ACOUSTICS OF DUCTSANDMUFFLERS

ACOUSTICS OF DUCTSANDMUFFLERSSecond Edition

M. L. MunjalDepartment of Mechanical EngineeringIndian Institute of ScienceBangalore, India

This edit ion first published 2014

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Library of Congress Cataloging-in-Publication Data

Munjal, M. L. (Manohar Lal), 1945-[Acoustics of ducts and mufflers with application to exhaust and

ventilation system design]Acoustics of ducts and mufflers / M. L. Munjal. – Second edition.

pages cmRevised edition of: Acoustics of ducts and mufflers with application to

exhaust and ventilation system design / M. L. Munjal. 1987.Includes bibliographical references and index.

ISBN 978-1-118-44312-5 (hardback)1. Air ducts–Acoustic properties. 2. Automobiles–Motors–Mufflers–Acoustic properties.

3. Ventilation–Noise. 4. Automobiles–Motors–Exhaust systems–Noise. I. Title.TH7683.D8M86 2014629.2502–dc23

2013041131

A catalogue record for this book is available from the British Library.

ISBN 9781118443125

Set in 10/12pt, Times-Roman by Thomson Digital, Noida, India

1 2014

To the memory of my parents

Contents

Preface xiii

1 Propagation of Waves in Ducts 11.1 Plane Waves in an Inviscid Stationary Medium 21.2 Three-Dimensional Waves in an Inviscid Stationary Medium 5

1.2.1 Rectangular Ducts 51.2.2 Circular Ducts 8

1.3 Waves in a Viscous Stationary Medium 121.4 Plane Waves in an Inviscid Moving Medium 161.5 Three-Dimensional Waves in an Inviscid Moving Medium 181.6 One-Dimensional Waves in a Viscous Moving Medium 201.7 Waves in Ducts with Compliant Walls (Dissipative Ducts) 23

1.7.1 Rectangular Duct with Locally Reacting Lining 241.7.2 Circular Duct with Locally Reacting Lining 271.7.3 Rectangular Duct with Bulk Reacting Lining 291.7.4 Circular Duct with Bulk Reacting Lining 32

1.8 Three-Dimensional Waves along Elliptical Ducts 34References 39

2 Theory of Acoustic Filters 412.1 Units for the Measurement of Sound 412.2 Uniform Tube 432.3 Radiation Impedance 462.4 Reflection Coefficient at an Open End 482.5 A Lumped Inertance 492.6 A Lumped Compliance 502.7 End Correction 512.8 Electroacoustic Analogies 512.9 Electrical Circuit Representation of an Acoustic System 522.10 Acoustical Filter Performance Parameters 53

2.10.1 Insertion Loss, IL 542.10.2 Transmission Loss, TL 55

2.10.3 Level Difference, LD 562.10.4 Comparison of the Three Performance Parameters 58

2.11 Lumped-Element Representations of a Tube 582.12 Simple Area Discontinuities 602.13 Gradual Area Changes 62

2.13.1 Conical Tube 622.13.2 Exponential Tube 642.13.3 Elliptical Tube 65

2.14 Extended-Tube Resonators 672.15 Helmholtz Resonator 692.16 Concentric Hole-Cavity Resonator 702.17 An Illustration of the Classical Method of Filter Evaluation 712.18 The Transfer Matrix Method 74

2.18.1 Definition of Transfer Matrix 742.18.2 Transfer Matrix of a Uniform Tube 762.18.3 A General Method for Derivation of Transfer Matrix 762.18.4 Transfer Matrices of Lumped Elements 772.18.5 Transfer Matrices of Variable Area Tubes 782.18.6 Overall Transfer Matrix of the System 812.18.7 Evaluation of TL in Terms of the Four-Pole Parameters 83

2.19 TL of a Simple Expansion Chamber Muffler 852.20 An Algebraic Algorithm for Tubular Mufflers 88

2.20.1 Development of the Algorithm 882.20.2 Formal Enunciation and Illustration of the Algorithm 89

2.21 Synthesis Criteria for Low-Pass Acoustic Filters 91References 94

3 Flow-Acoustic Analysis of Cascaded-Element Mufflers 973.1 The Exhaust Process 973.2 Finite Amplitude Wave Effects 1013.3 Mean Flow and Acoustic Energy Flux 1023.4 Aeroacoustic State Variables 1053.5 Aeroacoustic Radiation 1083.6 Insertion Loss 1113.7 Transfer Matrices for Tubular Elements 112

3.7.1 Uniform Tube 1133.7.2 Extended-Tube Elements 1143.7.3 Simple Area Discontinuities 1183.7.4 Physical Behaviour of Area Discontinuities 118

3.8 Perforated Elements with Two Interacting Ducts 1193.8.1 Concentric-Tube Resonator 1243.8.2 Cross-Flow Expansion Element 1243.8.3 Cross-Flow Contraction Element 1253.8.4 Some Remarks 125

3.9 Acoustic Impedance of Perforates 1263.10 Matrizant Approach 129

viii Contents

3.11 Perforated Elements with Three Interacting Ducts 1313.11.1 Three-Duct Cross-Flow Expansion Chamber Element 1353.11.2 Three-Duct Reverse Flow Expansion Chamber Element 136

3.12 Other Elements Constituting Cascaded-Element Mufflers 137References 143

4 Flow-Acoustic Analysis of Multiply-Connected Perforated Element Mufflers 1474.1 Herschel-Quincke Tube Phenomenon 1474.2 Perforated Element with Several Interacting Ducts 1514.3 Three-Pass Double-Reversal Muffler 1544.4 Flow-Reversal End Chambers 1584.5 Meanflow Lumped Resistance Network Theory 1634.6 Meanflow Distribution and Back Pressure Estimation 169

4.6.1 A Chamber with Three Interacting Ducts 1694.6.2 Three-Pass Double-Reversal Chamber 1714.6.3 A Complex Muffler Configuration 173

4.7 Integrated Transfer Matrix Approach 1754.7.1 A Muffler with Non-Overlapping Perforated Ducts and a Baffle 1764.7.2 Muffler with Non-Overlapping Perforated Elements,

Baffles and Area Discontinuities 1804.7.3 A Combination Muffler 184

References 186

5 Flow-Acoustic Measurements 1875.1 Impedance of a Passive Subsystem or Termination 187

5.1.1 The Probe-Tube Method 1885.1.2 The Two-Microphone Method 1935.1.3 Transfer Function Method 1975.1.4 Comparison of the Various Methods for a Passive Subsystem 202

5.2 Four-Pole Parameters of a Flow-Acoustic Element or Subsystem 2035.2.1 Theory of the Two Source-Location Method 2045.2.2 Theory of the Two-Load Method 2075.2.3 Comparison of the Two Methods 2085.2.4 Experimental Validation 208

5.3 An Active Termination – Aeroacoustic Characteristics of a Source 2105.3.1 Direct Measurement of Source Impedance 2115.3.2 Indirect Measurement of Source Characteristics 2145.3.3 Numerical Evaluation of the Engine Source Characteristics 2245.3.4 A Comparison of the Various Methods for Measuring Source

Characteristics 228References 229

6 Dissipative Ducts and Parallel Baffle Mufflers 2336.1 Acoustically Lined Rectangular Duct with Moving Medium 2346.2 Acoustically Lined Circular Duct with Moving Medium 2396.3 Transfer Matrix Relation for a Dissipative Duct 241

Contents ix

6.4 Transverse Wave Numbers for a Stationary Medium 2446.5 Normal Impedance of the Lining 2456.6 Transmission Loss 2496.7 Effect of Protective Layer 2516.8 Parallel Baffle Muffler 2576.9 The Effect of Mean Flow 2596.10 The Effect of Terminations on the Performance of Dissipative Ducts 2606.11 Lined Bends 2616.12 Plenum Chambers 2616.13 Flow-Generated Noise 2626.14 Insertion Loss of Parallel Baffle Mufflers 263References 264

7 Three-Dimensional Analysis of Mufflers 2677.1 Collocation Method for Simple Expansion Chambers 268

7.1.1 Compatibility Conditions at Area Discontinuities 2697.1.2 Extending the Frequency Range 2727.1.3 Extension to Other Muffler Configurations 275

7.2 Finite Element Methods for Mufflers 2757.2.1 A Single Element 2767.2.2 Variational Formulation of Finite Element Equations 2817.2.3 The Galerkin Formulation of Finite Element Equations 2857.2.4 Evaluation of Overall Performance of a Muffler 2887.2.5 Numerical Computation 290

7.3 Green’s Function Method for a Rectangular Cavity 2927.3.1 Derivation of the Green’s Function 2927.3.2 Derivation of the Velocity Potential 2957.3.3 Derivation of the Transfer Matrix 2977.3.4 Validation Against FEM 299

7.4 Green’s Function Method for Circular Cylindrical Chambers 3017.5 Green’s Function Method for Elliptical Cylindrical Chambers 3037.6 Breakout Noise 306

7.6.1 Breakout Noise from Hoses 3067.6.2 Breakout Noise from Rectangular Ducts 3107.6.3 Breakout Noise from Elliptical Ducts 3127.6.4 Breakout Noise from Mufflers 315

References 316

8 Design of Mufflers 3218.1 Requirements of an Engine Exhaust Muffler 3218.2 Simple Expansion Chamber 3228.3 Double-Tuned Extended-Tube Expansion Chamber 3248.4 Tuned Concentric Tube Resonator 3268.5 Plug Mufflers 3278.6 Side-Inlet Side-Outlet Mufflers 3298.7 Designing for Insertion Loss 331

x Contents

8.8 Three-Pass Double-Reversal Chamber Mufflers 3388.9 Perforated Baffle Muffler 3478.10 Forked Dual Muffler System 3498.11 Design of Short Elliptical and Circular Chambers 353

8.11.1 Short Chamber Configurations 3548.11.2 Theory 3558.11.3 Results and Discussion 3568.11.4 Acoustical Equivalence of Short Expansion and

Flow-reversal End Chambers 3618.12 Back-Pressure Considerations 3628.13 Practical Considerations 3658.14 Design of Mufflers for Ventilation Systems 3678.15 Active Sound Attenuation 369References 375

Appendix A: Bessel Functions and Some of Their Properties 377

Appendix B: Entropy Changes in Adiabatic Flows 381B.1 Stagnation Pressure and Entropy 381B.2 Pressure, Density, and Entropy 382

Appendix C: Nomenclature 385

Index 389

Contents xi

Preface

Engine exhaust noise pollutes the street environment and ventilation fan noise entersdwellings along with the fresh air. Work on the analysis and design of mufflers for both theseapplications has been going on since the early 1920s. However, it gained momentum in the1970s as people became more and more conscious of their working and living environment.Governments of many countries have responded to popular demand with mandatoryrestrictions on sound emitted by automotive engines. The exhaust system being the primarysource of engine noise (combustion-induced structural vibration sound is the next inimportance for diesel engines), exhaust mufflers have received most attention fromresearchers.

The text deals with the theory of exhaust mufflers for internal combustion engines of thereciprocating type, air-conditioning and ventilation ducts, ventilation and access openings ofacoustic enclosures, and so on. The common feature of all these systems is wave propagationin a moving medium. The function of a muffler is to muffle the sound through impedancemismatch or to dissipate the incoming sound into heat, while allowing the mean flow to gothrough almost unimpeded. The former type of muffler is called reflective, reactive, ornondissipative mufflers, while the latter are called dissipative or absorptive mufflers, orsimply silencers. However, every muffler contains some impedance mismatch and someacoustic dissipation. Therefore, this book deals with mufflers that are primarily reflective aswell as with those that are primarily dissipative. However, combination mufflers have alsobeen discussed.

The monograph ‘Acoustics of Ducts and Mufflers’, the original edition of which waspublished in 1987, has continued to be the only monograph on this subject. Elsewhere in theliterature, the theory of exhaust mufflers is generally covered in a chapter or two in books onindustrial noise control, a treatment which is too superficial for a student, researcher, orprospective designer; the actual design of practical exhaust and ventilation systems is treatedin a simplistic, textbook style approach.

Over the last 26 years, since publication of the original edition of this monograph, therehave been several significant developments in this important area. The mainframe computers,desktops, personal computers and laptops have got much faster with the random accessmemory (RAM) being several orders larger. Very efficient algorithms have been developedfor analysis of one-dimensional (1D) as well as three-dimensional (3D) analysis of mufflers.Commercially available softwares with user-friendly graphic user interface (GUI) have madethe analysis (and thence design) of mufflers much more convenient as well as faster.

Concurrently, efficient algorithms have been developed for plane wave analysis of multiply-connected perforated-element mufflers as well as the cascaded-element mufflers. Methodshave been developed to estimate the meanflow pressure drop across complex mufflerconfigurations making use of the lumped-element flow resistance networks or the flownetwork models, without resorting to tedious and time-consuming computational fluiddynamics (CFD) modeling. It is now possible to estimate the unmuffled exhaust as well asintake noise spectrum of internal combustion reciprocating engines making use of empiricalexpressions of acoustic source characteristics. Recently 3D as well as 1D models have beendeveloped for analysis of multiport elements as well as two-port elements.

The break-out noise radiating from the shell as well as end plates of a muffler can now bepredicted reasonably well, thereby enabling prediction of net transmission loss (TL) andinsertion loss (IL) of mufflers. All these developments are reflected in this second edition ofthe monograph.

Like the original edition, this second edition of the monograph is an outcome of myresearch in the analysis and design of mufflers for over 40 years, and of a course that I havebeen offering to graduate students all along. My experience in industrial consultancy is amplyreflected in the application orientated treatment of the subject. Although a bias in favor of themethods developed by me and my students over the years is unavoidable, every effort hasbeen made to offer the best to the reader. Emphasis is on the latest and/or the best methodsavailable, and not on the coverage of all the methods available in the current literature on anyparticular topic or aspect. A substantial portion of the book represents recent unpublishedmaterial. References have been cited from all over the English-language literature, but noeffort has been made to make the lists (at the end of every chapter) exhaustive.

The symbols of parameters used throughout the book are presented in an appendix.Otherwise, every chapter has been prepared so that it is complete in itself. Generally, symbolsare accompanied by the names of the parameters they represent in order to make the readingas smooth as possible.

The text starts with the propagation of waves in ducts, which forms the base for subsequentchapters. Exhaustive treatment of the one-dimensional theory of acoustic filters is followed byaeroacoustics of exhaust mufflers where the convective as well as dissipative effects ofmoving medium are incorporated.

The time-domain approach (method of characteristics) in Chapter 4 of the original editionhas been replaced with flow-acoustic analysis of multiply-connected perforated-elementmufflers, where estimation of back pressure making use of the flow resistance networkapproach has been discussed apart from the 1D frequency domain acoustic analysis.

Experimental methods needed for supplementing analysis corroborating analytical results,and verifying the efficacy of a muffler configuration, are discussed in Chapter 5. Dissipativeducts or mufflers are dealt with in Chapter 6. These days, a variety of muffler configurationshave come into commercial use in which three-dimensional effects predominate. Theseconfigurations can be analyzed best by means of the finite element methods. In this secondedition, 3D analytical methods have also been introduced in Chapter 7 in order to account forthe higher-order modes for prediction of the transverse TL (relevant to break-out noise) aswell as axial TL, and thence the net TL.

The last chapter is devoted exclusively to the design of mufflers for various applications.Active noise control in a duct has also been touched upon.

xiv Preface

This book is addressed to designers and graduate students specializing in technicalacoustics or engineering acoustics. Researchers will find in it a state-of-the-art account ofmuffler theory. An effort has been made to make the book complete in itself, that is,independently readable. Engine exhaust systems and ventilation systems are the primarytargets. However, methods discussed here can be applied to the inlet and discharge systems ofreciprocating compressors as well.

I owe my first interest in vibrations and dynamical systems to my former teacher, M. V.Narasimhan who has continued to be my friend, philosopher, and guide over the last fourdecades. I benefited greatly from my association and discussions with A. V. Sreenath, B.S.Ramakrishna, M. Heckl, B. V. A. Rao, Colin H. Hansen, Mats Abom, Hans Boden, AhmetSelamet, J. E. Sneckenberger, M. G. Prasad, Istvan L. Ver, Larry J. Eriksson, S.Soundranayagam, V. R. Sonti, C. S. Jog, Rudra Pratap, Joseph W. Sullivan, and S.Anantharamu, among others.

I have drawn heavily from the published work of my former students, Prakash T. Thawani,M. L. Kathuriya, V. B. Panicker, Mohan D. Rao, K. Narayana Rao, U. S. Shirahatti, A. D.Sahasrabudhe, V. H. Gupta, V. Easwaran, G. R. Gogate, T. S. S. Narayana, V. Bhujanga Rao,S. N. Panigrahi, Trinath Kar, R. N. Hota, B. Venkatesham, N. K. Mukherjee, P. Chaitanya,M. Harikrishna Reddy, N. K. Vijayasree, Akhilesh Mimani, D. Veerababu, Ramya Teja andAbhishek Verma. Some of them, who used the original version of the monograph, pointed outseveral typos and made useful suggestions that have been gratefully addressed in this secondedition of the monograph.

I wish to particularly thank R. Mangala and C. Srinivasa for their help with typing drawingsand formatting of the manuscript through several revisions and corrections.

This book took nearly two years of preparation, writing, and processing. If it is completedtoday, it is largely due to the constant forbearance, understanding, and cooperation of mywife, Vandana alias Bhuvnesh.

This manuscript has been catalyzed and supported by the Department of Science andTechnology (DST), under its Utilization of Scientific Expertise of Retired Scientists (USERS)Scheme.

M. L. MunjalBangalore, India

July 2013

Preface xv

1Propagation of Waves in Ducts

Exhaust noise of internal combustion engines is known to be the biggest pollutant of thepresent-day urban environment. Fortunately, however, this noise can be reduced sufficiently(to the level of the noise from other automotive sources, or even lower) by means of a well-designed muffler (also called a silencer). Mufflers are conventionally classified as dissipativeor reflective, depending on whether the acoustic energy is dissipated into heat or is reflectedback by area discontinuities.

However, no practical muffler or silencer is completely reactive or completely dissipative.Every muffler contains some elements with impedance mismatch and some with acousticdissipation. In fact, combination mufflers are getting increasingly popular with designers.

Dissipative mufflers consist of ducts lined on the inside with an acoustically absorptivematerial. When used on an engine, such mufflers lose their performance with time because theacoustic lining gets clogged with unburnt carbon particles or undergoes thermal cracking.Recently, however, better fibrous materials such as sintered metal composites have beendeveloped that resist clogging and thermal cracking and are not so costly. Besides, long strandunglued glass fibers can stand high temperatures. Nevertheless, no such problems areencountered in ventilation ducts, which conduct clean and cool air. The fan noise that wouldpropagate through these ducts can well be reduced during propagation if the walls of theconducting duct are acoustically treated. For these reasons the use of dissipative mufflers ismuch more common in air-conditioning systems.

Reflective mufflers, being nondissipative, are also called reactive mufflers. A reflectivemuffler consists of a number of tubular elements of different transverse dimensions joinedtogether so as to cause, at every junction, impedance mismatch and hence reflection of asubstantial part of the incident acoustic energy back to the source. Most of the mufflerscurrently used on internal combustion engines, where the exhaust mass flux varies strongly,though periodically, with time, are of the reflective or reactive type. In fact, even the mufflerof an air-conditioning system is generally provided with a couple of reflective elements at oneor both ends of the acoustically dissipative duct.

Clearly, a tube or pipe or duct is the most basic and essential element of either type ofmuffler. A study of the propagation of waves in ducts is therefore central to the analysis of amuffler for its acoustic performance (transmission characteristics). This chapter is devoted to

Acoustics of Ducts and Mufflers, Second Edition. M. L. Munjal. 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd.

the derivation and solution of equations for plane waves and three-dimensional waves alongrectangular ducts, circular tubes and elliptical shells without and with mean flow, without andwith viscous friction, with rigid unlined walls and compliant or acoustically lined walls. Westart with the simplest case and move gradually to the more general and involved cases.

1.1 Plane Waves in an Inviscid Stationary Medium

In the ideal case of a rigid-walled tube with sufficiently small cross dimensions* filled with astationary ideal (nonviscous) fluid, small-amplitude waves travel as plane waves. The acousticpressure perturbation (on the ambient static pressure) p and particle velocity u at all points of across-section are the same. The wave front or phase surface, defined as a surface at all pointsof which p and u have the same amplitude and phase, is a plane normal to the direction ofwave propagation, which in the case of a tube is the longitudinal axis.

The basic linearized equations for the case are:Mass continuity

ro∂u∂z �

∂r∂t � 0; (1.1)

Dynamical equilibrium

ro∂u∂t �

∂p∂z � 0; (1.2)

Energy equation (isentropicity)

∂p∂r

� �s

� g po � p� �ro � r

≅gporo

� c2o(say) (1.3)

where

z is the axial or longitudinal coordinate,p and r are acoustic perturbations on pressure and density,po and ro are ambient pressure and density of the medium,s is the entropy,p/po ¿ 1; r/ro ¿ 1

Equation 1.3 implies that

dr � dp

c2o;

∂r∂t �

1c2o

∂p∂t ;

∂r∂z �

1c2o

∂p∂z . (1.4)

The equation of dynamical equilibrium is also referred to as momentum balance equation,or simply, momentum equation. Similarly, the equation for mass continuity is commonlycalled continuity equation.

*The specific limits on the cross dimensions as a function of wave length are given in the next section.

2 Acoustics of Ducts and Mufflers, Second Edition

Substituting Equation 1.4 in Equation 1.1 and eliminating u from Equations 1.1 and 1.2 bydifferentiating the first with respect to (w.r.t.) t, the second with respect to z, and subtracting,yields

∂2

∂t2 � c2o∂2

∂z2

� �p � 0. (1.5)

This linear, one-dimensional (that is, involving one space coordinate), homogeneous partialdifferential equation with constant coefficients (co is independent of z and t) admits a generalsolution:

p z; t� � � C1f z� cot� � � C2g z� cot� �. (1.6)

If the time dependence is assumed to be of the exponential form e jwt, then the solution (1.6)becomes

p z; t� � � C1ejw t�z/co� �

� C2ejw t�z/co� �

. (1.7)

The first part of this solution equals C1 at z � t � 0 and also at z � cot. Therefore, itrepresents a progressive wave moving forward unattenuated and unaugmented with a velocityco. Similarly, it can be readily observed that the second part of the solution represents aprogressive wave moving in the opposite direction with the same velocity, co.

Thus, co is the velocity of wave propagation, Equation 1.5 is a wave equation, andsolution (1.7) represents a standing wave defined as superposition of two progressive waveswith amplitudes C1 and C2 moving in opposite directions.

Equation 1.5 is called the classical one-dimensional wave equation, and the velocity ofwave propagation co is also called phase velocity or sound speed. As acoustic pressure p islinearly related to particle velocity u or, for that matter, velocity potential f defined by therelations

u � ∂f∂z ; p � �ro

∂f∂t ; (1.8)

the dependent variable in Equation 1.5 could as well be u or f. In view of this generality, thewave character of Equation 1.5 lies in the differential operator

L � ∂2

∂t2 � c2o∂2

∂z2 ; (1.9)

which is called the classical one-dimensional wave operator.Upon factorizing this wave operator as

∂2

∂t2 � c2o∂2

∂z2 �∂∂t � co

∂∂z

� �∂∂t � co

∂∂z

� �; (1.10)

one may realize that the forward-moving wave [the first part of solution (1.6) or (1.7)] is thesolution of the equation

∂p∂t � co

∂p∂z � 0; (1.11)

Propagation of Waves in Ducts 3

and the backward-moving wave [the second part of solution (1.6) or (1.7)] is the solution ofthe equation

∂p∂t � co

∂p∂z � 0; (1.12)

Equation 1.7 can be rearranged as

p z; t� � � C1e�jkz � C2e

�jkz�

e jwt; (1.13)

where

k � w/co � 2p/l;k is called the wave number or propagation constant, and l is the wavelength.

As particle velocity u also satisfies the same wave equation, one can write

u z; t� � � C3e�jkz � C4e

�jkz�

e jwt. (1.14)

Substituting Equations 1.13 and 1.14 in the dynamical equilibrium equation (1.2) yields

C3 � C1/roco; C4 � �C2/roco ;

and therefore

u z; t� � � 1Zo

C1e�jkz � C2e

�jkz� �

e jwt; (1.15)

where Zo � roco is the characteristic impedance of the medium, defined as the ratio of theacoustic pressure and particle velocity of a plane progressive wave.

For a plane wave moving along a tube, one could also define a volume velocity vv (� Su)and mass velocity

v � roSu; (1.16)

where S is the area of cross-section of the tube. The corresponding values of characteristicimpedance (defined now as the ratio of the acoustic pressure and the said velocity of a planeprogressive wave) would then be as follows:

For particle velocity u; characteristic impedance � p/u � roco; (1.17a)

For volume velocity vv; characteristic impedance � p/vv � roco/S; (1.17b)

For mass velocity v; characteristic impedance � p/v � co/S. (1.17c)

For the latter two cases, the characteristic impedance involves the tube area S. As it is not aproperty of the medium alone, it would be more appropriate to call it characteristic impedanceof the tube. For tubes conducting hot exhaust gases, it is more appropriate to deal with

4 Acoustics of Ducts and Mufflers, Second Edition

acoustic mass velocity v. The corresponding characteristic impedance is denoted in thesepages by the symbol Y for convenience:

Yo � co/S. (1.18)

Equations 1.15, 1.16 and 1.18 yield the following expression for acoustic mass velocity:

v z; t� � � 1Yo

C1e�jkz � C2e

�jkz� �

e jwt (1.19)

Subscript 0 with Y and k indicates nonviscous conditions. Constants C1 and C2 inEquations 1.13 and 1.19 are to be determined by the boundary conditions imposed by theelements that precede and follow the particular tubular element under investigation. This has tobe deferred to the next chapter, where we deal with a system of elements or an acoustic filter.

1.2 Three-Dimensional Waves in an Inviscid Stationary Medium

In order to appreciate the limitations of the plane wave theory, it is necessary to consider thegeneral 3D (three-dimensional) wave propagation in tubes. The basic linearized equationscorresponding to Equations 1.1 and 1.2 for waves in stationary nonviscous medium areobtained by replacing ∂/∂z with the 3D gradient operator r. Thus,

Mass continuity: ror ? u� ∂r∂t � 0; (1.20)

Dynamical equilibrium: ro∂u∂t �rp � 0. (1.21)

The third equation is the same as Equations 1.3 or 1.4. On making use of this equation inEquation 1.20, differentiating Equation 1.20 w.r.t. to t, taking divergence of Equation 1.21and subtracting, one gets the required 3D wave equation,

∂2

∂t2 � c2or2

� �p � 0; (1.22)

where the Laplacianr2 is given as follows.Cartesian coordinate system (for rectangular ducts)

r2 � ∂2

∂x2 �∂2

∂y2 �∂2

∂z2 ; (1.23)

Cylindrical polar coordinate system (for circular tubes)

r2 � ∂2

∂r2 �1r

∂∂r �

1r2

∂2

∂q2� ∂2

∂z2 . (1.24)

1.2.1 Rectangular Ducts

For harmonic time dependence, making use of separation of variables, the general solution ofthe 3D wave equation (1.22) with the Laplacian given by Equation 1.23 can be seen to be

Propagation of Waves in Ducts 5

p x; y; z; t� � � C1e�jkzz � C2e

�jkzz� �

e�jkxx � C3e�jkxx

� �e�jkyy � C4e

�jkyy� �

e jwt; (1.25)

with the compatibility condition

k2x � k2y � k2z � k2o. (1.26)

Here, kx; ky and kz are wave numbers in the x, y and z direction, respectively. In the limitingcase of plane waves, kx � ky � o. Then, Equation 1.26 yields kz � ko and Equation 1.25reduces to Equation 1.13.

It may be noted from Equation 1.25 that x-dependent factor involves two unknownskx and c3, and the y-dependent factor involves the unknowns ky and C4. These may beevaluated from the relevant boundary conditions as follows.

For a rigid-walled duct of breadth b and height h (Figure 1.1), the boundary conditions are

∂p∂x � 0 at x � 0 and x � b (1.27a)

and

∂p∂y � 0 at y � 0 and y � h. (1.27b)

Substituting these boundary conditions in Equation 1.25 yields, respectively,

C3 � 1; kx � mpb

; m � 0; 1; 2; . . . . . . . (1.28a)

and

C4 � 1; ky � nph; n � 0; 1; 2; . . . . . . .; (1.28b)

and Equation 1.25 then becomes

p x; y; z; t� � �X∞m�0

X∞n�0

cosmpxb

�cos

npyh

�C1;m;ne

�jkz;m;nz � C2;m;ne�jkz;m;nz

� �e jwt; (1.29)

z

h

y

b

x

Figure 1.1 A rectangular duct and the Cartesian coordinate system (x, y, z)

6 Acoustics of Ducts and Mufflers, Second Edition

where, as per Equation 1.26, the transmission wave number for the (m, n) mode, kz;m;n is givenby the relation

kz;m;n � k2o � mp/b� �2 � np/h� �2h i1/2

. (1.30)

In order to evaluate axial particle velocity corresponding to the (m, n) mode, we make useof the z-component of the momentum equation (1.21)

ro∂uz;m;n∂t � ∂p

∂z � 0; (1.31)

which yields

uz;m;n��∂p/∂zjwro

� kz;m;nkoroco

C1;m.ne�jkz;m;nz � C2;m;ne

�jkz;m;nz�

cosmpxb

�cos

npyh

�e jwt.

(1.32)

Now, mass velocity can be evaluated by integration over the area of cross-section inFigure 1.1:

vz;m;n� ro

Z h

o

Z b

ouz;m;ndxdy

�Z b

ocos

mpxb

�dx

Z h

ocos

npyh

�dy

kz;m;nw

C1;m;ne�jkz;m;nz � C2;m;ne

�jkz;m;nz�

e jwt;

(1.33)

which yields

vz;m;n�0 for m ≠ 0 and=or n ≠ 0

� bh

coC1e

�jkoz � C2e�jkoz

� e jwt for m � n � 0.

(1.34)

Thus, acoustic mass velocity is nonzero only for the plane wave or (0, 0) mode for whichEquation 1.19 is recovered. Incidentally, it shows that the concept of acoustic volume velocityor mass velocity does not have any significance for higher-order modes. Equation 1.32 showsthat for the same acoustic pressure, amplitude of the particle velocity for the (m, n) mode isless than kz;m;n/ko times

� �that for the plane wave. It can be noted that for the (0, 0) mode,

kz;m;n � ko and Equation 1.29 reduces to Equation 1.13. Thus, plane wave corresponds to the(0, 0) mode solution in Equation 1.29.

Any particular mode (m, n) would propagate unattenuated if kz;m;n is greater than zero.Then, use of Equation 1.30 yields

k2o �mpb

�2 � nph

�2> 0 (1.35a)

Propagation of Waves in Ducts 7

or

l <2

m

b

�2 � n

h

�2� �1/2(1.35b)

Obviously, a plane wave of any wavelength can propagate unattenuated, whereas a highermode can propagate only insofar as inequality (1.35b) is satisfied. Thus, if h > b, the firsthigher mode (0, 1) would get cut-on (that is, it would start propagating) if

l < 2h or f >co2h

. (1.36)

In other words, only a plane wave would propagate (all higher modes, even if present,being cut-off, that is, attenuated exponentially) if the frequency is small enough so that

l > 2h or f <co2h

; (1.37)

Thus, the cut-off frequency of a rectangular duct (Figure 1.1) is given by

f co � co2h

; (1.38)

where h is the larger of the two transverse dimensions of the rectangular duct.

1.2.2 Circular Ducts

The wave equation (1.22), with the Laplacian given by Equation 1.24 governs wavepropagation in circular tubes (see Figure 1.2). Upon making use of the method of separationof variables, and writing time dependence as e jwt and q dependence as e jmq, one gets

p r; q; z; t� � �X

mRm r� �e jmqZ z� �e jwt. (1.39)

z

θ

y

x

r

P

Figure 1.2 A cylindrical duct/tube and the cylindrical polar coordinate system (r, q, z)

8 Acoustics of Ducts and Mufflers, Second Edition

Assuming the z-dependence function Z(z) as in Equation 1.25 with

d2Z

dz2� �k2z Z (1.40)

and substituting Equations 1.39 and 1.40 in the wave equation, one gets a Bessel equation forR(r):

d2Rm

dr2� 1

r

dRm

dr� k2o � k2z �

m2

r2

� �Rm � 0. (1.41)

As indicated in Appendix A, Equation 1.41 has a general solution

Rm � C3Jm krr� � � C4Nm krr� �; (1.42)

where the radial wave number kr is given by

k2r � k2o � k2z ; (1.43)

and Jm( ? ) and Nm( ? ) are Bessel function and Neumann function, respectively.Nm krr� � tends to infinity at r � 0 (the axis). But acoustic pressure everywhere has got to be

finite. Therefore, the constant C4 must be zero.Again, the radial velocity at the walls r � ro� � must be zero. Therefore,

dJm krr� �dr

� 0 at r � ro. (1.44)

Thus, kr takes only such discrete values as satisfy the equation

J 0m krro� � � 0. (1.45)

Upon denoting the value of kr corresponding to the nth root of this equation as kr;m;n; onegets

p r; q; t� � �X∞m�0

X∞n�1

Jm kr;m;nr� �

e jmqe jwt � C1;m;ne�jkz;m;nz � C2;m;ne

�jkz;m;nz� �

; (1.46)

where

kz;m;n � k2o � k2r;m;n

�1/2. (cf: Eq: 1.30) (1.47)

As the first zero of J 0o (or that of J1) is zero, kr;0;1 � 0 and kz;0;1 � ko. Thus, for the (0, 1)mode, Equation 1.46 reduces to Equation 1.13, the equation for the plane wave propagation.Hence, the plane wave corresponds to the (0, 1) mode of Equation 1.40 and propagatesunattenuated.

Propagation of Waves in Ducts 9

In most of the literature [1–3], n represents the number of the zero of the derivativeJ 0m krro� � as per Equation 1.45. This introduces a dissimilarity between the notation forrectangular ducts and circular ducts. In rectangular ducts, m and n represent the number ofnodes in the transverse pressure distribution as shown in Figure 1.3. A similar picture couldemerge for circular ducts if n were to denote the number of circular nodes in the transversepressure distribution. This is shown in Figure 1.4. With this notation [4,5], the plane modewould have the (0, 0) label in circular as well as rectangular ducts, and m and n would havethe same connotation, that is, the number of nodes (in respective directions) in the transversepressure distribution.

This new notation is adopted here henceforth. According to this, n� 0 wouldrepresent the first root of Equation 1.45 and n would represent the (n� 1)st root thereof.In Equation 1.46, the summation n � 1 to ∞ would read n � 0 to ∞ as in Equation 1.29for rectangular ducts.

The first two higher-order modes (1, 0) and (0, 1) will get cut-on if kz;1;0 and kz;0;1 are real,that is, if ko > kr;1;0 and kr;0;1. The first zero of J 01 occurs at 1.84 and the second zero of J 0ooccurs at 3.83. Thus, the cut-on wave numbers would be 1.84/ro and 3.83/ro, respectively. Inother words, the first azimuthal or diametral mode starts propagating at koro� 1.84 and thefirst axisymmetric mode at koro� 3.83. If the frequency is small enough (or wave length islarge enough) such that

koro < 1.84; or l >p

1.84D; or f <

1.84pD

co; (1.48)

2

1n

0

0 1

m

2

+ + + +

+ +

+

+ +

+

_ _

_

_ __

_

_

+

+ +

+

+

+

+

+

++

_ _

_

_

_ _

_

_

Figure 1.3 Nodal lines for transverse pressure distribution in a rectangular duct up to m� 2, n� 2(Reproduced with permission from [5])

10 Acoustics of Ducts and Mufflers, Second Edition

where D is the diameter 2ro, then only the plane waves would propagate. Thus the cut-offfrequency of a circular tube is given by

f co � 1.84p

coD

� 0.5857coD

(cf: Eq: 1.38) (1.49)

Fortunately, the frequencies of interest in exhaust noise of internal combustion engines arelow enough so that for typical maximum transverse dimensions of exhaust mufflers Equation1.49 is generally satisfied. Therefore, plane wave analysis has proved generally adequate. Inthe following pages, as indeed in most of the current literature on exhaust mufflers, one-dimensional wave propagation has been used throughout, with only a passing reference to theexistence of higher modes or three-dimensional effects. In practice, muffler configurations aredesigned making use of the 1D analysis, and 3D analysis is used for a final check.

Substituting the m; n� � mode component of Equation 1.46 in the equation of dynamicalequilibrium for the axial direction, that is,

ro∂uz∂t � ∂p

∂z � 0; (1.50)

yields

uz;m;n � �∂p/∂zjwro

m

0 1.84 3.05

0 1 2

0

1

2

6.715.333.83

9.978.547.02

++

+

+

__

_

_+ +

_ _ +_

+_

+ _ _+

+ +_ + _ + _ + _ ___ ++ ++

_+ +__

n

Figure1.4 Nodal lines for transversepressuredistribution inacircularductup tom� 2,n� 2 (Reproducedwith permission from [5])

Propagation of Waves in Ducts 11

or

uz;m;n � Jm kr;m;nr� �

e jmqe jwtkz;m;nkoroco

� C1;m;ne�jkz;m;nz � C2;m;ne

�jkz;m;nz�

. (1.51)

Thus, as compared to the plane wave, acoustic particle velocity for the (m, n) mode iskz;m;n/ko times, for the same acoustic pressure. Of course, as just shown for rectangular ducts,volume or mass velocity does not have a meaning for higher order modes.

1.3 Waves in a Viscous Stationary Medium

The analysis of wave propagation in a real (viscous) fluid with heat conduction from the wallsof the tube is originally due to Kirchhoff [6,7]. The presence of viscosity brings into play acoupling between the axial and radial motions of the particle in a circular tube. Even if onewere to assume axisymmetry (freedom from q dependence), the wave propagation in acircular tube would be two-dimensional.

Neglecting heat conduction in the first instance, the basic equations governing axisym-metric wave propagation in stationary medium are [8]:

Mass continuity

∂r∂t � r0

urr� ∂ur

∂r � ∂uz∂z

� �� 0; (1.52)

Dynamical equilibrium (Navier–Stokes equations)

r0∂uz∂t � ∂p

∂z � m∂2uz∂r2 � 1

r

∂uz∂r � ∂2uz

∂z2

� �� m

3∂2ur∂r∂z�

1r

∂ur∂z � ∂2uz

∂z2

� �; (1.53)

r0∂ur∂t � ∂p

∂r � m∂2uz∂r2 � 1

r

∂ur∂r � ur

r2� ∂2ur

∂z2

� �� m

3∂2ur∂r2 � 1

r

∂ur∂r � ur

r2� ∂2uz∂z∂r

� �. (1.54)

The thermodynamic process being isentropic for small-amplitude waves, Equation 1.3 isthe third equation.

Eliminating r from Equation 1.52 with the help of Equation 1.3, and using the resultingequation to eliminate p from Equations 1.53 and 1.54 yields

∂2uz∂t2 � c20

∂2uz∂z2 � 1

r

∂ur∂z � ∂2ur

∂z∂r

� �� ∂∂t

mr0

∂2uz∂r2 � 1

r

∂uz∂r � 1

3∂2ur∂r∂z�

131r

∂ur∂z � 4

3∂2uz∂z2

� �� �;

(1.55)

∂2ur∂t2 � c20

∂2ur∂r2 � 1

r

∂ur∂r � ur

r2� ∂2uz∂r∂z

� �� ∂∂t

mr0

∂2ur∂z2 � 1

3∂2uz∂z∂r�

43∂2ur∂r2 � 4

3r∂ur∂r � 4

3urr2

� �� �.

(1.56)

12 Acoustics of Ducts and Mufflers, Second Edition