Despite their novelty, wavelets have a tremendous impact on a number of modern scientific disciplines, particularly on signal and image analysis. Because of their powerful underlying mathematical theory, they offer exciting opportunities for the design of new multi-resolution processing algorithms and effective pattern recognition systems. This book provides a much-needed overview of current trends in the practical application of wavelet theory. It combines cutting edge research in the rapidly developing wavelet theory with ideas from practical signal and image analysis fields. Subjects dealt with include balanced discussions on wavelet theory and its specific application in diverse fields, ranging from data compression to seismic equipment. In addition, the book offers insights into recent advances in emerging topics such as double density DWT, multiscale Bayesian estimation, symmetry and locality in image representation, and image fusion. Audience: This volume will be of interest to graduate students and researchers whose work involves acoustics, speech, signal and image processing, approximations and expansions, Fourier analysis, and medical imaging.

This self-contained book is devoted to the study of the acoustic wave equations and the Maxwell system, the two most common waves equations that are encountered in physics or in engineering. It presents a detailed analysis of their mathematical and physical properties. In particular, the author focuses on the study of the harmonic exterior problems, building a mathematical framework which provides the existence and uniqueness of the solutions. This book will serve as a useful introduction to wave problems for graduate students in mathematics, physics, and engineering.

In the past few years we have written and edited several books in the area of acousticandspeechsignalprocessing. Thereasonbehindthisendeavoristhat there were almost no books available in the literature when we ?rst started while there was (and still is) a real need to publish manuscripts summarizing the most useful ideas, concepts, results, and state-of-the-art algorithms in this important area of research. According to all the feedback we have received so far, we can say that we were right in doing this. Recently, several other researchers have followed us in this journey and have published interesting books with their own visions and perspectives. The idea of writing a book on Microphone Array Signal Processing comes from discussions we have had with many colleagues and friends. As a c- sequence of these discussions, we came up with the conclusion that, again, there is an urgent need for a monograph that carefully explains the theory and implementation of microphone arrays. While there are many manuscripts on antenna arrays from a narrowband perspective (narrowband signals and narrowband processing), the literature is quite scarce when it comes to s- sor arrays explained from a truly broadband perspective. Many algorithms for speech applications were simply borrowed from narrowband antenna - rays. However, a direct application of narrowband ideas to broadband speech processing may not be necessarily appropriate and can lead to many m- understandings.

"Completing Transition: The Main Challenges" was the topic around which the Oesterreichische Nationalbank and the Joint Vienna Institute organized a high-level conference in 2000, in a continuation of long-standing efforts to promote the dialogue and understanding between various regions in Europe. Given the heterogeneity of the transition countries of Central and Eastern Europe and the heterogeneity of progress toward convergence, the outlook for finishing transition is divergent. However, what will generally be important is corporate governance and institutional reform to sufficiently underpin macroeconomic success, plus a definite commitment of the responsible institutions in the transition countries to follow the chosen policies consistently.

This illustrated guide to 100 of the world's most important concert halls and opera houses examines their architecture and engineering and discusses their acoustical quality as judged by conductors and music critics. The descriptions and photographs will serve as a valuable guide for today's peripatetic performers and music lovers. With technical discussions relegated to appendices, the book can be read with pleasure by anyone interested in musical performance. The photographs (specially commissioned for this book) and architectural drawings (all to the same scale) together with modern acoustical data on each of the halls provide a rich and unmatched resource on the design of halls for presenting musical performances. Together with the technical appendices, the data and drawings will serve as an invaluable reference for architects and engineers involved in the design of spaces for the performance of music.

These Conference Proceedings are intended to summarise the latest developments in diffraction and scattering theory as reported at the IU TAM Symposium on Diffraction and Scattering in Fluid Mechanics and Elasticity held in Manchester, England on 16-20 July 2000. This in formal meeting was organised to discuss mathematical advances, both from the theoretical and more applied points of view. However, its pri mary goal was to bring together groups of researchers working in dis parate application areas, but who nevertheless share common models, phenomenological features arising in such problems, and common math ematical tools. To this end, we were delighted to have four Plenary Speakers, Professors Allan Pierce, Ed Kerschen, Roger Grimshaw and John Willis FRS, who are undisputed leaders in the four thematic ar eas of our meeting (these are respectively acoustics, aeroacoustics, water or other free surface waves, elasticity). These Proceedings should offer an excellent vehicle for continuing the dialogue between these groups of researchers. The participants were invited because of their expertise and recent contributions to this field. Collectively, there were around 90 contrib utors to the Symposium from some 13 countries located all around the world. These included 45 speakers, 35 co-authors and about 10 other delegates. Individuals came from many of the major international cen tres of excellence in the field of scattering theory."

This volume presents the Proceedings of the Seventh International Conference on Vibration Problems, held in Istanbul, Turkey, September 5-9, 2005. The main objective being to stimulate a broad interdisciplinary research. The topics covered in the book vary from the effect of ground motion on the stochastic response of suspension bridges to coupling effects between different vibrations in rotor-blade systems.

Engineering the Guitar: Theory and Practice uniquely describes the mechanics of the guitar for engineers and craftsmen alike. Complete with informative illustrations, this popular musical volume describes the underlying mechanical concepts behind the guitar, supported by theory and test. A detailed description of guitar electronics paired with an analysis of sound quality appeals to scientific audiences as well as musicians technically apt. Readers will gain an understanding of the technical behavior of the instrument with respect to structural and component dynamics, in addition to the informative treatment of analytical models. Hand made and mass produced techniques are also examined in a chapter devoted to manufacturing processes.

Audiences interested in mechanics, acoustics, and instrument making will find Engineering the Guitar: Theory and Practice an informative and enjoyable read.

Detailed report on a topic that has already attracted much popular interest.

Provides fascinating reading for physicists, biologists and general readers alike.

Trees can reduce noise by sound reflection and absorption and this is the first book bringing together the widely scattered literature on noise abatement by urban trees. The book will interest those concerned with environmental management, noise control, and urban forestry. It is an invaluable source of information for environmental managers, foresters, acousticians, engineers, architects, scientists, and students.

Users of signal processing systems are never satis?ed with the system they currently use. They are constantly asking for higher quality, faster perf- mance, more comfort and lower prices. Researchers and developers should be appreciative for this attitude. It justi?es their constant e?ort for improved systems. Better knowledge about biological and physical interrelations c- ing along with more powerful technologies are their engines on the endless road to perfect systems. This book is an impressive image of this process. After "Acoustic Echo 1 and Noise Control" published in 2004 many new results lead to "Topics in 2 Acoustic Echo and Noise Control" edited in 2006 . Today - in 2008 - even morenew?ndingsandsystemscouldbecollectedinthisbook.Comparingthe contributions in both edited volumes progress in knowledge and technology becomesclearlyvisible: Blindmethodsandmultiinputsystemsreplace"h- ble" low complexity systems. The functionality of new systems is less and less limited by the processing power available under economic constraints. The editors have to thank all the authors for their contributions. They cooperated readily in our e?ort to unify the layout of the chapters, the ter- nology, and the symbols used. It was a pleasure to work with all of them. Furthermore, it is the editors concern to thank Christoph Baumann and the Springer Publishing Company for the encouragement and help in publi- ing this book.

Noise is everywhere and in most applications that are related to audio and speech, such as human-machine interfaces, hands-free communications, voice over IP (VoIP), hearing aids, teleconferencing/telepresence/telecollaboration systems, and so many others, the signal of interest (usually speech) that is picked up by a microphone is generally contaminated by noise. As a result, the microphone signal has to be cleaned up with digital signal processing tools before it is stored, analyzed, transmitted, or played out. This cleaning process is often called noise reduction and this topic has attracted a considerable amount of research and engineering attention for several decades. One of the objectives of this book is to present in a common framework an overview of the state of the art of noise reduction algorithms in the single-channel (one microphone) case. The focus is on the most useful approaches, i.e., filtering techniques (in different domains) and spectral enhancement methods. The other objective of Noise Reduction in Speech Processing is to derive all these well-known techniques in a rigorous way and prove many fundamental and intuitive results often taken for granted. This book is especially written for graduate students and research engineers who work on noise reduction for speech and audio applications and want to understand the subtle mechanisms behind each approach. Many new and interesting concepts are presented in this text that we hope the readers will find useful and inspiring.

TheCarg'eseSummerSchool"Sound-?owinteractions"washeldinthe- stitutd'EtudesScienti?quesdeCarg'eseinCorsica,Francefrom19thJune to1stJuly,2000. Theunderstandingofsoundand?owinteractionshasmadesomerema- ableprogresssincethepioneeringworksoftheRussianandBritishschools, inthe1950s. Inaddition,thegrowingavailabilityduringthepast10years ofsophisticatedcomputer/electronics/materialstechniquesallowsforthe- velopmentofagrowingnumberofapplicationsaswellasthepossibilityof addressingnewfundamentalproblems. Thecouplingbetweenacousticwaves and?owmotionisbasicallynonlinear,sothatthesoundpropagationand generationismodi?edbythe?owandthe?owcanalsobemodi?edbythe sound. Asaresult,thisproblemisinvestigatedinmanydi?erentscienti?c communities,suchasappliedmathematics,acousticsand?uidmechanics, amongothers. Inouropinion,thetimehadcometotrytogatherthe- searchersinthedi?erentcommunitiestogetherinatutorialenvironemnt. So, thisschoolbroughttogetherworldwidespecialistsinordertopresentvarious aspectsofsound-?owinteractions,andshareexpertiseandmethodologiesso astopromotecross-fertilisation. ThebasicknowledgeintheareaisintroducedbyA. HirschbergandC. Schram. Hepresentstheaeroacousticsofinternal?owinaverylivelyway withalotofillustrationdevices. Heintroducesaeroacousticanalogiesand applicationslikemusicalinstruments,theRijketube,speechproductionetc. M. S. Howeintroducesthetheoryofvortexsoundinaverydidacticway. From Lighthill'sacousticanalogy,heshowshowvorticityandentropy?uctuations canbeseenassourcesofsound. Then,usingthecompactGreen'sfunctions, heshowshowtocomputethevortexsound. Asanexampleofthemethod presented,heappliesthistheorytopressuretransientsgeneratedbyhi- speedtrains. F. Lundgivesthebasicequationsofsound-?owinteractions. Thenheintroducesveryclearlythescatteringofsoundbecauseofvorticity andgivesthemostrecentresultsonultrasoundpropagationthroughadis- dered?ow. V. Ostashevpresentsgeometricalacousticsinmovingmediaand theimportantpracticalproblemofsoundpropagationinturbulence(at- sphere,ocean). A. Fabrikantexaminestheplasma-hydrodynamicsanalogies includingtheresonantwave-?owinteractioninshear?ows,wavesofnegative VI Preface energyandover-re?ectionandacousticoscillatorsin?uid?ows. P. J. Mor- sondescribesthedynamicsofthecontinuousspectrumwhichoccursinshear ?ow. Theresultsareinterpretedinthecontextofin?nitedimensionalHam- toniansystemstheory. G. Chagelishvilipresentsnewlinearmechanismsof acousticwavegenerationinsmoothshear?owsusinganon-modalstudy. N. Peakepresents?uid-structureinteractionsinthepresenceofmean?ows, includingtheproblemsofinstabilityandcausality. Finally,W. Lauterborn presentsnonlinearacousticswithapplicationstosonoluminescenceandto acousticchaos. InthisCarg'eseSummerSchool,54studentsfrom12nations,and11l- turersfrom7nationsparticipated. Aknowledgements. TheSummerSchoolandthispublicationwouldnot havebeenpossiblewithout: *?nancialsupportfromtheEuropeanUnion,theCentreNationaldela RechercheScienti?que,theMinist'eredesA?airesEtrang'eres,theM- ist'eredel'EducationNationale,delaRechercheetdelaTechnologieand theGroupementdeRecherche"Turbulence"; *the guidance of Elisabeth Dubois-Violette, director of the Institut d'EtudesScienti?quesdeCarg'ese; *thehelpofChantalAriano,NathalieBedjai,BrigitteCassegrain,Pierre- EricGrossiandthewholeteaminpreparingandhostingofthisschool. Finally,wewishtothankthelecturersforgivingsomuchtimeinprep- ingthelecturesandwritingthemup,aswellasmakingthemselvesavailable fordiscussionsduringtheschool. 1 LeMans,Paris,Lyon YvesAur'egan , 2 September2001 Agn'esMaurel , 1 VincentPagneux , 3 Jean-Fran,coisPinton . 1 Laboratoired'Acoustiquedel'Universit'eduMaine,UMRCNRS6613, Av. OMessiaen,72085LeMansCedex9,France 2 LaboratoireOndesetAcoustique,UMRCNRS7587, ESPCI,10rueVauquelin,75005Paris,France 3 LaboratoiredePhysique,UMRCNRS1325, EcoleNormaleSup'erieuredeLyon,46all'eed'Italie,69007Lyon,France Preface VII SomeofthelecturersoftheCarg'eseSchool,fromlefttoright:M. S. Howe,A. Hirschberg,P. Morrison,W. Lauterborn,V. Ostashev,A. Fabrikant,N. Peake, T. Colonius(PhotoC. Schram) SomeoftheparticipantsoftheCarg'eseSchool(PhotoC. Schram) TableofContents APrimitiveApproachtoAeroacoustics AvrahamHirschberg,ChristopheSchram...1 1 Introduction ...1 2 FluidDynamics ...2 3 Lighthill'sAnalogy...4 4 JetNoise ...7 5 Thermo-Acoustics ...9 6 AcousticalEnergy ...10 7 Rijke-Tube...11 8 Vortex-SoundTheory ...14 9 ChoiceoftheGreen'sFunction...17 10 Howe'?owinteractions,andshareexpertiseandmethodologiesso astopromotecross-fertilisation. ThebasicknowledgeintheareaisintroducedbyA. HirschbergandC. Schram. Hepresentstheaeroacousticsofinternal?owinaverylivelyway withalotofillustrationdevices. Heintroducesaeroacousticanalogiesand applicationslikemusicalinstruments,theRijketube,speechproductionetc. M. S. Howeintroducesthetheoryofvortexsoundinaverydidacticway. From Lighthill'sacousticanalogy,heshowshowvorticityandentropy?uctuations canbeseenassourcesofsound. Then,usingthecompactGreen'sfunctions, heshowshowtocomputethevortexsound. Asanexampleofthemethod presented,heappliesthistheorytopressuretransientsgeneratedbyhi- speedtrains. F. Lundgivesthebasicequationsofsound-?owinteractions. Thenheintroducesveryclearlythescatteringofsoundbecauseofvorticity andgivesthemostrecentresultsonultrasoundpropagationthroughadis- dered?ow. V. Ostashevpresentsgeometricalacousticsinmovingmediaand theimportantpracticalproblemofsoundpropagationinturbulence(at- sphere,ocean). A. Fabrikantexaminestheplasma-hydrodynamicsanalogies includingtheresonantwave-?owinteractioninshear?ows,wavesofnegative VI Preface energyandover-re?ectionandacousticoscillatorsin?uid?ows. P. J. Mor- sondescribesthedynamicsofthecontinuousspectrumwhichoccursinshear ?ow. Theresultsareinterpretedinthecontextofin?nitedimensionalHam- toniansystemstheory. G. Chagelishvilipresentsnewlinearmechanismsof acousticwavegenerationinsmoothshear?owsusinganon-modalstudy. N. Peakepresents?uid-structureinteractionsinthepresenceofmean?ows, includingtheproblemsofinstabilityandcausality. Finally,W. Lauterborn presentsnonlinearacousticswithapplicationstosonoluminescenceandto acousticchaos. InthisCarg'eseSummerSchool,54studentsfrom12nations,and11l- turersfrom7nationsparticipated. Aknowledgements. TheSummerSchoolandthispublicationwouldnot havebeenpossiblewithout: *?nancialsupportfromtheEuropeanUnion,theCentreNationaldela RechercheScienti?que,theMinist'eredesA?airesEtrang'eres,theM- ist'eredel'EducationNationale,delaRechercheetdelaTechnologieand theGroupementdeRecherche"Turbulence"; *the guidance of Elisabeth Dubois-Violette, director of the Institut d'EtudesScienti?quesdeCarg'ese; *thehelpofChantalAriano,NathalieBedjai,BrigitteCassegrain,Pierre- EricGrossiandthewholeteaminpreparingandhostingofthisschool. Finally,wewishtothankthelecturersforgivingsomuchtimeinprep- ingthelecturesandwritingthemup,aswellasmakingthemselvesavailable fordiscussionsduringtheschool. 1 LeMans,Paris,Lyon YvesAur'egan , 2 September2001 Agn'esMaurel , 1 VincentPagneux , 3 Jean-FranccoisPinton . 1 Laboratoired'Acoustiquedel'Universit'eduMaine,UMRCNRS6613, Av. OMessiaen,72085LeMansCedex9,France 2 LaboratoireOndesetAcoustique,UMRCNRS7587, ESPCI,10rueVauquelin,75005Paris,France 3 LaboratoiredePhysique,UMRCNRS1325, EcoleNormaleSup'erieuredeLyon,46all'eed'Italie,69007Lyon,France Preface VII SomeofthelecturersoftheCarg'eseSchool,fromlefttoright:M. S. Howe,A. Hirschberg,P. Morrison,W. Lauterborn,V. Ostashev,A. Fabrikant,N. Peake, T. Colonius(PhotoC. Schram) SomeoftheparticipantsoftheCarg'eseSchool(PhotoC. Schram) TableofContents APrimitiveApproachtoAeroacoustics AvrahamHirschberg,ChristopheSchram...1 1 Introduction ...1 2 FluidDynamics ...2 3 Lighthill'sAnalogy...4 4 JetNoise ...7 5 Thermo-Acoustics ...9 6 AcousticalEnergy ...10 7 Rijke-Tube...11 8 Vortex-SoundTheory ...14 9 ChoiceoftheGreen'sFunction...17 10 Howe'sEnergyCorollary ...20 11 TheOpenPipeTerminationofanUn?angedPipe ...21 12 Whistler-NozzleandHumanWhistling ...25 13 Conclusion...27 References...28 LecturesontheTheoryofVortex-Sound MichaelS. Howe...31 1 AerodynamicSound...31 1. 1 Lighthill'sAcousticAnalogy(1952)...31 1. 2 AerodynamicSoundfromLow-Mach-NumberTurbulence ofUniformMeanDensity...34 1. 3 AerodynamicSoundfromLow-Mach-NumberTurbulence ofVariableMeanDensity...35 2 VorticityandEntropyFluctuations asSourcesofSound...37 2. 1 TheRoleofVorticityinLighthill'sTheory...37 2. 2 AcousticAnalogyinTermsoftheTotalEnthalpy...39 2. 3 VorticityandEntropySources...40 3 FundamentalSolutionsoftheWaveEquation...43 3. 1 TheHelmholtzEquation...43 3. 2 TheWaveEquation...46 4 GeneralSolutionoftheInhomogeneousWaveEquation...47 4. 1 GeneralSolutionintheFrequency-Domain...47 X TableofContents 4. 2 GeneralSolutionintheTime-Domain...49 5 CompactGreen'sFunctions...

New material treats such contemporary subjects as automatic speech recognition and speaker verification for banking by computer and privileged (medical, military, diplomatic) information and control access.

The book also focuses on speech and audio compression for mobile communication and the Internet. The importance of subjective quality criteria is stressed.

The book also contains introductions to human monaural and binaural hearing, and the basic concepts of signal analysis.

Beyond speech processing, this revised and extended new edition of Computer Speech gives an overview of natural language technology and presents the nuts and bolts of state-of-the-art speech dialogue systems.

This book brings together many advanced topics in network and acoustic echo cancellation aimed towards enhancing the echo cancellation performance of next-generation telecommunication systems. The resulting compendium provides a coherent treatment of such topics not found otherwise in journals or other books.

Signal Processing for Wireless Communication Systems brings together in one place important contributions and up-to-date research results in this fast moving area. The Contributors to this work were selected from leading researchers and practitioners in this field.
The book's 18 chapters are divided into three areas: systems, Networks, and Implementation Issues; Channel Estimation and Equalization; and Multiuser Detection. The Work, originally published as Volume 30, Numbers 1-3 of the Journal of VLSI Signal Processing Systems for Signal, Image, and Video Technology, will be valuable to anyone working or researching in the field of wireless communication systems. It serves as an excellent reference, providing insight into some of the most challenging issues being examined today.

Humans have always been fascinated by marine life, from extremely small diatoms to the largest mammal that inhabits our planet, the blue whale. However, studying marine life in the ocean is an extremely difficult propo- tion because an ocean environment is not only vast but also opaque to most instruments and can be a hostile environment in which to perform expe- ments and research. The use of acoustics is one way to effectively study animal life in the ocean. Acoustic energy propagates in water more efficiently than almost any form of energy and can be utilized by animals for a variety of purposes and also by scientists interested in studying their behavior and natural history. However, underwater acoustics have traditionally been in the domain of physicists, engineers and mathematicians. Studying the natural history of animals is in the domain of biologists and physiologists. Und- standing behavior of animals has traditionally involved psychologists and zoologists. In short, marine bioacoustics is and will continue to be a diverse discipline involving investigators from a variety of backgrounds, with very different knowledge and skill sets. The inherent inter-disciplinary nature of marine bioacoustics presents a large challenge in writing a single text that would be meaningful to various investigators and students interested in this field. Yet we have embarked on this challenge to produce a volume that would be helpful to not only beginning investigators but to seasoned researchers.

This is an introduction to noise, describing fundamental noise sources and basic circuit analysis, discussing characterization of low-frequency noise and offering practical advice that bridges concepts of noise theory and modelling, characterization, CMOS technology and circuits. The text offers the latest research, reviewing the most recent publications and conference presentations. The book concludes with an introduction to noise in analog/RF circuits and describes how low-frequency noise can affect these circuits.

This book presents a new method of asymptotic analysis of boundary-layer problems, the Successive Complementary Expansion Method (SCEM). The first part is devoted to a general presentation of the tools of asymptotic analysis. It gives the keys to understand a boundary-layer problem and explains the methods to construct an approximation. The second part is devoted to SCEM and its applications in fluid mechanics, including external and internal flows.

Since 1968, the International Acoustical Imaging Symposium has provided a unique forum for advanced research, promoting the sharing of technology, developments, methods and theory among all areas of acoustics. Volume 28 of the Proceedings offers an excellent collection of papers presented in six major categories, offering both a broad perspective on the state of the art in the field as well as an in-depth look at its leading edge research.

This work is the first and only book on the fundamentals of ultrasonic machining. It presents the foundations of dynamic and control for ultrasonic processing systems and considers ultrasonic systems as special vibratory machines that function by exploiting nonlinear dynamic processes. Recommendations are given for designing and tuning ultrasonic machines. The ultrasonic machines analyzed are predominantly concerned with the processing of solids.

This volume constitutes the proceedings of the 1997 IUTAM Symposium, where invited researchers in acoustics, aeronautics, elastodynamics, electromagnetics, hydrodynamics, and mathematics discussed non-reflecting computational boundaries. The participants formulated benchmark problems for evaluating computational boundaries, as described in the first article.

Recent Developments in Time-Frequency Analysis brings together in one place important contributions and up-to-date research results in this fast moving area. Recent Developments in Time-Frequency Analysis serves as an excellent reference, providing insight into some of the most challenging research issues in the field.

Computational Mechanics of the Classical Guitar describes a new dynamic paradigm in instrument acoustics based on time-dependent transient analysis and simulation of complete musical instruments. It describes the current state of theoretical and experimental research into the guitar for engineers, instrument makers and musicians. This includes a summary of the basic equations for the mechanics of vibrating bodies and a presentation of the FDM (finite difference method) model with which the true vibrational behaviour of the instrument as an entire system can be understood for the first time. This monograph presents various new theoretical and experimental results and insights into guitar playing such as the coupling between the strings and the top plate or a description of the finger noise made when the fingers slide over the strings before plucking.

The study of incompressible ?ows is vital to many areas of science and te- nology. This includes most of the ?uid dynamics that one ?nds in everyday life from the ?ow of air in a room to most weather phenomena. Inundertakingthesimulationofincompressible?uid?ows, oneoftentakes many issues for granted. As these ?ows become more realistic, the problems encountered become more vexing from a computational point-of-view. These range from the benign to the profound. At once, one must contend with the basic character of incompressible ?ows where sound waves have been analytically removed from the ?ow. As a consequence vortical ?ows have been analytically "preconditioned," but the ?ow has a certain non-physical character (sound waves of in?nite velocity). At low speeds the ?ow will be deterministic and ordered, i.e., laminar. Laminar ?ows are governed by a balance between the inertial and viscous forces in the ?ow that provides the stability. Flows are often characterized by a dimensionless number known as the Reynolds number, which is the ratio of inertial to viscous forces in a ?ow. Laminar ?ows correspond to smaller Reynolds numbers. Even though laminar ?ows are organized in an orderly manner, the ?ows may exhibit instabilities and bifurcation phenomena which may eventually lead to transition and turbulence. Numerical modelling of suchphenomenarequireshighaccuracyandmostimportantlytogaingreater insight into the relationship of the numerical methods with the ?ow physics.