Fundamentals of Noise and Vibration Analysis for Engineers
`
`Noise and vibration affects all kinds of engineering structures, and is fast becoming an integral part
`of engineering courses at universities and colleges around the world. In this second edition, Michael
`Norton’s classic text has been extensively updated to take into account recent developments in the
`field. Much of the new material has been provided by Denis Karczub, who joins Michael as second
`author for this edition.
`This book treats both noise and vibration in a single volume, with particular emphasis on wave–
`mode duality and interactions between sound waves and solid structures. There are numerous case
`studies, test cases and examples for students to work through. The book is primarily intended as a
`text book for senior level undergraduate and graduate courses, but is also a valuable reference for
`practitioners and researchers in the field of noise and vibration.
`
`Page 1 of 79
`
`GOOGLE EXHIBIT 1018
`
`

`

`Page 2 of 79
`
`Page 2 of 79
`
`

`

`Fundamentals of Noise
`and Vibration Analysis
`for Engineers
`Second edition
`
`M. P. Norton
`
`School of Mechanical Engineering, University of Western Australia
`
`and
`D. G. Karczub
`
`S.V.T. Engineering Consultants, Perth, Western Australia
`
`Page 3 of 79
`
`

`

`CAMBRIDGE UNIVERSITY PRESS
`Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
`
`Cambridge University Press
`The Edinburgh Building, Cambridge CB2 8RU, UK
`
`Published in the United States of America by Cambridge University Press, New York
`
`www.cambridge.org
`Information on this title: www.cambridge.org/9780521495616
`
`© First edition Cambridge University Press 1989
`© Second edition M. P. Norton and D. G. Karczub 2003
`
`This publication is in copyright. Subject to statutory exception
`and to the provisions of relevant collective licensing agreements,
`no reproduction of any part may take place without the written
`permission of Cambridge University Press.
`
`First edition published 1989
`Reprinted 1994
`Second edition published 2003
`
`A catalogue record for this publication is available from the British Library
`
`ISBN 978-0-521-49561-6 hardback
`ISBN 978-0-521-49913-2 paperback
`
`Transferred to digital printing 2007
`
`Page 4 of 79
`
`

`

`To
`our parents,
`the first author’s wife Erica,
`and his young daughters Caitlin and Sarah
`
`Page 5 of 79
`
`

`

`Page 6 of 79
`
`Page 6 of 79
`
`

`

`Contents
`
`Preface
`Acknowledgements
`Introductory comments
`
`page xv
`xvii
`xviii
`
`1
`
`Mechanical vibrations: a review of some fundamentals
`
`1.1 Introduction
`1.2 Introductory wave motion concepts – an elastic continuum viewpoint
`1.3 Introductory multiple, discrete, mass–spring–damper oscillator concepts –
`a macroscopic viewpoint
`1.4 Introductory concepts on natural frequencies, modes of vibration, forced
`vibrations and resonance
`1.5 The dynamics of a single oscillator – a convenient model
`1.5.1 Undamped free vibrations
`1.5.2 Energy concepts
`1.5.3 Free vibrations with viscous damping
`1.5.4 Forced vibrations: some general comments
`1.5.5 Forced vibrations with harmonic excitation
`1.5.6 Equivalent viscous-damping concepts – damping in real systems
`1.5.7 Forced vibrations with periodic excitation
`1.5.8 Forced vibrations with transient excitation
`1.6 Forced vibrations with random excitation
`1.6.1 Probability functions
`1.6.2 Correlation functions
`1.6.3 Spectral density functions
`1.6.4 Input–output relationships for linear systems
`1.6.5 The special case of broadband excitation of a single oscillator
`1.6.6 A note on frequency response functions and transfer functions
`1.7 Energy and power flow relationships
`
`1
`
`1
`3
`
`8
`
`10
`12
`12
`15
`16
`21
`22
`30
`32
`33
`37
`38
`39
`41
`42
`50
`52
`52
`
`vii
`
`Page 7 of 79
`
`

`

`viii
`
`Contents
`
`1.8 Multiple oscillators – a review ofsome general procedures
`1.8.1 A simple two-degree-of-freedom system
`1.8.2 A simple three-degree-of-freedom system
`1.8.3 Forced vibrations of multiple oscillators
`1.9 Continuous systems – a review of wave-types in strings, bars and plates
`1.9.1 The vibrating string
`1.9.2 Quasi-longitudinal vibrations of rods and bars
`1.9.3 Transmission and reflection of quasi-longitudinal waves
`1.9.4 Transverse bending vibrations of beams
`1.9.5 A general discussion on wave-types in structures
`1.9.6 Mode summation procedures
`1.9.7 The response of continuous systems to random loads
`1.9.8 Bending waves in plates
`1.10 Relationships for the analysis of dynamic stress in beams
`1.10.1 Dynamic stress response for flexural vibration of a thin beam
`1.10.2 Far-field relationships between dynamic stress and structural
`vibration levels
`1.10.3 Generalised relationships for the prediction of maximum
`dynamic stress
`1.10.4 Properties of the non-dimensional correlation ratio
`1.10.5 Estimates of dynamic stress based on static stress and
`displacement
`1.10.6 Mean-square estimates for single-mode vibration
`1.10.7 Relationships for a base-excited cantilever with tip mass
`1.11 Relationships for the analysis of dynamic strain in plates
`1.11.1 Dynamic strain response for flexural vibration of a constrained
`rectangular plate
`1.11.2 Far-field relationships between dynamic stress and structural
`vibration levels
`1.11.3 Generalised relationships for the prediction of maximum
`dynamic stress
`1.12 Relationships for the analysis of dynamic strain in cylindrical shells
`1.12.1 Dynamic response of cylindrical shells
`1.12.2 Propagating and evanescent wave components
`1.12.3 Dynamic strain concentration factors
`1.12.4 Correlations between dynamic strain and velocity spatial
`maxima
`References
`Nomenclature
`
`56
`56
`59
`60
`64
`64
`72
`77
`79
`84
`85
`91
`94
`96
`96
`
`100
`
`102
`103
`
`104
`105
`106
`108
`
`109
`
`112
`
`113
`113
`114
`117
`119
`
`119
`122
`123
`
`Page 8 of 79
`
`

`

`ix
`
`2
`
`Contents
`
`Sound waves: a review of some fundamentals
`
`2.1 Introduction
`2.2 The homogeneous acoustic wave equation – a classical analysis
`2.2.1 Conservation of mass
`2.2.2 Conservation of momentum
`2.2.3 The thermodynamic equation of state
`2.2.4 The linearised acoustic wave equation
`2.2.5 The acoustic velocity potential
`2.2.6 The propagation of plane sound waves
`2.2.7 Sound intensity, energy density and sound power
`2.3 Fundamental acoustic source models
`2.3.1 Monopoles – simple spherical sound waves
`2.3.2 Dipoles
`2.3.3 Monopoles near a rigid, reflecting, ground plane
`2.3.4 Sound radiation from a vibrating piston mounted in a rigid baffle
`2.3.5 Quadrupoles – lateral and longitudinal
`2.3.6 Cylindrical line sound sources
`2.4 The inhomogeneous acoustic wave equation – aerodynamic sound
`2.4.1 The inhomogeneous wave equation
`2.4.2 Lighthill’s acoustic analogy
`2.4.3 The effects of the presence of solid bodies in the flow
`2.4.4 The Powell–Howe theory of vortex sound
`2.5 Flow duct acoustics
`References
`Nomenclature
`
`3
`
`Interactions between sound waves and solid structures
`
`3.1 Introduction
`3.2 Fundamentals of fluid–structure interactions
`3.3 Sound radiation from an infinite plate – wave/boundary matching
`concepts
`3.4 Introductory radiation ratio concepts
`3.5 Sound radiation from free bending waves in finite plate-type structures
`3.6 Sound radiation from regions in proximity to discontinuities – point and
`line force excitations
`
`128
`
`128
`131
`134
`136
`139
`140
`141
`143
`144
`146
`147
`151
`155
`157
`162
`164
`165
`167
`174
`177
`180
`183
`187
`188
`
`193
`
`193
`194
`
`197
`203
`207
`
`216
`
`Page 9 of 79
`
`

`

`x
`
`Contents
`
`3.7 Radiation ratios of finite structural elements
`3.8 Some specific engineering-type applications of the reciprocity principle
`3.9 Sound transmission through panels and partitions
`3.9.1 Sound transmission through single panels
`3.9.2 Sound transmission through double-leaf panels
`3.10 The effects of fluid loading on vibrating structures
`3.11 Impact noise
`References
`Nomenclature
`
`4
`
`Noise and vibration measurement and control procedures
`
`4.1 Introduction
`4.2 Noise and vibration measurement units – levels, decibels and spectra
`4.2.1 Objective noise measurement scales
`4.2.2 Subjective noise measurement scales
`4.2.3 Vibration measurement scales
`4.2.4 Addition and subtraction of decibels
`4.2.5 Frequency analysis bandwidths
`4.3 Noise and vibration measurement instrumentation
`4.3.1 Noise measurement instrumentation
`4.3.2 Vibration measurement instrumentation
`4.4 Relationships for the measurement of free-field sound propagation
`4.5 The directional characteristics of sound sources
`4.6 Sound power models – constant power and constant volume sources
`4.7 The measurement of sound power
`4.7.1 Free-field techniques
`4.7.2 Reverberant-field techniques
`4.7.3 Semi-reverberant-field techniques
`4.7.4 Sound intensity techniques
`4.8 Some general comments on industrial noise and vibration control
`4.8.1 Basic sources of industrial noise and vibration
`4.8.2 Basic industrial noise and vibration control methods
`4.8.3 The economic factor
`4.9 Sound transmission from one room to another
`4.10 Acoustic enclosures
`4.11 Acoustic barriers
`4.12 Sound-absorbing materials
`4.13 Vibration control procedures
`
`221
`227
`230
`232
`241
`244
`247
`249
`250
`
`254
`
`254
`256
`256
`257
`259
`261
`263
`267
`267
`270
`273
`278
`279
`282
`282
`283
`287
`290
`294
`294
`295
`299
`301
`304
`308
`313
`320
`
`Page 10 of 79
`
`

`

`xi
`
`Contents
`
`4.13.1 Low frequency vibration isolation – single-degree-of-freedom
`systems
`4.13.2 Low frequency vibration isolation – multiple-degree-of-freedom
`systems
`4.13.3 Vibration isolation in the audio-frequency range
`4.13.4 Vibration isolation materials
`4.13.5 Dynamic absorption
`4.13.6 Damping materials
`References
`Nomenclature
`
`5
`
`The analysis of noise and vibration signals
`
`5.1 Introduction
`5.2 Deterministic and random signals
`5.3 Fundamental signal analysis techniques
`5.3.1 Signal magnitude analysis
`5.3.2 Time domain analysis
`5.3.3 Frequency domain analysis
`5.3.4 Dual signal analysis
`5.4 Analogue signal analysis
`5.5 Digital signal analysis
`5.6 Statistical errors associated with signal analysis
`5.6.1 Random and bias errors
`5.6.2 Aliasing
`5.6.3 Windowing
`5.7 Measurement noise errors associated with signal analysis
`References
`Nomenclature
`
`6
`
`Statistical energy analysis of noise and vibration
`
`6.1 Introduction
`6.2 The basic concepts of statistical energy analysis
`6.3 Energy flow relationships
`6.3.1 Basic energy flow concepts
`6.3.2 Some general comments
`6.3.3 The two subsystem model
`
`322
`
`325
`327
`330
`332
`334
`335
`336
`
`342
`
`342
`344
`347
`347
`351
`352
`355
`365
`366
`370
`370
`372
`374
`377
`380
`380
`
`383
`
`383
`384
`387
`388
`389
`391
`
`Page 11 of 79
`
`

`

`xii
`
`Contents
`
`6.3.4 In-situ estimation procedures
`6.3.5 Multiple subsystems
`6.4 Modal densities
`6.4.1 Modal densities of structural elements
`6.4.2 Modal densities of acoustic volumes
`6.4.3 Modal density measurement techniques
`6.5 Internal loss factors
`6.5.1 Loss factors of structural elements
`6.5.2 Acoustic radiation loss factors
`6.5.3 Internal loss factor measurement techniques
`6.6 Coupling loss factors
`6.6.1 Structure–structure coupling loss factors
`6.6.2 Structure–acoustic volume coupling loss factors
`6.6.3 Acoustic volume–acoustic volume coupling loss factors
`6.6.4 Coupling loss factor measurement techniques
`6.7 Examples of the application of S.E.A. to coupled systems
`6.7.1 A beam–plate–room volume coupled system
`6.7.2 Two rooms coupled by a partition
`6.8 Non-conservative coupling – coupling damping
`6.9 The estimation of sound radiation from coupled structures using total
`loss factor concepts
`6.10 Relationships between dynamic stress and strain and structural vibration
`levels
`References
`Nomenclature
`
`7
`
`Pipe flow noise and vibration: a case study
`
`7.1 Introduction
`7.2 General description of the effects of flow disturbances on pipeline noise
`and vibration
`7.3 The sound field inside a cylindrical shell
`7.4 Response of a cylindrical shell to internal flow
`7.4.1 General formalism of the vibrational response and sound
`radiation
`7.4.2 Natural frequencies of cylindrical shells
`7.4.3 The internal wall pressure field
`7.4.4 The joint acceptance function
`7.4.5 Radiation ratios
`
`393
`395
`397
`397
`400
`401
`407
`408
`410
`412
`417
`417
`419
`420
`421
`423
`424
`427
`430
`
`431
`
`433
`435
`437
`
`441
`
`441
`
`443
`446
`451
`
`451
`454
`455
`458
`460
`
`Page 12 of 79
`
`

`

`xiii
`
`Contents
`
`7.5 Coincidence – vibrational response and sound radiation due to higher
`order acoustic modes
`7.6 Other pipe flow noise sources
`7.7 Prediction of vibrational response and sound radiation characteristics
`7.8 Some general design guidelines
`7.9 A vibration damper for the reduction of pipe flow noise and vibration
`References
`Nomenclature
`
`8
`
`Noise and vibration as a diagnostic tool
`
`8.1 Introduction
`8.2 Some general comments on noise and vibration as a diagnostic tool
`8.3 Review of available signal analysis techniques
`8.3.1 Conventional magnitude and time domain analysis techniques
`8.3.2 Conventional frequency domain analysis techniques
`8.3.3 Cepstrum analysis techniques
`8.3.4 Sound intensity analysis techniques
`8.3.5 Other advanced signal analysis techniques
`8.3.6 New techniques in condition monitoring
`8.4 Source identification and fault detection from noise and vibration
`signals
`8.4.1 Gears
`8.4.2 Rotors and shafts
`8.4.3 Bearings
`8.4.4 Fans and blowers
`8.4.5 Furnaces and burners
`8.4.6 Punch presses
`8.4.7 Pumps
`8.4.8 Electrical equipment
`8.4.9 Source ranking in complex machinery
`8.4.10 Structural components
`8.4.11 Vibration severity guides
`8.5 Some specific test cases
`8.5.1 Cabin noise source identification on a load–haul–dump vehicle
`8.5.2 Noise and vibration source identification on a large induction
`motor
`Identification of rolling-contact bearing damage
`8.5.3
`8.5.4 Flow-induced noise and vibration associated with a gas pipeline
`
`461
`467
`471
`477
`479
`481
`483
`
`488
`
`488
`489
`493
`494
`501
`503
`504
`507
`511
`
`513
`514
`516
`518
`523
`525
`527
`528
`530
`532
`536
`539
`541
`541
`
`547
`550
`554
`
`Page 13 of 79
`
`

`

`xiv
`
`Contents
`
`8.5.5 Flow-induced noise and vibration associated with a racing
`sloop (yacht)
`8.6 Performance monitoring
`8.7 Integrated condition monitoring design concepts
`References
`Nomenclature
`
`Problems
`Appendix 1: Relevant engineering noise and vibration control journals
`Appendix 2: Typical sound transmission loss values and sound absorption
`coefficients for some common building materials
`Appendix 3: Units and conversion factors
`Appendix 4: Physical properties of some common substances
`
`Answers to problems
`
`Index
`
`557
`557
`559
`562
`563
`
`566
`599
`
`600
`603
`605
`
`607
`
`621
`
`Page 14 of 79
`
`

`

`2 Sound waves: a review of some
`fundamentals
`
`2.1
`
`Introduction
`
`Sound is a pressure wave that propagates through an elastic medium at some charac-
`teristic speed. It is the molecular transfer of motional energy and cannot therefore pass
`through a vacuum. For this wave motion to exist, the medium has to possess inertia and
`elasticity. Whilst vibration relates to such wave motion in structural elements, noise
`relates to such wave motion in fluids (gases and liquids). Two fundamental mechanisms
`are responsible for sound generation. They are:
`(i) the vibration of solid bodies resulting in the generation and radiation of sound
`energy – these sound waves are generally referred to as structure-borne sound;
`(ii) flow-induced noise resulting from pressure fluctuations induced by turbulence and
`unsteady flows – these sound waves are generally referred to as aerodynamic sound.
`With structure-borne sound, the regions of interest are generally in a fluid (usually air)
`at some distance from the vibrating structure. Here, the sound waves propagate through
`the stationary fluid (the fluid has a finite particle velocity due to the sound wave, but
`a zero mean velocity) from a readily identifiable source to the receiver. The region of
`interest does not therefore contain any sources of sound energy – i.e. the sources which
`generated the acoustic disturbance are external to it. A simple example is a vibrating
`electric motor. Classical acoustical theory (analysis of the homogeneous wave equation)
`can be used for the analysis of sound waves generated by these types of sources. The
`solution for the acoustic pressure fluctuation, p, describes the wave field external to
`the source. This wave field can be modelled in terms of combinations of simple sound
`sources. If required, the source can be accounted for in the wave field by considering
`the initial, time-dependent conditions.
`With aerodynamic sound, the sources of sound are not so readily identifiable and
`the regions of interest can be either within the fluid flow itself or external to it. When
`the regions of interest are within the fluid flow, they contain sources of sound energy
`because the sources are continuously being generated or convected with the flow (e.g.
`turbulence, vortices, etc.). These aerodynamic sources therefore have to be included in
`the wave equation for any subsequent analysis of the sound waves in order that they can
`
`128
`
`Page 15 of 79
`
`

`

`129
`
`2.1 Introduction
`
`Fig. 2.1. A schematic model of internal aerodynamic sound and external structure-borne sound in a
`gas pipeline.
`
`be correctly identified. The wave equation is now inhomogeneous (because it includes
`these source terms) and its solution is somewhat different to that of the homogeneous
`wave equation in that it now describes both the source and the wave fields.
`It is very important to be aware of and to understand the difference between the
`homogeneous and the inhomogeneous wave equations for the propagation of sound
`waves in fluids. The vast majority of engineering noise and vibration control relates
`to sources which can be readily identified, and regions of interest which are outside
`the source region – in these cases the homogeneous wave equation is sufficient to
`describe the wave field and the subsequent noise radiation. Most machinery noise, for
`instance, is associated with the vibration of solid bodies. Engineers should, however, be
`aware of the existence of the inhomogeneous wave equation and of the instances when
`it has to be used in place of the more familiar (and easier to solve!) homogeneous wave
`equation.
`A good industrial example that combines both types of noise generation mechanisms
`is high speed gas flow in a pipeline. Such pipelines are typically found in oil refineries
`and liquid-natural-gas plants. A typical section of pipeline is illustrated in Figure 2.1.
`Inside the pipe there are pressure fluctuations which are caused by turbulence, sound
`waves generated at the flow discontinuities (e.g. bends, valves, etc.), and vortices which
`are convected downstream of some buff body such as a butterfly valve splitter-plate.
`These pressure fluctuations result in internal pipe flow noise which is aerodynamic in
`
`Page 16 of 79
`
`

`

`130
`
`2 Sound waves: a review of some fundamentals
`
`Fig. 2.2. Typical noise and vibrations paths for a machinery source. (Adapted from Pickles2.10.)
`
`nature – the source of sound is distributed along the whole length of the inside of the pipe.
`If an analysis of the sound sources within the pipe is required, the inhomogeneous wave
`equation would have to be used. This internal aerodynamic noise excites the structure
`internally and the vibrating structure subsequently radiates noise to the surrounding
`external medium. The source of sound is not in the region of space under analysis
`for the external noise radiation, and this problem can therefore be handled with the
`homogeneous wave equation. A knowledge of the internal source field (the wall pressure
`fluctuations) allows for a prediction of the external sound radiation; the converse is,
`however, not true. A point which is sometimes overlooked is that a description of the
`external wave field does not contain sufficient information for the source to be identified,
`but, once the source has been identified and described, the sound field can be predicted.
`Pipe flow noise is discussed in chapter 7 as a case study.
`
`Page 17 of 79
`
`

`

`131
`
`2.1 Introduction
`
`Fig. 2.3. Schematic illustration of structure-borne sound in a building.
`
`Typical machinery noise control problems in industry involve (i) a source, (ii) a path,
`and (iii) a receiver. There is always interaction and feedback between the three, and there
`are generally several possible noise and vibration energy transmission paths for a typical
`machinery noise source. An internal combustion engine, for instance, generates both
`aerodynamic and mechanical energy, each with several possible transmission paths.
`This is illustrated schematically in Figure 2.2. The two sources of sound energy are (i) the
`aerodynamic energy associated with the combustion process and the exhaust system,
`and (ii) mechanical vibration energy associated with the various functional requirements
`of the engine. Source modification to reduce the aerodynamic noise component would
`require changes in the combustion process itself or in the design of the exhaust system.
`Source modification to reduce the mechanical vibration energy would require a re-
`design of the moving parts of the engine itself. Various options are open for the reduction
`of path noise. These include muffling the exhaust noise, structural modification such
`as adding mass, stiffness or damping to the various radiating panels, providing anti-
`vibration mounts, enclosing the engine, and providing acoustic barriers. Finally, the
`receiver could be provided with personal protection such as an enclosure or hearing
`protectors.
`A specific example of structure-borne sound is the vibration and stop–start shocks
`that can emanate from a lift if it is not properly isolated. This is illustrated in Figure 2.3.
`In cases such as these, the vibrations are transmitted throughout the building – the
`waves are carried for large distances without being significantly attenuated. There are
`
`Page 18 of 79
`
`

`

`132
`
`2 Sound waves: a review of some fundamentals
`
`Fig. 2.4. Schematic illustration of aerodynamic sound emanating from a jet nozzle.
`
`no sources of sound in the ambient air in the building, and any acoustic analysis would
`only require usage of the homogeneous wave equation. The problem could be overcome
`by isolating the winding machinery from the rest of the structure or by separating the
`lift shaft and the winding machinery from the remainder of the building.
`A specific example of aerodynamic noise is the formation of turbulence in the mixing
`region at the exhaust of a jet nozzle such as the nozzle of a jet used for cleaning
`machine components with compressed air. The jet noise increases with flow velocity
`and the strength of the turbulence is related to the relative speed of the jet in relation to
`the ambient air. By introducing a secondary, low velocity, air-stream, as illustrated in
`Figure 2.4, and thus reducing the velocity profile across the jet, significant reductions
`in radiated noise levels can be achieved. Hence, compound nozzles are sometimes used
`in industry – here, the velocity of the core jet remains the same but its noise radiating
`characteristics are reduced by the introduction of a slower outer stream.
`
`2.2
`
`The homogeneous acoustic wave equation – a classical analysis
`
`Three methods are available for approaching problems in acoustics. They are (i) wave
`acoustics, (ii) ray acoustics and (iii) energy acoustics.
`
`Page 19 of 79
`
`

`

`133
`
`2.2 The homogeneous wave equation
`
`Wave acoustics is a description of wave propagation using either molecular or par-
`ticulate models. The general preference is for the particulate model, a particle being
`a fluid volume large enough to contain millions of molecules and small enough such
`that density, pressure and temperature are constant. Ray acoustics is a description of
`wave propagation over large distances, e.g. the atmosphere. Families of rays are used to
`describe the propagation of sound waves and inhomogeneities such as temperature gra-
`dients or wind have to be accounted for. Over large distances, the ray tracing procedures
`are preferred because they approximate and simplify the exact wave approach. Finally,
`energy acoustics describes the propagation of sound waves in terms of the transfer
`of energy of various statistical parameters where techniques referred to as statistical
`energy analysis (or S.E.A.) are used.
`The wave acoustics approach is probably the most fundamental and important ap-
`proach to the study of all disciplines of acoustics. The ray acoustics approach generally
`relates to outdoor or underwater sound propagation over large distances and is therefore
`not directly relevant to industrial noise and vibration control. The S.E.A. approach is
`fast becoming popular for quick and effective answers to complex industrial noise and
`vibration problems. The wave acoustics approach will thus be adopted for the better
`part of this book, and this chapter is devoted to some of the more important fundamental
`principles of sound waves. The subject of ray acoustics is not discussed in this book,
`but the concepts and applications of statistical energy analysis techniques are discussed
`in chapter 6.
`Sound waves in non-viscous (inviscid) fluids are simply longitudinal waves and
`adjacent regions of compression and rarefaction are set up – i.e. the particles oscillate
`to and fro in the wave propagation direction, hence the acoustic particle velocity is in
`the same direction as the phase velocity. The pressure change that is produced as the
`fluid compresses and expands is the source of the restoring force for the oscillatory
`motion. There are four variables that are of direct relevance to the study of sound
`waves. They are pressure, P, velocity, (cid:1)U , density, ρ, and temperature, T . Pressure,
`density and temperature are scalar quantities whilst velocity is a vector quantity (i.e. an
`arrow over a symbol denotes a vector quantity). Each of the four variables has a mean
`and a fluctuating component. Thus,
`P((cid:1)x , t) = P0((cid:1)x) + p((cid:1)x , t),
`(cid:1)U ((cid:1)x , t) = (cid:1)U 0((cid:1)x) + (cid:1)u((cid:1)x , t),
`ρ((cid:1)x , t) = ρ0((cid:1)x) + ρ(cid:2)
`((cid:1)x , t),
`T ((cid:1)x , t) = T0((cid:1)x) + T
`((cid:1)x , t).
`(cid:2)
`The wave equation can thus be set up in terms of any one of these four variables.
`In acoustics, it is the pressure fluctuations, p((cid:1)x , t), that are of primary concern – i.e.
`noise radiation is a fluctuating pressure. Thus it is common for acousticians to solve
`the wave equation in terms of the pressure as a dependent variable. It is, however, quite
`valid to solve the wave equation in terms of any of the other three variables. Also,
`
`(2.1a)
`(2.1b)
`(2.1c)
`(2.1d)
`
`Page 20 of 79
`
`

`

`134
`
`2 Sound waves: a review of some fundamentals
`
`generally, (cid:1)U 0((cid:1)x) is zero (i.e. the ambient fluid is stationary) and therefore (cid:1)U ((cid:1)x , t) =
`(cid:1)u((cid:1)x , t).
`As for wave propagation in solids, several simplifying assumptions need to be made.
`They are:
`(1) the fluid is an ideal gas;
`(2) the fluid is perfectly elastic – i.e. Hooke’s law holds;
`(3) the fluid is homogeneous and isotropic;
`(4) the fluid is inviscid – i.e. viscous-damping and heat conduction terms are neglected;
`(5) the wave propagation through the fluid media is adiabatic and reversible;
`(6) gravitational effects are neglected – i.e. P0 and ρ0 are assumed to be constant;
`(7) the fluctuations are assumed to be small – i.e. the system behaves linearly.
`In order to develop the acoustic wave equation, equations describing the relationships
`between the various acoustic variables and the interactions between the restoring forces
`and the deformations of the fluid are required. The first such relationship is referred to as
`continuity or the conservation of mass; the second relationship is referred to as Euler’s
`force equation or the conservation of momentum; and the third relationship is referred
`to as the thermodynamic equation of state. In practice, sound waves are generally three-
`dimensional. It is, however, convenient to commence with the derivation of the above
`equations in one dimension and to subsequently extend the results to three dimensions.
`
`2.2.1
`
`Conservation of mass
`
`The equation of conservation of mass (continuity) provides a relationship between the
`density, ρ((cid:1)x , t), and the particle velocity, (cid:1)u((cid:1)x , t) – i.e. it relates the fluid motion to its
`compression.
`Consider the mass flow of particles in the x-direction through an elemental, fixed,
`control volume, dV , as illustrated in Figure 2.5. For mass to be conserved, the time rate of
`change of the elemental mass has to equal the nett mass flow into the elemental volume.
`Because the flow is one-dimensional, the vector notation is temporarily dropped. It will
`be re-introduced later on when the equations are extended to three-dimensional flow.
`Note that
`(cid:1)k,
`(cid:1)j + uz
`(cid:1)i + u y
`(cid:1)u = ux
`where ux , u y and uz are the particle velocities in the x-, y- and z-directions, respectively.
`For flow in the x-direction only:
`(i) the elemental mass is ρ A dx (where A = dy dz);
`(ii) the mass flow into the elemental volume is (ρu A)x ;
`(iii) the mass flow out of the elemental volume is (ρu A)x+dx .
`For the conservation of mass,
`= (ρu A)x − (ρu A)x+dx .
`∂(ρ A dx)
`∂t
`
`(2.2)
`
`Page 21 of 79
`
`

`

`135
`
`2.2 The homogeneous wave equation
`
`Fig. 2.5. Mass flow of particles in the x-direction through an elemental, fixed, control volume.
`
`(cid:1)
`
`Using a Taylor series expansion,
`(ρu A)x − (ρu A)x − ∂(ρu A)x
`=
`∂ x
`
`∂(ρ A dx)
`∂t
`
`(cid:2)
`
`dx
`
`.
`
`(2.3)
`
`= 0.
`
`(2.4)
`
`(2.5)
`
`(2.6)
`
`Hence,
`+ ∂(ρux )
`∂ρ
`∂t
`∂ x
`Equation (2.4) represents the one-dimensional conservation of mass in the x-direction.
`It can be extended to three dimensions, and the three-dimensional equation of conser-
`vation of mass is therefore
`+ (cid:1)∇ · ρ(cid:1)u = 0,
`∂ρ
`∂t
`where (cid:1)∇ is the divergence operator, i.e.
`(cid:1)
`(cid:2)
`(cid:1)i + ∂
`(cid:1)j + ∂
`(cid:1)k
`(cid:1)∇ =
`∂y
`∂z
`
`∂
`∂ x
`
`.
`
`Equation (2.5) is thus a vector representation for
`+ ∂(ρux )
`+ ∂(ρu y)
`+ ∂(ρuz)
`= 0.
`∂ x
`∂y
`∂z
`
`∂ρ
`∂t
`
`(2.7)
`
`The equation of conservation of mass (equations 2.5 or 2.7) is a scalar quantity. It is
`also non-linear because the mass flow terms involve products of two small fluctuating
`components ((cid:1)u and ρ(cid:2)
`). These terms have second-order effects as far as the propagation
`of sound waves is concerned – i.e. the equation can be linearised. Substituting for
`ρ((cid:1)x , t) = ρ0((cid:1)x) + ρ(cid:2)
`((cid:1)x , t) into equation (2.7) and deleting second- and higher-order
`terms yields
`∂ρ(cid:2)
`+ ρ0
`∂t
`
`∂ux
`∂ x
`
`+ ρ0
`
`∂u y
`∂y
`
`+ ρ0
`
`∂uz
`∂z
`
`= 0,
`
`(2.8)
`
`Page 22 of 79
`
`

`

`136
`
`2 Sound waves: a review of some fundamentals
`
`Fig. 2.6. Momentum balance in the x-direction for an elemental, fixed, control volume.
`
`+ ρ0
`
`or
`∂ρ(cid:2)
`∂t
`Equation (2.9) is the linearised equation of conservation of mass (continuity).
`
`(cid:1)∇ · (cid:1)u = 0.
`
`(2.9)
`
`2.2.2
`
`Conservation of momentum
`
`The equation of conservation of momentum provides a relationship between the pres-
`sure, P((cid:1)x , t), the density, ρ((cid:1)x , t), and the particle velocity, (cid:1)u((cid:1)x , t). It can be obtained
`either by observing the stated law of conservation of momentum with respect to an
`elemental, fixed, control volume, dV , in space, or by a direct application of Newton’s
`second law with respect to the fluid particles that move through the elemental, fixed,
`control volume. To a purist both procedures are identical! It is, however, instructive to
`consider them both.
`Consider the first approach – consider the momentum flow through an elemental,
`fixed, control volume, dV , as illustrated in Figure 2.6. For momentum to be conserved
`the time rate of change of momentum contained in the fixed volume plus the nett rate
`of flow of momentum through the surfaces of the volume are equal to the sum of all
`the forces acting on the volume. Once again, because the flow is one-dimensional,
`the vector notation is temporarily dropped. Also, body forces are neglected – i.e. only
`pressure forces act on the body. For flow in the x-direction only:
`(i) the momentum of the control volume is ρux A dx;
`(ii) the momentum flow into the control volume is (ρu2 A)x ;
`(iii) the momentum flow out of the control volume is (ρu2 A)x+dx ;
`(iv) the force at position x is (P A)x ;
`(v) the force at position x + dx is − (P A)x+dx .
`For the conservation of momentum
`= (ρu2 A)x − (ρu2 A)x+dx + (P A)x − (P A)x+dx .
`∂(ρu x A) dx
`∂t
`
`(2.10)
`
`Page 23 of 79
`
`

`

`137
`
`2.2 The homogeneous wave equation
`
`If a Taylor series expansion is used for (ρu2 A)x+dx and (P A)x+dx , equation (2.10)
`simplifies to
`
`(cid:4)
`
`(cid:3)
`
`ρu2
`x
`∂ x
`
`− ∂ P
`∂ x
`
`.
`
`(cid:2)
`
`(2.11)
`
`(2.12)
`
`∂ux
`∂ x
`
`+ ∂ P
`∂ x
`
`= 0,
`
`= − ∂
`(cid:1)
`
`∂(ρux )
`∂t
`
`ρ
`
`∂ux
`∂t
`
`∂ρ
`∂t
`
`∂ρ
`∂ x
`
`∂ux
`∂ x
`
`Equation (2.11) can be re-arranged as
`+ ux
`+ ux
`+ ρ
`+ ρux
`
`where the term in brackets is the continuity equation. Thus, equation (2.12) sim-
`plifies to
`+ ρux
`
`ρ
`
`∂ux
`∂t
`
`∂ux
`∂ x
`
`+ ∂ P
`∂ x
`
`= 0.
`
`(2.13)
`
`Equation (2.13) represents the one-dimensional conservation of momentum in the
`x-direction. Similar expressions can be obtained for the y- and z-directions. The three-
`dimensional equation of conservation of momentum is therefore obtained by intro-
`ducing the divergence operator, (cid:1)∇. It is
`(cid:1)
`(cid:2)
`∂ (cid:1)u
`+ ((cid:1)u · (cid:1)∇)(cid:1)u
`+ (cid:1)∇ P = 0.
`∂t
`
`(2.14)
`
`ρ
`
`This eq