn treinkaartje. Gelukkig maar. Danielle, bedankt!

Oktober 2000, Enschede

Levensloop

De auteur werd geboren op 13 december 1971 in Reutum, gemeente Tubbergen.Na het VWO op College Noetsele in Nij verdal te hebben doorlopen, startte hij in1990 met de studlie werktuigbouwkunde. Die keuze was vrij makkelijk gemaaktondanks, of misschien wel dankzij, het feit dat er in de directe familie geentechneut (lees: fietsenmaker) aanwezig was. De auteur kijkt graag terug naar diezesjarige periode waar hij onder meer deelnam aan een multidisciplinaireontwerpwedstrijd, Kreatech, stage hep op de Amerikaanse universiteit Auburn inAlabama en aistudeerde vanuit de vakgroep Technische Mechanica bij hetNationaal Lucht- en Ruimtevaartcentrum, het NLR, op het gebied van oneindige-eindige numerieke akoestische elementen.

De eerdere kennismaking met zowel de vakgroep als de akoestiek was demotivatie orn in 1996 te beginnen met het promotieonderzoek op het gebied vanhet modelleren van geluid absorberende materialen. Het onderzoek is een mix vanrekenen en meten en heeft geleid tot een aantal artikelen, congresbijdragen, postersen een patentaanvraag.

215

216

Wat is de kern, meneer?Wat is de zin van dit verhaal?¡k vraag het nog een keer,Waarom die woorden allemaal?Wat deel ik mee,Wat wil ik zeggen,Waarom draait wat ik vertel in het bifzonder?

Wat is de kern, meneer?Wat is de zin van dit verhaal?1k vraag het nog een keer,Waarom die woorden allemaal?Wat deel ik mee,Wat wil ik zeggen,Waarom draait wat ik vertelde, kort en goed?

Nou?

Deel uit 'Als het maar kort is', gezongen door Boudewijn de GrootLied uit de musical 'Tsjechov'Tekst: Dimitri Frenkel Frank, Robert LongMuziek: Robert Long