UNIVERSITE

PIERRE & MARIE

CURIE

LA SCIENCE A P ARIS

` ´ PARIS 6 THESE de DOCTORAT de l’UNIVERSITE Sp´ecialit´e : Acoustique, Traitement du Signal et Informatique Appliqu´es a` la Musique

The Physics of Double-reed Wind Instruments and its Application to Sound Synthesis

Pr´esent´ee par :

Andr´e Almeida

pour obtenir le titre de Docteur de l’Universite´ Paris 6

A soutenir le 26 Juin 2006 devant le jury compose´ de : M. M. M. M. M. M. M.

Xavier RODET Ren´e CAUSSE Christophe VERGEZ Jean-Pierre DALMONT Murray CAMPBELL Benoˆıt FABRE Gary SCAVONE

Directeur de the` se (Universit´e Paris 6, Paris) Directeur de th`ese (IRCAM, Paris) Encadrant (LMA, Marseille) Rapporteur (LAUM, Le Mans) Rapporteur (Universite´ d’Edimbourg) Examinateur (LAM, Paris) Examinateur (Universite´ McGill, Montr´eal)

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To my parents, To my sister, Ana,

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Acknowledgments A thesis is not only the product of the PhD student but of many others that followed the thesis and contributed in a greater or lesser extent to the completion of this project. To all of them I wish to express my profound gratitude. In particular, I want to thank Xavier Rodet, who accepted to supervise this thesis and always trusted my scientific skills throughout this work.To Rene´ Causs´e I wish to thank the co-supervision of this thesis, to have hosted me in the Musical Acoustics team at IRCAM, and for his continuous implication in this thesis especially in the experimental work. To Christophe Vergez, I cannot thank enough, for continuously supporting me throughout this work, for the countless and fruitful scientific discussions both in the more theoretical parts as in the experimental part, they were always the source of interesting ideas that contributed to the accomplishment of my work. Thank you also for the support during the most difficult times of this thesis, and for not having given up during these moments. To the wonderful colleague, the vigorous scientist and the joyful friend, I hope we can share many more years of team work! Still at IRCAM, I would like to leave my word of gratitude to all the colleagues and friends that accompanied me during this journey. Firstly, a huge and warm thank you to Claudia Fritz, with whom I shared the experience of the PhD during a great part of this work, thank you for sharing your ideas, for the great ambiance at ‘‘labo 7’’ and for the friendship that lasted beyond the scientific work. Many thanks also to Aude Liz e´ e, for the joy she brought to our team during her ‘‘DEA’’ internship as well as for her scientific contribution. To the interns that contributed directly to parts of this work, I thank their commitment to the subject, Cle´ m´ent Vern, who contributed to the measurements of the reed dynamics, to Matthias Coulon, who improved the image analysis technique, worked in the measurements of the elastic properties of the reed and gave a substantial help in the anemometry measurements, and finally to Sylvain Hourcade, who contributed in widening the variety of reeds for which the elastic properties were measured. Big thanks to Alain Terrier, always ready to improve the experimental setups with new ideas and for the brilliant execution of many parts needed for the experiments, to Ge´ rard Bertrand for his help with electronics devices and execution of required apparatuses, ideas for experiments and hours of friendly discussions. Thanks to all the colleagues at IRCAM for discussions or help in several specific problems, in particular Thomas H e´ lie, Joël Bensoam, Axel Roebel, and all those at the IRCAM research team whom I can’t thank in particular for lack of space. . . Many other scientists, musicians and instrument makers contributed to this work through discussions in conferences, visits at IRCAM or collaborations in other laboratories. In particular I wish to thank Benoit Fabre, whose collaboration was essential for

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the experiments related to the flow, and who helped in many scientific issues in this thesis. The list continues with Jean-Pierre Dalmont and Joël Gilbert for the interest they showed for this subject and my work since the beginning of this thesis. Thanks to Avraham Hirschberg, who contributed with many fruitful discussions and ideas for the models, to Jean Kergomard, Kees Nederveen and Murray Campbell. I am also very grateful for the interest shown by the musicians, in particular, David Rachor, Jim Kopp and Franck Leblois, whose advises and ideas were very important to interpret and relate many scientific results to the actual use of the double-reed and the instrument. The support for such a long-term project does not only come from the scientific or technical domains. Although most of the people referred to above also contributed with personal support, many other people helped in encouraging me throughout this work. I can only mention here some persons who provided me with constant human support, they are my ‘‘parisian family’’, Bruno, Paula and Beatriz, Andre´ , Vale and Caterina, Rui and Cris (even in Lisbon, a true parisian support!), Ze´ , Catarina and Luca. Thank you to Kristina for the sincere professional concern and advises. Thank you so much to my parents who never let me give up and who always have a kind and effective word to encourage me. Finally, a special thought to Leonor, whose life is probably, after mine, the most influenced by this project. To you, thank you for all the words of encouragement and your belief in my work!

Contents I 1

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Introduction The double-reed 1.1 Generic description of the double reed 1.2 General working principles . . . . . . . 1.3 Fabrication of the double-reed . . . . . 1.4 Utilisation of the double-reed . . . . . 1.4.1 Soaking . . . . . . . . . . . . . 1.4.2 Embouchure . . . . . . . . . . 1.4.3 Articulation and dynamics . . .

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Reed models 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Overview of the chapter . . . . . . . . . . . 2.2 Self-sustained instruments . . . . . . . . . . . . . 2.2.1 Passive and active parts of the instrument . 2.2.2 Wind instruments . . . . . . . . . . . . . . 2.2.3 Reed instruments . . . . . . . . . . . . . . . 2.3 The resonator . . . . . . . . . . . . . . . . . . . . . 2.3.1 Wave equation and wave propagation . . . . 2.3.2 Propagation inside a cylindrical waveguide . 2.3.3 Diffusion and dispersion of the waves . . . . 2.3.4 Resonator termination and radiation . . . . 2.3.5 A complete cylindrical resonator . . . . . . . 2.3.6 Conical resonator models . . . . . . . . . . 2.3.7 Complex geometries . . . . . . . . . . . . . 2.4 The exciter . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Double reeds as opposed to single reeds . . 2.4.2 Description of the double reed . . . . . . . . 2.4.3 Dynamic effects of the reed . . . . . . . . . . 2.4.4 Considerations about the elementary model 2.5 State-of-the-art in double-reed physical modelling .

Characterisation of the double-reed Geometrical and Mechanical aspects of the reed

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3.1 Introduction . . . . . . . . . . . . . . . . . . . 3.1.1 Overview of the chapter . . . . . . . . 3.2 Cross-section profile of the double-reed . . . . 3.3 Geometry of the reed opening . . . . . . . . . 3.3.1 Experimental approaches . . . . . . . 3.3.2 Observations . . . . . . . . . . . . . . 3.3.3 Quantitative analysis . . . . . . . . . . 3.3.4 Symmetry . . . . . . . . . . . . . . . . 3.3.5 Conclusions on the reed opening shape 3.4 Reed equivalent stiffness . . . . . . . . . . . . 3.5 Viscoelastic effects . . . . . . . . . . . . . . . 3.5.1 Synthetic reed . . . . . . . . . . . . . . 3.5.2 Humidified natural cane reed . . . . . 3.6 Partial conclusions . . . . . . . . . . . . . . . 4

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The nonlinear characteristics of the reed 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Context . . . . . . . . . . . . . . . . . . . . . 4.2 Principles of measurement and practical issues . . . 4.2.1 Volume flow measurements . . . . . . . . . . 4.2.2 Practical issues and solutions . . . . . . . . . 4.2.3 Experimental set-up and calibrations . . . . . 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Double-reeds used in this study and operating 4.3.2 The pressure vs flow characteristics . . . . . . 4.3.3 Repeatability . . . . . . . . . . . . . . . . . . 4.3.4 Effect of the added mass . . . . . . . . . . . . 4.3.5 Effect of humidity . . . . . . . . . . . . . . . . 4.3.6 Variations between reeds . . . . . . . . . . . . 4.3.7 Differences between bassoon and oboe reeds . 4.3.8 Comparison with single-reeds . . . . . . . . . 4.3.9 Comparison with synthetic reeds . . . . . . . 4.4 Partial conclusions . . . . . . . . . . . . . . . . . . . 4.4.1 Differences relative to the elementary model . 4.4.2 Other remarks . . . . . . . . . . . . . . . . .

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Details of the flow inside the reed 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . 5.2 Overview . . . . . . . . . . . . . . . . . . . . . . . 5.3 Difficulties in measuring a complete velocity field 5.4 Visualisation of the flow at the reed output . . . . 5.5 Hot-wire measurements . . . . . . . . . . . . . . 5.5.1 Experimental device . . . . . . . . . . . . 5.5.2 Calibration of the hot wire . . . . . . . . . 5.5.3 Double-reeds used in these measurements

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5.6 Velocity profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Variation of diametrical profiles with the mouth pressure . . . 5.6.2 Variation of diametrical profiles with the reed opening . . . . . 5.6.3 Comparison of two perpendicular diameters . . . . . . . . . . 5.7 General discussion on the flow profiles . . . . . . . . . . . . . . . . . 5.8 Volume flow calculations . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Volume flow deduced from the maximum of the velocity profile 5.8.2 Variation of flow with pressure . . . . . . . . . . . . . . . . . . 5.9 Average flows deduced from characteristics measurements . . . . . . 5.9.1 Small openings . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.2 Mean flow velocity estimations . . . . . . . . . . . . . . . . . . 5.10 Static flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.1 Vena contracta . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.2 Singular losses . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.3 Conical diffuser . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.4 Complete model . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Conclusion on the static flow model . . . . . . . . . . . . . . . . . . .

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Auto-oscillations in the double-reed 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Overview of the chapter . . . . . . . . . . . . 6.2 Discussion on dynamic flow effects . . . . . . . . . . 6.2.1 Effects of inertia in the air flow . . . . . . . . 6.2.2 Compressibility . . . . . . . . . . . . . . . . . 6.2.3 Fluid-structure coupling . . . . . . . . . . . . 6.3 Periodic motion of the reed . . . . . . . . . . . . . . . 6.3.1 Two-state movement . . . . . . . . . . . . . . 6.3.2 Time relations between open and closed states 6.3.3 Closed to open transition . . . . . . . . . . . . 6.3.4 Open to closed transition . . . . . . . . . . . . 6.3.5 Oscillations during the open state . . . . . . . 6.3.6 Effect of biting . . . . . . . . . . . . . . . . . . 6.4 Dynamic flow characterization . . . . . . . . . . . . . 6.4.1 Procedure . . . . . . . . . . . . . . . . . . . . 6.4.2 Experimental conditions . . . . . . . . . . . . 6.4.3 Obtained data . . . . . . . . . . . . . . . . . . 6.4.4 Sketch of the flow evolution . . . . . . . . . . 6.4.5 Conjectures on the flow in a coupled reed . .

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Retrospective 7.1 Summary of part II . . . . . . . . . . . . . . . . 7.1.1 Geometrical aspects of the reed opening 7.1.2 Elastic properties . . . . . . . . . . . . . 7.1.3 Vibrational properties . . . . . . . . . . 7.1.4 Non-linear characteristics . . . . . . . .

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7.1.5 Flow inside the reed . . . . . . 7.1.6 Dynamic regimes . . . . . . . . 7.1.7 Dynamic flow . . . . . . . . . . 7.2 Retained facts for a double-reed model

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III

Towards a numerical model of a double reed instrument

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Synthesis 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Historical background . . . . . . . . . . . . . 8.1.2 A digital model of a reed instrument . . . . . . 8.1.3 Overview of the chapter . . . . . . . . . . . . 8.2 Digital model . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Implementation of cylindrical resonators . . . 8.2.2 Implementation of conical resonators . . . . . 8.2.3 Other elements of the bore . . . . . . . . . . . 8.2.4 Exciter model . . . . . . . . . . . . . . . . . . 8.2.5 Coupling the exciter to the resonator . . . . . 8.3 Application of experimental data . . . . . . . . . . . . 8.4 Control of the model . . . . . . . . . . . . . . . . . . 8.5 Results of the simulation . . . . . . . . . . . . . . . . 8.5.1 General remarks on the time-evolution curves

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Conclusion 9.1 Quasi-static model of the reed . . . . . . . . . 9.1.1 The double reed as a nonlinear exciter 9.1.2 Reed opening geometry . . . . . . . . . 9.1.3 Elastic model . . . . . . . . . . . . . . 9.1.4 Sound synthesis . . . . . . . . . . . . 9.2 Open questions . . . . . . . . . . . . . . . . . 9.3 Perspectives . . . . . . . . . . . . . . . . . . .

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A Organology 157 A.1 Classification of double reed instruments . . . . . . . . . . . . . . . . . . 157 A.2 Variability of reed shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 B

Preliminary modal analysis B.1 Mechanical response of the double-reed B.2 Modes of vibration of the reed . . . . . . B.3 An oscillator driven by an external force B.4 Frequency analysis of reed vibration . . B.4.1 Experimental approaches . . . . B.4.2 Data analysis . . . . . . . . . . . B.4.3 Comparison of excitation methods B.5 Admittance spectra of bassoon reeds . .

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B.5.1 Comparison of two different cane reeds . . . . . . . . . . B.5.2 Effect of humidity . . . . . . . . . . . . . . . . . . . . . . B.5.3 Comparison between plastic and cane reeds . . . . . . . B.5.4 Oboe reeds . . . . . . . . . . . . . . . . . . . . . . . . . B.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.6.1 Importance of the dynamic aspects of the reed . . . . . . B.6.2 Importance of higher-order modes of vibration of the reed C Momentum C.1 Differential form . . . . . . . . . . . . . C.2 Integral form . . . . . . . . . . . . . . C.3 Vena Contracta . . . . . . . . . . . . . C.4 Singular losses . . . . . . . . . . . . . C.5 Potential theory for the Vena contracta C.5.1 Ecoulements par des orifices . . C.5.2 Borda tube ( = −π) . . . . . .

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Viscous effects 183 D.1 How important are viscous losses? . . . . . . . . . . . . . . . . . . . . . 183 D.2 Turbulent flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 D.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

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Effects of gravity

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Elasticity data

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G Artificial mouth G.1 Artificial mouth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.2 Stroboscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.2.1 Aim of the device . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I

Image Analysis H.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . H.1.1 Image analysis in musical instrument acoustics . H.1.2 Challanges of automatic image analysis . . . . . . H.2 Lighting balance . . . . . . . . . . . . . . . . . . . . . . H.3 Binarization . . . . . . . . . . . . . . . . . . . . . . . . . H.4 Identification of the reed opening . . . . . . . . . . . . . H.4.1 Removing thin connections or cracks . . . . . . . H.4.2 Reducing the number of regions . . . . . . . . . . H.4.3 Segmentation and region identification . . . . . . H.5 Geometrical measurements . . . . . . . . . . . . . . . . H.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . H.6.1 Difficulties encountered during image processing H.7 Notions on mathematical morphology . . . . . . . . . .

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Signal processing 205 I.1 Signal synchronisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

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I.1.1 I.1.2 I.1.3 I.1.4 J

Splitting the periods of a signal . . . Period statistics . . . . . . . . . . . . Image analysis and reed opening area Flow reversal . . . . . . . . . . . . .

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Topography methods 211 J.1 Hand measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 J.2 Artisanal scanner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

List of Figures 1.1

Modern double-reed instruments

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Double-reeds

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Terms used to describe the regions of double-reeds

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5

1.4

Reed making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.5

Thickness regions in oboe reed . . . . . . . . . . . . . . . . . . . . . . .

8

2.1

Cylindrical resonator and associated coordinates and variables . . . . .

17

2.2

Reflection coefficient

19

2.3

Impedance of a cylindrical resonator

. . . . . . . . . . . . . . . . . . .

19

2.4

Coordinates and variables used on conical resonator description . . . .

20

2.5

Impedance of a cylindrical resonator

. . . . . . . . . . . . . . . . . . .

22

2.6

Non-linear characteristics of a reed . . . . . . . . . . . . . . . . . . . .

25

3.1

oboe reed internal profiles . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.2

Reed opening designs . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.3

Pressure distribution in the double-reed

. . . . . . . . . . . . . . . . .

35

3.4

Front view of an oboe reed during closure . . . . . . . . . . . . . . . . .

37

3.5

Area vs inter-blade distance . . . . . . . . . . . . . . . . . . . . . . . .

39

3.6

Time evolution of the area swept by each blade . . . . . . . . . . . . . .

41

3.7

Stiffness measurements (S vs ∆p) . . . . . . . . . . . . . . . . . . . . .

42

3.8

S(t ) and ∆p(t ) during stiffness measurements . . . . . . . . . . . . . .

43

3.9

Relaxation of the reed . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.10

Relaxation in a characteristics measurement . . . . . . . . . . . . . . .

46

4.1

Diaphragm mounted on the reed . . . . . . . . . . . . . . . . . . . . . .

49

4.2

Reed and reed plus diaphragm characteristics compared . . . . . . . .

50

4.3

Addition of masses to the oboe reed . . . . . . . . . . . . . . . . . . . .

52

4.4

Device used for characteristics measurements. . . . . . . . . . . . . . .

52

4.5

Diaphragm geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

4.6

Diaphragm calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiv

List of Figures

4.7

Pressure acquisitions plotted against the time . . . . . . . . . . . . . .

55

4.8

Reed opening and pressure acquisitions plotted against time . . . . . .

56

4.9

A typical flow vs pressure characteristics . . . . . . . . . . . . . . . . .

58

4.10

Repeatability of the characteristics measurements . . . . . . . . . . . .

59

4.11

Non dimensional comparison of characteristics for a same reed . . . . .

59

4.12

Effect of the added mass on the reed characteristics . . . . . . . . . . .

60

4.13

Effect of humidity on a reed characteristics . . . . . . . . . . . . . . . .

62

4.14

The characteristics of two different reeds compared . . . . . . . . . . .

63

4.15

Comparison of the characteristics for different instrument reeds . . . .

64

4.16

Normalised characteristics of different reeds . . . . . . . . . . . . . . .

65

4.17

The characteristics of natural and synthetic reeds compared . . . . . .

66

5.1

Flow at the reed output (Schlieren)

. . . . . . . . . . . . . . . . . . . .

72

5.2

Experimental device . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5.3

Reed positioning device . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

5.4

Definition of the measuring region . . . . . . . . . . . . . . . . . . . . .

74

5.5

Velocity profiles vs pressure . . . . . . . . . . . . . . . . . . . . . . . .

76

5.6

normalised velocity profiles vs pressure . . . . . . . . . . . . . . . . . .

76

5.7

Velocity profiles vs reed opening . . . . . . . . . . . . . . . . . . . . . .

78

5.8

Perpendicular diametral velocity profiles

. . . . . . . . . . . . . . . . .

78

5.9

Differences between the measured profile and the real profile . . . . . .

80

5.10

Velocity profile in a cylindrical pipe . . . . . . . . . . . . . . . . . . . .

81

5.11

Volume flow q vs umax

83

5.12

Volume flow q vs pressure pm

. . . . . . . . . . . . . . . . . . . . . . .

84

5.13

Input flow velocity vs pressure . . . . . . . . . . . . . . . . . . . . . . .

84

5.14

average velocity u vs pressure pm . . . . . . . . . . . . . . . . . . . . .

85

5.15

Volume flows (hot-wire compared to diaphragm) . . . . . . . . . . . . .

86

5.16

Corrected volume flows from hot-wire measurements . . . . . . . . . .

87

5.17

Extrapolation of the residual opening area . . . . . . . . . . . . . . . .

88

5.18

Flow velocity at the reed entrance . . . . . . . . . . . . . . . . . . . . .

89

5.19

Flow regions in the reed

. . . . . . . . . . . . . . . . . . . . . . . . . .

90

5.20

Pressure recovery vs contraction coefficient CD . . . . . . . . . . . . . .

92

5.21

Reed pressure vs mouth pressure . . . . . . . . . . . . . . . . . . . . .

94

5.22

Comparison of flow models . . . . . . . . . . . . . . . . . . . . . . . . .

95

5.23

Recovery coefficient in the staple . . . . . . . . . . . . . . . . . . . . . .

95

5.24

Reed tip non-linear characteristics

96

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

List of Figures

xv

5.25

Non-linear characteristics with and without Vena Contracta

. . . . . .

97

5.26

Non-linear characteristics with conical diffuser . . . . . . . . . . . . . .

98

5.27

Pressure recovery deduced from experimental characteristics . . . . . .

98

5.28

Reed tip non-linear characteristics determined from experimental data

99

5.29

Stiffness compared in flow and flow-less systems . . . . . . . . . . . . . 100

5.30

Mach number of the flow inside the reed . . . . . . . . . . . . . . . . . 102

6.1

Measured opening area S(t ) . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2

S(t ) for two different bitings . . . . . . . . . . . . . . . . . . . . . . . . 111

6.3

Experimental device for dynamic anemometry . . . . . . . . . . . . . . 112

6.4

Impedance curves calculated for the reed duct (reed and staple)

6.5

Time evolution of a diametrical velocity profile . . . . . . . . . . . . . . 115

6.6

Time evolution of an axial velocity profile . . . . . . . . . . . . . . . . . 117

6.7

Snapshots of diametrical profiles for a reed with lips . . . . . . . . . . . 118

6.8

Influence of lips on the flow evolution . . . . . . . . . . . . . . . . . . . 119

6.9

Influence of lips on the opening area evolution . . . . . . . . . . . . . . 120

6.10

Sketch of the flow evolution

8.1

Scattering junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.2

Scattering junction used for a tone hole . . . . . . . . . . . . . . . . . . 140

8.3

Simulated pressure and reed position at note attack . . . . . . . . . . . 145

8.4

Simulated pressure and reed position . . . . . . . . . . . . . . . . . . . 146

8.5

Simulated pressure and reed position . . . . . . . . . . . . . . . . . . . 146

8.6

Simulation for variable mouth pressures . . . . . . . . . . . . . . . . . 148

8.7

Simulation for variable dampings . . . . . . . . . . . . . . . . . . . . . 148

A.1

Reed dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

B.1

Response of a bassoon reed

B.2

Comparison of two different cane reeds. . . . . . . . . . . . . . . . . . . 167

B.3

Comparison between different levels of humidification.

B.4

Comparison between plastic and cane reeds. . . . . . . . . . . . . . . . 168

B.5

Mechanical response of an oboe reed. . . . . . . . . . . . . . . . . . . . 169

C.1

Flow into a Borda tube . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

C.2

Singular losses

C.3

Flow in the complex spatial plane (z)

C.4

Hodograph — Set of velocities (u) found in the flow

. . . . 114

. . . . . . . . . . . . . . . . . . . . . . . . 120

. . . . . . . . . . . . . . . . . . . . . . . . 166 . . . . . . . . . 168

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 . . . . . . . . . . . . . . . . . . . 176 . . . . . . . . . . . 177

xvi

List of Figures

C.5

Transformed hodograph (ζ ) . . . . . . . . . . . . . . . . . . . . . . . . . 178

C.6

Flow potential set (w . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

C.7

Auxiliary variable v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

C.8

Jet profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

C.9

Flow shape for the Borda tube . . . . . . . . . . . . . . . . . . . . . . . 181

F.1

Fibercane reed stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

F.2

Dry cane reed stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

F.3

Fibercane reed stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

G.1

Artificial mouth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

G.2

Reed plcaed inside the artificial mouth . . . . . . . . . . . . . . . . . . 192

H.1

Image analysis procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 196

H.2

Original image and binarisataion

H.3

Morphological operations . . . . . . . . . . . . . . . . . . . . . . . . . . 198

H.4

Inversion and flood-filling . . . . . . . . . . . . . . . . . . . . . . . . . . 199

H.5

Problems arising for small openings . . . . . . . . . . . . . . . . . . . . 199

H.6

Complete image analysis procedure . . . . . . . . . . . . . . . . . . . . 201

H.7

Fitting an ellipse to the reed opening

I.1

Flow velocity cartography . . . . . . . . . . . . . . . . . . . . . . . . . . 208

I.2

Unfolding the velocity signal . . . . . . . . . . . . . . . . . . . . . . . . 209

J.1

Scanner setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

J.2

Analysis of a scanner image . . . . . . . . . . . . . . . . . . . . . . . . 212

J.3

Three-dimensional reconstitution of a reed cast

. . . . . . . . . . . . . . . . . . . . . 197

. . . . . . . . . . . . . . . . . . . 202

. . . . . . . . . . . . . 213

List of Symbols Description

Dimensions

Observations, Typical values

α

Coefficient of Vena Contracta

–

α=

f

Frequency

s−1

h

Reed opening height

m

0 < hoboe . 10−3

k

Reed stiffness

N m−1 = kg s−2

?

ν

Kinematic viscosity

m2 s−1

νair = 1.44 × 105 , νCO2 = 7.6 × 104

Reed mass

kg

?

µ

Dynamic viscosity

kg m−1 s−1

µair = 1.8 × 105 , µCO2 = 1.48 × 105

ω

Angular frequency

rad/s

ω = 2πf

Atmospheric pressure

Pa (kg m−1 s−2 )

patm ' 105 Pa

Pa (kg m−1 s−2 )

0 . p m . PM

Pa (kg m−1 s−2 )

Symbol

m

patm pm

PM

Reed beating pressure

Pa (kg m−1 s−2 )

PT

maximum-flow2 pressure

Pa (kg m−1 s−2 )

-

∆p

Pressure drop in the reed

Pa

∆p , pM − pR

Pa

Pressure at the reed output

Pressure drop in aphragm (chap. 4)

the

1

di-

(in the oboe)

q

Volume flow

m3 s−1

r

Reed damping

kg s−1

(∆p)d , pR − patm   1/ 2 k 1/2 S03/2 0 < q . 332/2 2ρ

Reynolds Number

–

Re =

Reed opening area

m2

0 < S . S0

(∆p)d

Re S

2

1

−pm . pr (t ) . pm PM ' 25kPa

pr

1

Mouth pressure

Sj S

relative to the atmospheric pressure in a non-linear characteristics curve

?

ud ν

xviii

List of Figures

S0

Equilibrium reed opening area

m2

Sa

Effective area of pressure application

0 < S0 ' 2 × 10−6

m2

Sa ' 50 × 10−6

Sj

Jet cross-section

m2

Diaphragm cross-section

m2

ρ

Volumic mass

kg m−3

u

Flow velocity

m s−1

Sdiaph

Zin

Resonator input impedance

kg

m−4

ρair = 1.2, ρCO2 = 1.82

10 – 200 m s−1 (in the oboe reed) s−1

Preface This work started as an attempt to propose a synthesis algorithm for double-reed instruments. In general, reed instruments, especially the clarinet have been the subject of numerous studies, starting off with [Backus, 1961]. A model of the clarinet based on basic physical assumptions about the reed and the flow, and the acoustic propagation in the bore was implemented as early as the 1980 by [Schumacher, 1981]. Since then, a lot of works have studied both the behavior of simple models, and the refinements to bring to these models to improve their sound quality and realism. Unfortunately, double-reed instruments such as the oboe or the bassoon were not the object of such a great number of studies as the clarinet. One of the reasons for this, is that these instruments are not as popular as single-reed instruments (starting with the saxophone, an to a lesser extent the clarinet), so that the industrial and scientific investment in this kind of instruments is much more scarce than the former. Oboes and bassoons (the most common double-reed instruments in music performances nowadays) use conical bores as their resonators. Since the acoustical behavior of conical resonators is more difficult to interpret than straight ones, it is natural that scientists base their work on cylindrical resonator instruments while researching for the details in the exciters, and studying the instrument behavior. Nevertheless, the interest in conical resonators is considerable, mainly because of the popular saxophone, so that there exist resonator models that can either replace the conical resonator with others of similar response , implement conical resonators or even more realistic models that take into account the deviations from standard shapes. Today, the most important difficulty that one faces when trying to implement a model of an oboe or bassoon, is the lack of experimental and theoretical data about the doublereed, or even arguments in favor or against the utilisation of a generic reed model for the case of double-reeds. Other than the lesser interest on double-reeds, there is a difficulty that explains this lack: the small dimensions of the reed. These make it very difficult to study both mechanical and aerodynamic properties of these exciters. It is thus natural that the work of this thesis followed a more experimental path, in order to fill in some of the gaps existing in scientific literature. It was soon realised that the interest in this kind of experimental data went beyond the application to sound synthesis. In fact, the particular relation of double-reed musicians with the reeds makes the art of reed making a passionating as much as puzzling subject. For musicians, reed-making is a matter of empirical learning, much as the instrument performance itself. Not only it is difficult to translate their experience on the techniques and their effects into scientific terms, it is also a fact that the technique is widely variable from one musician to another, and in certain cases, one particular gesture can be described as having opposite effects, when different musicians are questioned.

xx

List of Figures

Due to the time constraints of a PhD work, this work had to concentrate more on general aspects of the double-reeds than on the details of different scrapping or gouging techniques, so that most of this work can seem insufficient to oboists and bassoonists. The author and the team that collaborate in this work are aware of this weakness, and of the need of bringing this work to a level of detail that can be useful to the musicians in their practice.

Overview This work is divided into three main parts, an introduction (part I) describing the doublereed and generic models of double-reed instruments, an experimental characterisation of the double-reed (part II) and finally an attempt to put the experimental data together with the classical model of reed instruments in order to propose a synthesis algorithm for double-reed instruments (part III). In chapter 1, we describe the double-reed, its fabrication and its use in musical contexts. Chapter 2 provides the generic model of reed instruments and a review of the studies found in scientific literature about the double-reed. Chapter 3 introduces some basic aspects about the double-reed, such as a characterisation of the reed opening, and its variations in different opening states, measurements of basic characteristics like effective stiffness and viscoelasticity effects, and finally an overview of the modes of vibration of the reed independently from the flow aspects. In chapter 4 the static non-linear characteristics of the reed is measured and compared for various kinds of reeds, and different environment conditions. Chapter 5 tries to investigate deeper the aspects of the static flow inside the reed and proposes a flow model to explain the measured non-linear characteristics. Chapter 6 briefly studies the periodic motion of the reed, and investigates the details of the flow in dynamic regimes. The main experimental results are finally collected in chapter 7, which also introduces the main features applied to the synthesis model. Finally, chapter 8 collects all the experimental data from the previous chapters, discusses the modifications to be brought to the generic model of the reed in order to build a model of the double-reed and finally presents the implementation of a double-reed instrument

Part I.

Introduction

1. The double-reed There are several instruments in the double reed family. Undoubtedly, the more wellknown ones are the modern oboe and bassoon, which take part in the modern orchestra (fig. 1.1). The resonators in both these instruments are mainly conical throughout most of their lengths, and this fact contributes to the specific timbre which we associate to double reed instruments. For instance, while in the clarinet most of the sound energy is concentrated in odd harmonics, giving it a round sound, notes played by oboes or bassoons present both even and odd harmonics in similar proportions.

Figure 1.1.: The oboe and the bassoon, the two most common double-reed instruments used in the modern orchestra The first question that should be asked concerns the particularity of double-reed instruments. They are particular from the morphologic point of view. But is it possible to associate a particular sound to these instruments? The fact is that their sound can be very variable. From the nasal sound of the oboe, to a more strident sound associated to instruments played without lip support (like the bombarde), or the surprisingly flutelike sound of the guanzi ( ), their timbre is all but homogeneous. This however does not imply that their physics is inherently different, because the way they are blow is also highly variable. However the conical resonator may not be the only explanation for the specific color of their sound. As an example, the saxophone is a case that mixes single-reeds to conical resonators, and still its sound is very easily distinguishable from instruments associating double-reeds to conical resonators. There are also examples of musical instruments using double reeds attached to straight bores, such as some chanters used in bagpipes, or ancient instruments such as the crumhorn (see appendix A.1 for a listing of double reed instruments using double reeds). Their sounds are usually very different from those of a clarinet, but they have some significant differences from orchestral double-reeds: in general the lips do not di-

4

Chap. 1: The double-reed

rectly touch the reed. Such instruments tend to have a brighter and more harsh sound than orchestral instruments, probably because of the damping and increased inertia introduced by the lips. It is also worth mentioning the development of single-reed mouthpieces for oboes and bassoons. These were commercialised until a few years ago [Bate, 1975], but to our knowledge are not fabricated any longer. However, one recent attempt by Nederveen 1 shows that they can be used to produce a sound almost indistinguishable from a real double-reed. It is interesting to notice however that although these mouthpieces are based on clarinet and saxophone ones, their dimensions approach those of oboe or bassoon double-reeds, and this may give some insight into for further considerations that will be made to the double-reed (sect. 2.4.1). The facts listed above may suggest that only the resonator determines the timbre of a reed instrument and not the kind of reed exciters used to play them. However differences may exist in the coupling between the exciter and the resonator, which is often more continuous in double-reed instruments.

1.1. Generic description of the double reed Despite the variety of instruments using double-reeds as their exciters, their basic configuration remains the same for all of them.

Figure 1.2.: Two double-reeds used in oboes (1) and bassoon (2). The term ‘‘double-reed’’ usually refers to the device used to modulate the air-flow entering a wind instrument such as the oboe or the bassoon. This device can be assimilated to a valve, which by the action of pressure controls the flow. The double-reed is traditionally constituted by two blades cut off natural cane (usually of the species Arundo Donax). The two blades are bound together by their base forming 1

Although no publication was made on this device, several musical acoustics colleagues had the chance to listen to and try it out at SMAC, 2003. Also, the commercialisation (by Runyon) of mouthpieces for bassoons using clarinet single-reeds has been reported.

1.1 Generic description of the double reed

5

Staple Cork

Butt

Oboe

Ligature Blades Rails

Tip

Wires

Bassoon

Figure 1.3.: Terms used to describe the regions of double-reeds

6

Chap. 1: The double-reed

a duct into which the air is driven by the difference of pressure between the mouth and the duct. The duct starts as an oblong cross-section in the tip (in the upstream part of the reed), progressively evolving into a quasi-circular cross-section towards the butt in the downstream region. This end is introduced in the main bore of the instrument (see fig. 1.3). The tip of a blase is thinner than the remaining part, and this makes it soft and flexible. In orchestral double-reeds, the musician gently presses its lips against the reed. When compared to some ancient double-reed instruments, such as the shawm, the crumhorn or popular ones such as some bagpipes, the latter have very bright, sometimes strident sounds, whereas oboes and bassoons have fewer high-frequency components, sounding mellower.

1.2. General working principles The softer part of the reed (at the tips) can be forced to bend by the action of an external pressure, imposed by the lips, or by a difference between the external and internal air pressure. At a given value of the applied pressure the reed ends up closing, so that ideally no flow should cross the reed. In a regular musical use, there are periodical variations of the pressure inside the reed, imposed by the resonator’s modes of vibration. These pressure oscillations drive the position of each blade and the flow crossing it. In turn, these flow variations regenerate the acoustic oscillations inside the instrument.

1.3. Fabrication of the double-reed The fabrication of the double reed is a complex issue, and the details involved in the preparation and tuning of the reed are the subject of a large number of works (see for example [Kopp, 2003] [Smith, 1992]). Such details are out of the scope of this document, and could be the subject of a complete new work involving the knowledge of instrumentists and reed makers, their experience on the effects of different reed scrapings and tunings of the reed, side to side with vibrational analysis of these scrapes, and its effects on the intonation of the instrument, as an example. To the subtleties of reed makers’ skills is added the fact that oboe and bassoon reeds are played between the instrumentist’s lips. This changes the vibrational properties of the reed, in a way that also depends on the instrumentist. An ideal scrapping to an oboist may thus not be convenient to another oboist. However, a quick glance at the procedure of fabrication can provide some general ideas on the physics behind the functionning of the double reed. Several stages can be identified in this procedure: The double reed is usually fabricated from a single sample of cane, which is a cylindrical segment split into equal strips using a splitting arrow (fig. 1.4, 1). For an oboe reed a dry tube of cane of 9.5 to 11 mm in diameter is split into 3 equal strips. For a bassoon the initial diameter is 24-27 mm and it is split into 4 strips [Ponthot, 1988].

1.3 Fabrication of the double-reed

7

1 Pith Bark

2

3

4

Figure 1.4.: Stages of reed making: 1 Raw tube of cane, with a strip cut off; 2 Folded cane;3 Shaped cane placed on the staple; 4 Cane bound to the staple

8

Chap. 1: The double-reed

The cane is then shaved (gouged) from the inside (the pith), in order to remove part of the soft matter, until the base thickness of the reed is achieved. Because the diameter of the gouge can be different from the cane diameter, the basic cross-section of the gouged strip consists of a crescent shape, uniform over the strip length [Yefchak, 2005]. The thickness of the gouge dictates the reed hardness, while the diameter influences the final opening of the reed [Goossens and Roxburgh, 1980]. After letting the reed soak for a few hours, an incision is made at half of the reed’s length in the exterior part of the reed, breaking the hard fibers of the outer cane shell (the bark). The cane is then folded at the middle (2), and placed around a shaper, a hard steel plate whose shape is roughly the final shape of each blade as seen from above. The shaper is used to guide the trimming of the blades’ sides. The two blades are now visible and present a tapering from the tip (in the middle of the strip) to the extremities. They are still connected at the tip by the softer matter in the reed (3). Depending on the instrument, the two blades are attached to each other by the butt (in the bassoon) or tied to the staple (in the oboe). At this extremity they are forced to increase their curvature, whereas at the tip, the junction between the two blades still forces them to be straight one over the other. The tip is then scraped to remove the bark, and cut at about 5 mm from the bending (dashed line in 3). The two blades separate from each other at the tip. Depending on the force attaching the two reeds together by the butt, the two blades are usually in contact with each other along the sides (rails). This provokes an initial pre-constraint on the reeds, and should assure air-tightness of the reed. The final stage is to scrape the blades’ tips in order to give them a decreasing thickness towards the tip of the reed. Usually the scraped part is limited to 7 to 9 mm near the tip in the case of an oboe reed [Goossens and Roxburgh, 1980].

Lay

Heart

Back

Figure 1.5.: Regions of thickness in [Goossens and Roxburgh, 1980])

an

oboe

blade

(after

This is a very rough description of the scraping phase. Usually there is a thicker part of the reed near the tip at the center called the ‘‘heart’’, and the corners of the reed (‘‘lay’’) are thinner than the remaining tip. Some musicians prefer to leave a thicker ‘‘spine’’ in

1.4 Utilisation of the double-reed

9

the center of the reed, and they refer to this as a ‘‘W’’ scrape, and others prefer to scrape more evenly the back of the reed (‘‘U’’ and ‘‘V’’ scrapes) [Goossens and Roxburgh, 1980]. However, the scraping technique and the final thickness profile of the reed depend much on the musician’s practice. There are extensive works dedicated to the art of making good reeds [Ledet, 1999] [Shalita, 2004], as an example. However, the perfect technique does not exist, it depends on the physiognomy of the musician, on his playing technique and on the learning path he followed. As a final remark, [Goossens and Roxburgh, 1980] stresses the fact that symmetry should remain a constant throughout the fabrication of the reed, so that for instance the reed opening should present a symmetrical shape.

1.4. Utilisation of the double-reed Due to the fragile nature of the double-reed, and the fact that it is used in straight interaction with the musician’s mouth (in most modern instruments), the technique used for playing a double-reed instrument plays a crucial role in the behavior of the instrument, and the timbre of the notes produced. The performance technique of both oboes and bassoons has been the object of a large number of works dedicated to beginner musicians. These are usually intended as a complement for the apprentice, because a close supervision of an experienced musician is required to transmit an empirical knowledge that can be very difficult to translate into words. The following sections will give a quick glance on the general performance technique, mainly as a review of what can be found in the musical literature, and intended to provide some understanding on how the action of the musician can affect the physics of the reed.

1.4.1. Soaking Before playing the instrument the reed is usually soaked for a few (about 5 [Rader, 2004]) minutes in water so that the reed softens and its shape assumes a correct opening shape (more closed than when dry for [Rader, 2004], however [Sprenkle, 1961] and others say that an over-soaked reed has too large a tip opening). Soaking also helps to seal the rails of the reed, so that the flow does not escape from the sides while playing the instrument [Sprenkle, 1961]. Eventual leaks can be corrected by wrapping the base of the reed in gold-beaters skin2 . The accuracy of advised soaking timings (never beyond 5 minutes) suggests that in fact the reed should not be saturated in water content in order to produce a good sound. In fact, over-soaking is considered to open the tip too wide and making the reed less responsive [Sprenkle, 1961] [Roscoe, ]. This has a double-sided effect of softening 2

the outer membrane of ox intestine. The advantages of this material are its strength and fairly uniform thickness.

10

Chap. 1: The double-reed

the reed too much and forcing the musician to bite the reed too hard, vitiating his embouchure. After this initial soaking the water content is maintained by the natural humidity provided by the musician’s breath. This is why an oboist or a bassoonist keeps the reed in its mouth even during pauses.

1.4.2. Embouchure

Embouchure is understood to be the basic position of the mouth (lips, teeth and tongue) that should be assumed to start playing the instrument. It should optimise the air supply to the reed, and a correct position should allow the musician to control such reed properties as the opening, its damping or inertia, as well as the amount of reed that is left free to vibrate inside the mouth. The contact between the lips and the reed should be air-tight (once again we see the absence of leakages in and around the reed as a crucial factor). The lips should roll over the teeth until the transition between red and white flesh is in contact when the two jaws are brought together [Sprenkle, 1961]. This provides a soft and damping material whose tension and hardness can be controlled via the lip muscles (changing the biting should be avoided and jaws left at a constant position [Goossens and Roxburgh, 1980]), which is referred to as ‘‘lip-pressure’’. Lip-pressure can vary according to pitch and dynamics. [Campbell et al., 2004] states that some players can hold the reed directly on their teeth in order to reach the highest notes, which suggests that these notes require reduced damping while maintaining a strict control over the reed opening. The contact between the lips and the reed should be symmetrical in both blades [Goossens and Roxburgh, 1980], so that the angle at which the instrument is held is important. A tip of 1 to 2 mm should be left inside the mouth out of lip contact [Goossens and Roxburgh, 1980], [Campbell et al., 2004], so that at least a short tip of the reed is able to vibrate freely. As a remark, we can suppose that the part of the reed which is in contact with the lips can also vibrate, although its motion is expected to be reduced by the damping and inertia of the lips. In fact, since the lips are not attached to the blades and they have no counterpart inside the reed, they should be able to separate from the lips during part of the motion induced by the reed tip. Given that only the lower jaw can move with respect to the remaining skull, it is this part of the face that can exert the greater control over the reed [Goossens and Roxburgh, 1980]. A special attention is thus paid to the lower jaw and lip. We can suppose that this remark is mostly intended for the control of the lip-pressure, because the symmetry of the reed implies an equally important contribution of both lips.

1.4.3. Articulation and dynamics In this section we shall discuss briefly the recommendations on the gestures used during intonation, both concerning transitions between notes, attacks of new notes from silence

1.4 Utilisation of the double-reed

11

and musical dynamics. In general, it is advised to maintain a good and constant breath support (by keeping the respiratory muscles under tension) even if the air pressure is to change inside the respiratory tract [Goossens and Roxburgh, 1980]. This also ensures a good steadiness of the force applied to the air even if the pressure is required to change in the mouth. Pressure changes occur very frequently while playing the instrument, because the average flow varies depending on the regime, for example, flow can be important before the oscillation starts but it decreases abruptly once the reeds starts to oscillate. While attacking a note, the position of the jaws and the lip-force should not change. The tongue is placed against the corner of the reed, preventing reed oscillations. Breath is asserted before starting the note by removing the tongue from the reed [Sprenkle, 1961]. The tongue should block the air flow before the note is attacked [Goossens and Roxburgh, 1980]. We conclude that before the tongue is removed the air pressure in the mouth is thus higher than once the note is about to start. Pressure should increase naturally as the oscillation is established, if the breath support is maintained constant. This was verified in analysis of the time-variations of the pressure during basic musical tasks [Fuks and Sundberg, 1996]. During articulation (transition between to notes), the tongue touches the reed gently but quickly, in order to stop the oscillation for a short interval. Breath support must remain constant throughout this procedure. Double-reeds are known to have a smaller dynamic range than other instruments [Goossens and Roxburgh, 1980]. Similarly as what was said for articulation and attacks, the breath support should be constant for different dynamics. The amplitude of the reed oscillations (and thus the sound produced by the instrument) can be controlled by the lip-pressure [Sprenkle, 1961]. A slight movement of the reed away from the mouth is allowed [Goossens and Roxburgh, 1980] in order to reduce the vibrating length of the reed during diminuendi.

12

Chap. 1: The double-reed

2. Reed models This chapter is dedicated to describing the general principles of functioning of reed instruments, and unveiling some clues about the particularities of double-reed instruments. In particular, we review some propositions and studies existing in the scientific literature.

2.1. Introduction Double-reed instruments share many features with other reed instruments, for instance, the resonance of an air column, the pressure-controlled valve... It is thus natural to start our study with a description of what is already known for reed instruments, providing a basic mathematical model that can describe the generic behavior of the instrument. In the way, we indicate some key-points where this model is more susceptible to fail for double-reeds. We will finish with an overview of some of the propositions already found in the literature.

2.1.1. Overview of the chapter The chapter starts with a general presentation on the physical approach to musical instrument modelling, gradually focusing, first on the large group of auto-oscillating instruments, then wind instruments and the more particular case of reed instruments. Section 2.3 introduces the physical model of resonators, both cylindrical and conical. Section 2.4 focuses on the part of the instrument that will be the central subject of this thesis: the exciter. Finally, section 2.5 reviews the present-state of knowledge on double-reeds.

2.2. Self-sustained instruments In a general way, sound production in musical instruments can be seen as a way of taking advantage of the natural oscillations of an object, or several objects in contact. Most objects can naturally vibrate according to different modes, depending on the method that is used to excite these vibrations. Reed instruments belong to a large group of instruments where oscillations are selfsustained. This means that the natural vibrations of the resonator (the air column present inside the bore of the instrument) are maintained throughout the playing of one note by the internal mechanisms of the instrument and a continuous energy supply. The sound produced by self-sustained instruments is characterized by a stable output of sound whose intensity can be maintained as long as the musician wishes.

14

Chap. 2: Reed models

2.2.1. Passive and active parts of the instrument In a classical approach, it is usual to distinguish a passive part of the instrument, called the resonator, which has its modes of vibration, and receives the energy supply from the exciter to set these modes into vibration and eventually maintain them through time (in the case of self-sustained instruments). This formal separation proves to be useful because the resonator can be described separately from the exciter and the two systems linked through a set of common variables. Usually, the two systems depend on each other, so that the time evolution of the state of the instrument can only be found by solving the two systems together. Only in some special cases, which do not concern our study, can the state of the exciter be found independently, providing an input signal which is transformed (filtered) by the resonator.

2.2.2. Wind instruments In instruments that concern this thesis’ study, the resonator is assimilated to the air column contained inside the instrument. It is limited by the walls of the bore, roughly a cylindrical or conical pipe. The length of the air column is usually much larger than the transverse dimensions (the diameter), a fact that simplifies the mathematical description of the resonator as will be seen below (sect. 2.3) When the instrument is sounded, some of the properties of the air-column oscillate around their values at rest. Pressure (p) and flow velocity (u) are the usual quantities that are used to describe the vibration of the air, but other properties periodically oscillate and transmit these oscillations to the neighboring regions of air. These are, for example the density (ρ) and temperature (T ) of the air. In general, however, the oscillations of these variables inside the air column are much stronger than those present in the propagation of sound in free-air. The modes of oscillation of these properties in the air-column determine roughly the frequency at which the instrument will sound, although the mechanical properties of the reed can also influence it to a lesser extent [Brown, 1990]. Several methods exist to create and maintain the oscillations of the air-column. Excitation can be described as a set of conditions imposed on the oscillating properties of the air-column in a determined region.

2.2.3. Reed instruments In reed and lip-driven instruments the excitation mechanism is a solid component that periodically blocks or limits the airflow into the instrument (a valve). The main difference between these two kinds of valves is that lips are blown-open (or outward-striking: an increase in the mouth pressure forces the valve to open) whereas reeds are blown-closed (or inward-striking). The next sections describe the general principles of functioning of a reed instrument, and how these can be transported into a physical model of the instrument. The usual

2.3 The resonator

15

conceptual separation between exciter and resonator is used. In practice, the resonator is the region where the air oscillations are sufficiently small to be described by the equations of acoustic propagation (a linearised version of the general equations of fluid motion for small variations of pressure, flow velocity, density and temperature around average values of these variables). Eventually, there may be some cases (for loud playing levels) where non-linearities may be important, because the variations of the referred variables may become too strong to be described by a linear model. Such non-linearities were shown to be important in brass instruments [Vergez, 2000], but recent studies [Gilbert et al., 2005] prove that in the case of woodwinds these non-linearities have no perceptible effects. The exciter is the part of the instrument where the approximations used in acoustics are not valid, whether it be due to the variations in the boundary conditions (movement of the reed) or to massive variations of the flows. This includes the vibrating reed and the acceleration of the flow into the instrument. It must be kept in mind that this separation between exciter and resonator is a conceptual one, which simplifies the mathematical description of the instrument. However, in the case of wind instruments, the air-column is also the continuation of the air flow involved in the excitation. This can make the separation between both systems delicate and somewhat artificial, requiring special care in the interpretation of the results of experiments and models when the exciter or the resonator are studied independently.

2.3. The resonator As stated above, the study of the resonator is based on the linear propagation of acoustic waves. The deduction of the acoustic wave equations from the equations of fluid motion can be found in any introductory acoustics book [Fletcher and Rossing, 1991], [Morse and Ingard, 1968] and will not be reproduced here. Our study of the resonator will start with simple solutions of this wave equation for the propagation of planar waves in a given direction.

2.3.1. Wave equation and wave propagation The solutions to the linear acoustic pressure wave equation (for plane waves) are a linear combination of the following: p+ (z, t ) = A(ω)e (ωt −kz )

(2.1)

(ωt +kz )

(2.2)

−

p (z, t ) = B(ω)e

In this equation, z is the direction of propagation of the wave (perpendicular to the wavefronts), ω represents the angular frequency measured in rad/s (omega = 2πf , where f is the frequency of the wave). k is the wave number which in ideal propagation is related to the angular frequency by the relation c = ωk , c representing the velocity of sound. It is found from the resolution of the linearized Navier-Stokes equation and the

16

Chap. 2: Reed models

adiabatic compression relation: c=

r

γpatm ρ

(2.3)

where γ = Cp /Cv = 1.4 is the ratio of constant pressure and constant volume specific heats, patm is the average atmospheric pressure and ρ is the density of the air. In equation (2.1), p+ represents a wave travelling from left to right (that is, in the positive direction of z)1 . The propagation of a pressure wave (p ± (t )) implies a propagating flow velocity wave ± (u (t )), which follows the pressure wave with a phase offset.

2.3.2. Propagation inside a cylindrical waveguide Inside a duct, the solid walls impose boundary conditions on the flow of the acoustic wave. In particular for an ideal fluid, the velocity component perpendicular to the wall has to be zero. In the case of a cylindrical duct, and in fact of any duct with straight, parallel walls (constant cross-section), one possibility for fulfilling these conditions is that the wave propagates parallel to the walls. Wave fronts are thus perpendicular to the walls. Since the acoustic wave is a longitudinal wave, the flow will thus always be parallel to the walls. Of course this is not the only possibility. Even if the propagation is perpendicular to the walls, standing waves can exist that have q nodes (no oscillation of the flow velocity) at the reed walls. For the study of wind instruments we shall restrict ourselves to waves that propagate longitudinally (along the axis of the bore). Due to the dimensions of the bore, transverse standing waves tend to correspond to frequencies much higher than usual playing frequencies of the instruments. In a waveguide, the acoustic flow is defined as q(z, t ) = S(z )u (z, t ), where u (z, t ) is the acoustic velocity, supposed to be homogeneous over a cross-section (S(z )) of the waveguide at position z (usually defined to start at one of the resonator’s ends, as long as only the axial modes are considered. Acoustic flow can then be related to the pressure waves according to the formula: q± (z, t ) = ± with Zc =

1 p± Zc (z )

ρc S

(2.4)

(2.5)

which is the so-called characteristic impedance of the acoustic wave, constant throughout the resonator length because its cross section S is constant. A wave of frequency ω propagates along the resonator with a velocity of c = ωk , but as we’ll see below (section 2.3.3) the wavenumber k may need to be considered as 1

This can be verified by following a constant phase (φ = ωt − kz): constant phase implies that dφ = 0, so that ωdt − kdz = 0 and ∂z = + ωk . ∂t

2.3 The resonator

17

y x r (z ) z

Figure 2.1.: Cylindrical resonator and associated coordinates and variables

depending on the frequency. The propagation can thus be described by the following formula: P ± (ω, x ) = e ∓ık (x −x0 ) P ± (ω, x0 )

(2.6)

2.3.3. Diffusion and dispersion of the waves In the previous section we described mathematically the propagation of an ideal wave. Equations (2.1) state that the amplitude of the wave is maintained through time and space, and the same is true for the acoustic flow. In practice, in a free field, some of the energy of the wave is lost to a temperature increase through viscosity. When confined to a cylindrical waveguide, to this energy dissipation must be added the dissipation through viscous effects due to the no-slip condition on the boundaries (friction against the resonator walls) and thermal conduction through the resonator walls which implies that compressions and expansions are not really adiabatic. Contributions from losses through the walls are much more important than through viscosity and conduction through the air, so that the latter will be neglected. For large resonator diameters (when compared to the boundary layer width), these effects can be introduced as perturbation (Kirchhoff’s theory [Rayleigh, 1945]), so that the propagating waves can still be described by equations (2.1) and (2.4) with a variable wave number: ω i 3/2 k (ω) = − ηcω1/2 (2.7) c 2 with η=

2 Rc 3/2



√

lv +



cp cv

−1



√

lt



(2.8)

and R is the bore radius, lt and lv the thermal and viscous characteristic lengths, and cp /cv the ratio of the constant volume to constant pressure specific heats.

18

Chap. 2: Reed models

2.3.4. Resonator termination and radiation When a travelling wave arrives at one end of the resonator, it does not follow unchanged in the same direction. A part of the wave is reflected back to the resonator (in the opposite direction) and the rest is transmitted to the free acoustic field outside of the resonator. The relationship between the reflected and the incident wave can be written in frequential form using the reflection coefficient (Rrad ): P + (ω) = Rrad (ω)P − (ω)

(2.9)

A particularly simple case of this situation exists when a boundary condition is imposed at the end which states that the pressure field outside of the resonator is constant (corresponding to the atmospheric pressure). In first approximation this is true, considering that the pressure fields inside the resonator are much higher than outside. In this case, the total acoustic pressure at the end of the resonator is zero, so that 0 = P + + P − , which means that the outgoing wave (p − ) is reflected back symmetrically (P + = −P − ). The key to describing the part of the wave which is radiated is to know the characteristic impedance of the radiated wave exactly at the resonator ending (transition from resonator to the free field). This radiated wave is assimilated to a wave produced by a plane circular piston with the same cross section as the resonator. In fact, there is no analytical solution to this problem with the particular boundary conditions of the cylindrical resonator (which are boundaries for the free field as well as for the internal field), but two particular solutions exist for the flanged and unflanged planar piston. For the flanged planar piston, the load impedance seen by the planar wave arriving at the termination is written [Levine and Schwinger, 1948]: Zrad =

ρc S

(Rr (2ka ) + ıXr (2ka ))

(2.10)

with Rr (x )

= 1−

Xr (x )

=

2J1 (x ) x

2H1 (x ) x

(2.11) (2.12)

and J1 (x ) the first order Bessel function of the first kind, and H1 (x ) the first order Struve function. S denotes the cross-section area of the tube at the exit, and k = ω/c the wavenumber of the sinusoidal wave. These models provide a radiation impedance Zrad (ω) which can be used to calculate the reflection and transmission coefficients seen by the outgoing wave (p − ) at the extremity of the bore: Zrad /Zc − 1 Rrad = (2.13) Zrad /Zc + 1 where Zc is the characteristic impedance of the planar wave inside the bore.

2.3 The resonator

19

Reflection Coeff.

1 0.8 0.6 0.4 0.2 0 100

1000 10000 Frequency HHzL

Figure 2.2.: Reflection coefficient corresponding to a flanged pipe termination (absolute value)

2.3.5. A complete cylindrical resonator

Impedance HdBL

These three key effects (acoustic propagation, visco-thermal losses and radiation) are essential to describe a complete resonator model. We are now able to calculate the input impedance of an ideal cylindrical resonator (without visco-thermal losses and ideal end reflection) to a more realistic resonator, including losses and radiation of a flanged piston.

160 150 140 130 120 110 100 90 0

1000 2000 3000 4000 Frequency HHzL

5000

Figure 2.3.: Cylindrical resonator impedances. In dashed red, the simplest model consisting of a perfect reflection at the bore’s end, and no propagation losses. In solid blue, a flanged pipe termination is supposed, and visco-thermal losses through the walls are taken into account

20

Chap. 2: Reed models

2.3.6. Conical resonator models Most modern double reed instruments such as oboes or bassoons use resonators whose shapes are conical throughout most of their length. Although these are slightly more complicated than the resonators described in section 8.2.1, an exact description using spherical waves is possible (neglecting visco-thermal losses, non-linearities and nonaxial modes).

y x r (z )

θ

z

Figure 2.4.: Coordinates and variables used on conical resonator description If we consider conical walls as boundary conditions of the wave equation, it can be written simply in spherical coordinates. If we only consider variations in the radial direction (along r, which is a similar restriction as the restriction to longitudinal propagation seen in 2.3.2), it can be written as:

1 ∂ r 2 ∂r



r2

∂p ∂r



=

1 ∂2 p c 2 ∂t 2

(2.14)

whose solutions are: rp(r, t ) = P (r, t ) = P + (r, t ) + P − (r, t )

(2.15)

and A(ω) (ωt −kr ) e r B(ω) (ωt +kr ) e p− (r, t ) = r p+ (r, t ) =

(2.16) (2.17)

The acoustic flow waves q ± (r, t ) are related to the pressure waves p ± (r, t ) by the following formulas:

1

q+ (z, t )

=

q− (z, t )

= −

Zc (ω, z )

p+ (z, t )

1 Z ∗ (ω, z ) c

p− (z, t )

(2.18) (2.19)

2.3 The resonator

21

Zc (ω, z ) =

ρc

1

1 S 1 + ıkz

(2.20)

These are the spherical travelling waves which will be used to describe the conical bore. Diffusion and dispersion The inclusion of viscosity and thermal diffusion terms in the spherical wave equation greatly complicates its resolution. While for the cylinder the analytical solutions can be written (section 2.3.3), for the cone no exact solution has been found yet, to our knowledge. However, if we consider that the tapering angle of the resonator is small, the planewave Kirchhoff theory should be a good approximation to the total losses by using an value of R which stands between the input and the output radii in equations (2.7) and (2.8). The best approximation for this value is the radius at one third of the length from the smaller radius. Resonator termination and radiation For conical resonators, and in a general way, for a resonator which terminates with a bell, the plane piston theory of section 2.3.4 is no longer exact, because the wavefronts on the edge of the bell are curved. In a conical waveguide, the flanged piston case turns into a pulsating dome (circular portion of the sphere) oscillating in the remaining rigid sphere. Thomas Helie’s theory [Helie and Rodet, 2003] provides radiation impedances for spherical wave terminations compatible with the one-dimensional descriptions of wind instrument resonators. These impedances (Zrad ) can be used to calculate the reflected and radiated waves in a similar way as given by equation (2.13), but since we are now working with spherical waves, the appropriate characteristic impedances have to be used, giving: Rrad =

Zrad /Zc − 1

Zrad /Zc∗ + 1

(2.21)

Impedance of a truncated cone In practice, a conical resonator needs to be truncated at its apex, because a non-zero input cross-section is required for the coupling to the exciter, or to a consecutive bore segment. Figure 2.5 shows the impedance of a truncated cone. In a truncated cone, impedance maxima are not harmonically related. For this reason, an instrument composed of a conical resonator associated to a reed is unstable, and usually some alterations are made to its conical shape to bring the impedance peaks back into a harmonic relation.

22

Chap. 2: Reed models

Impedance HdBL

180 170 160 150 140 130 120 0

1000 2000 3000 4000 Frequency HHzL

5000

Figure 2.5.: Impedances of a truncated conical resonator. In dashed red, the simplest model consisting of a perfect reflection at the bore’s end, and no propagation losses. In solid blue, a flanged pipe termination is supposed, and viscothermal losses through the walls are taken into account In real instruments, in addition to the geometrical corrections to the bore there are further aspects that balance this inharmonicity. In particular for reed instruments, the cavity inside the mouthpiece, the reed damping and the flow induced by the reed motion (‘‘pumped flow’’) are found to provide the necessary correction for the overall instrument impedance so that resonance frequencies are harmonically related [Nederveen, 1998]. In a mathematical model this correction can be introduced by adding a small cylindrical section between the exciter and the conical resonator [Scavone, 2002].

2.3.7. Complex geometries In general, the resonator of a wind instrument has a more complex cross-section profile than that presented in the previous sections. This profile can be described by a function S(x ), the cross-section area as a function of the position along the resonator (x). One way to implement an arbitrary cross-section is to divide the whole resonator into segments where it can be approximated by a conical or cylindrical profile. These are then coupled together by assimilating the acoustic variables (p and q) in the extremities in contact between two segments. This is the approach used in our synthesis models (see sect. 8.2.3 for details). A different approach consists in considering that the wavefront is approximately spherical throughout the resonator, and writing the wave equation as a function of the resonator profile S(x ). This kind of wave propagation description is called the Webster equation. Recently, it was shown [Helie, 2003] that the spherical wavefront approximation can be dropped, and the wavefront shape can be determined simultaneously to the wave equation, providing better results for the calculation of input impedances or reflection functions, for instance.

2.4 The exciter

23

2.4. The exciter The main object of this thesis remains the double-reed used as an exciter for double-reed instruments (such as oboes and bassoons). There is no obvious way to distinguish the exciter from the rest of the instrument. This should be regarded as a conceptual distinction, so that it is not always possible to associate a physical part of the instrument to the exciter or to the resonator alone, but it may be possible to distinguish some processes that are clearly related to the creation and maintenance of the oscillation (needed for all self-sustained instruments, as described in section 2.2).

2.4.1. Double reeds as opposed to single reeds The name ‘‘double-reed’’ suggests that the number of oscillating reeds has an important influence in the behavior of the exciter. However, the strong coupling between both blades in double-reeds, imply that their behavior is in fact closer to that of a single vibrating solid than to two independent vibrating structures. In fact, this will be the subject of some observations shown in section 3.3.4 for example, where we give experimental evidence of the joint behavior of the two blades that compose the double reed. A distinction has to be made between the object traditionally associated with the double-reed (because it is detachable from the remaining instrument as a single piece) and the exciter as functionally independent from the resonator. In fact, blowing into an oboe double-reed for instance reveals the acoustical influence of the duct downstream of the vibrating reeds, allowing us to classify this part of the duct as a resonator. For example, shortening the staple has obvious consequences on the frequency of vibration of the reed, and the sound produced by it, very much like the consequences of changing the length of the main bore of an oboe, for example.

2.4.2. Description of the double reed At the end of the fabrication of the reed, the tip is open showing an oblong eye shape. This opening is due to the natural curvature of the cane from which the reed was made and to the pre-constraint induced by tying the reeds together at the butt extremity, but the radius of curvature of each blade is usually less pronounced than that of the original cane, because of the force of contact between the sides of the blade. The elasticity of the cane allows the reed to close when the blades are forced against each other. This can be done either by pressing the fingers against the reed, or by applying a pressure difference between the outside (pm ) and the inside of the reed (pr ). This behavior is similar to that of a spring, whose displacement from the position at rest (x0 ) increases with the force applied to it. The simplest way of describing the relation between the force and the displacement is to suppose a linear and instantaneous relation between both quantities. This model, which will be tested in section 3.4, can be written in the form of a simple equation:

24

Chap. 2: Reed models

pm − pr = kS (x − xO )

(2.22)

where the stiffness ks is given per unit surface. Given its surface-averaged value, ks can be defined from the traditional stiffness k used in the study of a spring by dividing it by the effective surface Seff over which the pressure is exerted: ks = and Seff =

k

(2.23)

Seff F

(2.24)

pm − p r

where F is the total force applied on the reed. Equation (2.24) provides a way of determining Seff , if necessary. In practice we will see that the whole equivalent oscillator can be described in terms of surface averaged parameters, such as ks . A difference in pressure between the outside and the inside causes the reed to change its opening. If this difference is sufficiently high, the two blades will bend until they are in contact with each other throughout the whole length of the reed opening. The velocity of the flow entering the reed also depends on the pressure difference between the inside and the outside of the reed. Using the simplest possible model, one can say that while entering the reed, the total energy of the air is conserved, leading to the Bernoulli condition: pm +

1 2 1 ρum = pr + ρur2 2 2

(2.25)

The volume flow entering the reed can be found by supposing that the velocity is homogeneous over a cross-section of the flow. In the reed we would have qr = Sr ur and in the mouth qm = Sm um . If the flow is uncompressible (which is evaluated in section 6.2.2) both volume flows are identical, because of the condition of mass conservation. Since the flow is distributed over a much bigger surface inside the mouth than at the reed entrance, the flow velocity is usually neglected (um ' 0). The flow law (after replacing the velocity ur by the corresponding flow expression Sqr ) from equation (2.25) can be combined with the elasticity law from equation (2.22) to obtain a curve relating the flow to the pressure difference applied to the reed (fig. 2.6), and described by the equation:

q=

pM − (∆p)r ks

s

2(∆p)r ρ

(2.26)

Rearranging the parameters in equation (2.26) it is possible to derive a simpler nondimensional formulation for the same model:

˜ = (1 − p ˜ )˜ q p1/2

(2.27)

2.4 The exciter

25

PT

PM

Flow

(l/s)

0.25 0.2 0.15 0.1 0.05 0 0

5

10 Pressure

15 (kPa)

20

Figure 2.6.: A typical non-linear characteristics curve for a reed of dimensions similar to an oboe reed with ks

r

ρ

˜ q

=

q

(2.28)

˜ p

= (∆p)r /pM

(2.29)

3/2

pM

2

This formula shows that the shape of the non-linear characteristic curve of the elementary model is independent of the reed and blowing parameters, although the curve is scaled along the pressure p and volume flow q axis by the stiffness ks and the beating pressure pM = kS0

2.4.3. Dynamic effects of the reed In a normal utilisation of the double-reed – as an exciter of a wind instrument – the reed system is far from a static case. In section 2.4 we have stated the simplest model possible for the reed elasticity. Beyond the fact that the relation between pressure and reed opening might not be linear, there are several other factors that are neglected in this expression in what concerns the vibration of the reed without the coupling to the flow. Due to the inertia and damping, the reed can be seen as a harmonic oscillator driven by the external forces, which are imposed by the difference between the nearly static mouth pressure applied by the musician pm 2 and the time-varying pressure inside the reed pr (t ). If we consider the external forces to be distributed along an effective surface S eff , the harmonic oscillator can be described in terms of an effective mass per unit surface 2

In fact time variations of the mouth pressure can be considered and can explain variations in timbre and attack times which musicians can control by varying the shape of their vocal tract [Fritz, 2004]

26

ms =

Chap. 2: Reed models m Seff

and damping rs =

r Seff

:

pr (t ) − pm = kS (x (t ) − xO ) + rs

∂x ∂t

+ ms

∂2 x ∂t 2

(2.30)

This equation can be somewhat simplified if we consider only the variations of x (t ) relative to its equilibrium position x0 , written as ∆x (t ) = x (t ) − x0 , and the pressure difference between the inside and the outside of the reed ∆p(t ) = pr (t ) − pm . Moreover, the whole equation can be divided by the surface mass ms :

∆p(t ) ms

= ωr2 ∆x (t ) + 2α

∂(∆x ) ∂t

+

∂2 (∆x ) ∂t 2

(2.31)

Equation (2.31) should replace equation (2.22) as a simplified mechanical description of the reed. Eventually, the reed will have to be studied and modelled as a vibrating solid body with its own modes and frequencies of vibration. A preliminary study is presented in appendix B, but higher modes of vibration will not be considered in the present work.

2.4.4. Considerations about the elementary model For the sake of simplicity, several factors have been disregarded in the former analysis, and we will try to discuss some of them in detail, giving an idea on whether or not they may be important in the description of the instrument, and why. As much as possible, experimental data accompaigns the analysis, trying to justify the importance or unimportance. In other cases, when experiments were not possible with the available equipment, possible hypothesis are advanced, based on theoretical or dimensional analysis.

2.5. State-of-the-art in double-reed physical modelling Reed instruments have been the subject of numerous studies in the past, and they still arise a continuing interest, mainly because of the relative simplicity of their modeling. In fact, for example the form of the non-linearity present in reed exciters is much simpler than that of bowed strings, a family included in the wide group of self-sustained instruments. Present studies vary from analysis of the dynamic system constituted by the reed and the resonator [Kergomard, 1995], period-doublings and route to chaos [Maganza et al., 1986] [Liz´ee, 2004], arbitrary resonator geometries for the resonator [Ducasse, 2001] [Helie, 2003], the effect of conical geometries [Scavone, 1997b] or the inclusion of non-linear effects in the propagation inside the resonator [Gilbert et al., 2005]. The particular aspect of the propagation in the vocal tract was also included in classical models [Fritz, 2004] in order to test its importance. In the particular domain of single-reed instruments, the exciter model has been refined for example with respect to the mechanical properties of the reed such as the viscoelastic properties of the material [Casadonte, 1995] [Obataya and Norimoto, 1999], the

2.5 State-of-the-art in double-reed physical modelling

27

vibrational properties of the reed [Pinard et al., 2003] or the effect of the particular geometry of the table which supports the reed [Gazengel, 1994]. Flow details were also addressed, investigating the effects of the formation of a jet at the reed input [Hirschberg et al., 1990]. However, the details of the exciter seem to have arisen a greater interest in singlereeds than for the particular case of double-reeds. One of the reasons for this condition is without doubt the relative difficulty of carrying out experiments and measurements in the relatively tight double-reed. A certain number of articles can be found about double-reed instruments though, on some particular aspects. The details of the bores and the reed used in double-reed instruments such as the oboe or the bassoon is discussed and compared to other woodwinds in [Nederveen, 1998]. Experimental measurements of the parameters of the double-reed exciter can be found for example in [Almeida et al., 2004b], with a first attempt to obtain the non-linear characteristics. Aspects connected to the performance of the oboe were approached by [Fuks, 1998]. Other aspects include the particular oscillation cycle of several double-reed instruments, which is very asymmetric when compared to that of the clarinet [Gokhshtein, 1981] ´ 2001]. This is a particularity of conical resonators, more than [Rocaboy, 1989] [Agullo, of the double-reed itself, and of the fact that in these kind of resonators the acoustical cycle is determined by a single symmetric reflection at the end of the resonator rather than a sequence of 3 reflections (at the end, at the top and again at the end) in cylindrical resonators [Rocaboy, 1989]. These authors agree that the fraction of the period during which the reed remains open is determined by the acoustical propagation in the resonator (variable according to the playing frequency) and that the time of closure is independent of the playing frequency. However, the authors diverge as to what determines this time. For some, it is the frequency of vibration of the volume of air trapped inside the reed [Rocaboy, 1989]. Others consider that it is the truncation length of the cone [Dalmont et al., 2002]. However, none of these justifications explain the increase in the closure time as the mouth pressure is increased [Gokhshtein, 1979]. [Brown, 1990] uses Rocaboy’s theory to explain the change in tuning that can be achieved by changing the embouchure parameters. According to him, these have a primary effect on the value in the pressure wave at which the transitions between open and closed regimes is observed. The existence of two symmetric movements of the reed motion was demonstrated for the simplified Helmholtz motion in conical resonators [Dalmont et al., 2002], inverting the durations of open and closed states. Moreover, the existence of other regimes of vibration (in particular with more than one closed period per cycle and quasi-periodic regimes) was observed in the bassoon [Shimizu et al., 1989]. In this article, the authors analyse the parameters that cause transitions between regimes. For the Tenora, a complete physical model has been proposed and simulated in the ´ 1989]. The reed model remains similar to the classical time-domain [Barjau and Agullo,

28

Chap. 2: Reed models

model presented in section 2.4, but the relation between the reed position and the opening area is changed from linear to quadratic. The simulated pressure waves are compared to waves measured in a real instrument. The reed motion, and its relation to the pressure signals inside the reed is analysed in [Gokhshtein, 1979]. The same author proposes a model for the pressure distribution inside the reed, and discusses the effects of changing the Vena Contracta at the reed input by rounding off the reed’s edges [Gokhshtein, 1981]. Lately, the flow model inside the reed was the subject of some studies [Wijnands and Hirschberg, 19 proposing that a constriction downstream of the reed increases the resistance seen by the flow. Although this model was proposed for some particular kinds of reeds [Hirschberg, 1995], it has been since then adopted as a model for double-reeds in the musical acoustics literature [Fletcher and Rossing, 1998]. An investigation on the effects of the hysteresis inherent to this model on simulated instruments is given in [Vergez et al., 2003].

Part II.

Characterisation of the double-reed

3. Geometrical and Mechanical aspects of the reed In this chapter we introduce some basic experimental data on the mechanical properties of the double reed, and the shape of the reed opening. Firstly, the geometry of the reed opening is studied as a function of pressures applied to it. Then, we address the elastic properties of the reed, including viscoelastic effects in the reed material. Finally, the vibrating properties of the reed are analysed through a scanning of its mechanical response.

3.1. Introduction The double-reed is a complicated system involving two main components that influence each other: the elastic reed blades, which can be deformed by external forces, and the air flowing between them [Hirschberg, 1995]. These two components can be modelled independently, and the interaction between both established through a set of coupling variables. The aim of this chapter is to gather some experimental informations about the solid, elastic reed, as much as possible as an independent system from the air flow that runs through it in normal utilisation. A first investigation concerns the variations in the reed opening shape for typical pressure variations expected in a musical use, and the following experiments are dedicated to the elastic and vibrational properties of the reed. Most of the experiments described in this chapter try to minimise the effects of the flow. However, for measurements of the evolution of the reed shape we compared static to dynamic regimes.

3.1.1. Overview of the chapter The chapter begins with measurements of the cross-section profile along the reed (sect. 3.2). In section 3.3 we study the variations in the shape of the reed opening for a range of pressures found in normal uses of the reed. The main aim is to identify a single variable that can describe the reed opening state in a one-to-one relation, during normal operating regimes. In particular, a model for the double-reed requires the knowledge of the relation between the opening area and the displacement of the reed. Section 3.4 is dedicated to measuring and discussing the relation between the pressure applied to the reed and the variation in the reed opening area, allowing to propose a spring model for the reed.

32

Chap. 3: Geometrical and Mechanical aspects of the reed

Section 3.5 discusses the non-instantaneous character of the previous relation, trying to link it to the viscoelasticity in the reed material. Finally, section B.1 investigates the vibrational properties of the reed for small driving forces and negligible induced flow effects.

3.2. Cross-section profile of the double-reed Along this work, it will be necessary to know the internal geometry of the reed. This information is important for instance when discussing the evolution of the flow inside the reed (chapter 5). The internal geometry of the reed was measured indirectly, by filling it with a casting material after having applied a de-moulding liquid, so that the cast would not stick to the reed. For the casts, several materials were tried out and finally, two revealed successful: one solidified completely into a hard mould requiring the destruction of the reed, while another more supple material (Rhodorsil) required only that the reed blades were unmounted. For the digitalisation of the cast surface, an automatic topography method was tried out (described in appendix J). The development of this method would require further investment in time and material, so that finally a semi-manual method was used to extract some key-points in the cast surface (see appendix J.1). The profiles extracted for a synthetic oboe double-reed are presented in figure 3.1. For natural cane-reeds, it is more difficult to assure that the material used for the casting does not deform the reed blades. The cross-section area is seen to increase constantly from the reed tip till the reed output, with a slight discontinuity at the staple input. In the upstream part of the reed, bounded by the blades, the width of the reed decreases from the tip till the staple input, whereas the height increases in the same direction.

3.3. Geometry of the reed opening In this section, we investigate the shape of the reed entrance for different opening states of the reed. The reed entrance, often called reed opening in this document, is the space between the tip of the two blades, through which the air flows in a normal utilisation of the instrument. Among other observations, we examine whether there is a characteristic shape of the reed opening for a particular opening area, and consequently we can describe the reed opening state through a single variable. These investigations are firstly driven by the interest of having a simple model of the reed input, so that ideally a single variable is sufficient to describe the shape of the reed opening. Its area can therefore be related to this variable, and eventually the flow entering the reed can also be linked to the same variable.

3.3 Geometry of the reed opening

33

8

dimension (mm)

height width 6

4

2

0

0

10

20

0

10

20

30 40 axial distance (mm)

50

60

70

50

60

70

area (mm2)

15

10

5

0

30

40

Figure 3.1.: Measurements of the cross-section profile on an oboe double reed

34

Chap. 3: Geometrical and Mechanical aspects of the reed

There is still a more general interest in characterising the reed opening shape. Musicians and reed-makers usually have an idea of the shape the reed should have in order to guide the preparation of the reed. In addition to this idea of the reed opening shape at rest, they test the reed opening shape by pressing the reed between the fingers and observing the variation of the shape. In literature dedicated to musicians and reed makers, some information can be found about the shape of the reed. No consensus seems to exist about an ‘‘ideal’’ shape of the reed opening. This subject, like many other issues on the reed making, seems to be greatly dependent on the kind of instrument, the formation of the musician, and the musician’s preferences as well. As an example, two modes of transition between closed and open states are shown in figure 3.2. These can be found in some works about reed manufacture [Smith, 1992].

h

h

1 – proportional

2 – diamond-shaped

Figure 3.2.: Transition between open and closed states of the reed opening for two different designs (according to Heinrich [Heinrich, 1979])

Still, some of the indications left by musicians in these works were transported into scientific works, for example in order to propose physical models of double-reed instru´ 1989]. In scientific works it is common to read [Nederveen, 1998] ments [Barjau and Agullo, that double-reeds differ intrinsically from single-reeds in the geometrical model of the flow input. In particular, it is proposed that while closing, both dimensions (width and height) of the reed opening are reduced, implying a 2nd order power law for the dependence of the area upon the reed height (S(h ) ∝ h 2 ).

3.3.1. Experimental approaches Because the reed is a flexible solid, the deformation of the reed blades depends on the distribution of the force over the reed surfaces. In a common utilisation of the reed, there is a constant pressure applied by the lips over a region of the reed. When blowing the reed, the pressure increases in a the part of the reed that remains inside of the mouth (fig. 3.3, 1). Moreover, when a note is played, between the two reed blades there are periodical variations of the pressure. When testing the reed during its preparation, the force is distributed differently over the blades (fig. 3.3, 2). Any of these ‘‘usual’’ distributions of pressure can be difficult to reproduce exactly in experiments, so that more homogeneous distributions were used in practice.

3.3 Geometry of the reed opening

pm

35

Lip force

Reed pressure

1 – Musical performance

2 – Preparation

pm pm Reed pressure

3 – experimental, non-oscillating

4 – experimental, oscillating

Figure 3.3.: Some common pressure distributions used in the double-reed, on top for common uses of the reed by musicians and on the bottom for our experiments. Some measurements and observations were tried out using artificial lips in the artificial mouth. For most experiments however, the pressure was evenly distributed along the reed external surface and left free to adapt itself in the internal surface (fig. 3.3, 4), and the results did not vary much from the results shown in the following sections. In fact, the force acting on the reed is expected to have a greater influence on the reed tip than on the remaining reed for two main reasons: the reed blades are thinner there, and since the reed can be seen as a cantilever beam fixated on the staple edge, the moment of the force increases towards the reed tip. Operating Regimes For the analysis of the reed opening shape, it was judged important to compare some of the observations for non-oscillating (static) and oscillating (dynamic) regimes, so that some conclusions can be applied later to the movement of the double reed, and eventually to a physical model of the double reed. Static regimes One way to access the whole range of reed openings in a statice regime is to prevent the reed oscillations by covering the input section of the reed with a plastic film (fig. 3.3, 3), blocking the airflow. The reed is thus submitted to a pressure difference which can be adjusted easily by increasing the blowing. One other way is to use a diaphragm at the reed output, which was done in the measurements of chapter 4. Similar observations can be made without the application

36

Chap. 3: Geometrical and Mechanical aspects of the reed

of a plastic film, however, flow effects are more important in this case. In fact, the plastic covering can influence the results in two particular aspects: • Increasing the actual force on the reed, because the surface upon which the pressure acts is slightly increased. However the difference can be considered small (the surface covering the reed opening is small when compared to the blades’ area). • Changing the distribution of forces on the reed, which could influence the way the reed closes. Nevertheless, observations of the reed geometry while closing with and without the plastic film covering (in a non-oscillating case) were compared to check that the geometry is basically the same. Because a light source placed in front of the reed would produce specular reflections on the plastic covering, the experiment is lit from behind. The reed opening thus appears as a bright region on a dark background (left-hand side of fig. 3.4), which has the additional advantage of producing an image with better contrast, and an easier identification during the subsequent image analysis (see sect. 3.3.3). Dynamic regimes For dynamic observations, the artificial mouth is also used to generate the pressure that drives the reed into self-sustained vibration, but now the input section of the reed is left uncovered. The reed is illuminated with a stroboscopic light, synchronised with the period of the sound wave produced by the instrument (or by the reed alone), as described in section G.2. Reeds used in this study The results presented in this section refer to natural cane oboe double-reeds, used as bought from Glottin (no further scraping applied). However, these are basic analysis that can be performed on any reed, so that most results apply and were observed on other kinds of oboe double-reeds. Substantial differences observed in certain reeds are mentioned when needed.

3.3.2. Observations Figure 3.4 shows frontal views of the reed opening using two different methods of applying the pressure. In sequence 1, a plastic film covers the reed so that no air flow can enter the reed. The reed does not oscillate during observations. In sequence 2, the photographs were captured while the reed was oscillating, with the help of a stroboscope synchronised with the sound produced by the instrument. These observations show that the contact between the blades is almost simultaneous along the reed opening length, in both static and dynamic regimes (see figure 3.4). In fact, as the reed opening closes, its shape changes as if it was scaled along its minor axis, emulating the behavior shown in the left of figure 3.2.

3.3 Geometry of the reed opening

1 – Non-oscillating

37

2 – Oscillating (strobe)

Figure 3.4.: Observations of the reed opening at different closing states. 1 – Variations are slow; 2 – different phases of a high frequency oscillation

38

Chap. 3: Geometrical and Mechanical aspects of the reed

Due to the utilisation of back-lighting, the left-hand side of figure 3.4 can highlight small openings that remain even for high mouth pressures, which are difficult to observe with front lighting. Moreover, a slight reduction of the reed opening length is observed in the start of this series, which is probably linked to the fact that the tip of some natural cane reeds are not in contact in the edges when no pressure is applied to them. In fact, a regular use of the reeds and the consequent humidification tends to bring the edges together at rest. After the first stages of the reed closure, the reed length remains practically constant, ´ 1989]. a different behavior from the model proposed in [Barjau and Agullo, The reed shape does not vary considerably from the shape of the reed opening at rest, suggesting that one single coordinate describing the distance between the two blades is sufficient to parameter the shape of the reed at any time, for example the central distance between blades h or the reed opening area itself S. One implication of this observation is that no transverse modes of vibration of the reed are observed. In fact, a transverse mode would imply that the shape of the reed opening changes throughout one period, with a higher frequency.

3.3.3. Quantitative analysis Although the observations above already provide interesting information about the reed opening geometry, methods described in appendix H allow us to determine the function relating the reed coordinate z (the maximum distance between the two blades at the reed tip) to the corresponding opening cross-section S(z ), and quantify the non-linearity in the geometric relationship S(z ). These measurements can usually be done on the same images as those used in qualitative observations. Figure 3.5 represents the reed opening area as a function of h, for different openings. A rectangular area in which only one of the dimensions is altered (like the clarinet) would produce a linear distribution, with the other dimension as slope. On the other hand, a variation like the one represented on the right hand of figure 3.2, would produce an almost quadratic distribution, because both dimensions vary at the same time when reeds close. For the double-reed, the obtained distribution (figure 3.5) is mostly linear, which confirms the direct observations described in the beginning of this section. The slope of the graph corresponds to the effective length of an equivalent rectangular reed opening (with the same area), which is smaller than the length measured on a real reed. The coefficient between effective and real lengths is a parameter to be used in mathematical models of the double-reed, since it controls the magnitude of the volume flow that enters the instrument for a given distance between the blades of the reed. Note that all the reeds (both natural and synthetic) showed a linear relationship between their opening area and width. However, certain reed makers claim that depending on the reed, both cases depicted in figure 3.2 are possible. In practice, should the behavior shown on the right part of figure 3.2 be observed, an exponent (usually between 1 and 2) would be included as a parameter for the fitting to the data in figure 3.5.

3.3 Geometry of the reed opening

39

−6

4

x 10

3.5 3

2

Area (m )

2.5 2 1.5 1 0.5

Area=5.183492e−03*width+−6.253252e−08

0 0

1

2

3 4 Width (m)

5

6

7 −4

x 10

−6

2.5

x 10

1.5

2

Area (m )

2

1

0.5

0 0

Area=5.354350e−03*width+−1.044430e−07

1

2

3 Width (m)

4

5 −4

x 10

Figure 3.5.: Opening area measurements on a natural cane reed (a, on top – without lips and b, with artificial lips).

40

Chap. 3: Geometrical and Mechanical aspects of the reed

3.3.4. Symmetry In oboe-like instruments, the two reed blades control the air flow into the instrument. One fundamental question is to know if the two blades oscillate synchronously and symmetrically in order to check whether both reeds can be considered as oscillators with identical mechanical parameters. With the help of the stroboscope, periodic reed motion can be inspected by eye. For relatively low frequencies (compared to the reed natural frequency), the two blades oscillate in phase, and their motion looks symmetric at first sight (see figure 3.4). This conclusion stands for many recorded sequences of images and was also observed directly at playing frequencies non-multiple of the camera recording rate, for which video recording is difficult. We propose to draw quantitative information from the image analysis. First, the main-axis of the reed is determined as the greatest straight line that can be traced inside the reed opening region. The sub-area on each side of the reed main axis are then calculated. The left sub-area and the opposite of the right sub-area are displayed in figure 3.6 against time (case of a plastic reed). It is obvious that for that particular reed, both blades oscillate synchronously and symmetrically. This results stands for other plastic reeds. However, in some natural cane reeds, asymmetry can be noted: the two blades motion, while synchronous, sometimes have different amplitudes. This indicates an ill-constructed reed, as mentioned in section 1.3. It is frequent in fact to observe that the two blades have slightly different curvatures at rest. Also, for other regimes (quasi-periodic, multiphonics or squeaks) symmetry is not always observed. However, these regimes are only used in special occasions, or not used at all in musical interpretation. Note that all observations were made in steady states, so that transients have been disregarded.

3.3.5. Conclusions on the reed opening shape The results shown in this section indicate that for a normal musical use it is accurate enough to describe the double-reed as a system with one degree of freedom. This does not necessarily mean that both blades are identical, but that the coupling between both is sufficiently strong to make them work as a single oscillator. Even in cases where the double-reed showed a clear asymmetry (results not shown in this section), the movement of both blades was observed to be simultaneous. However, as mentioned in section 1.3, such asymmetric reed-openings should be considered to be ill-designed or unfinished. None of the observed reeds has shown a tendency for the quadratic law relating the distance between reeds (h) and the reed opening are S(h ), which was proposed by ´ 1989] for the Tenora (however Tenora reeds were not studied by [Barjau and Agullo, us). Such quadratic laws for S(h ) are also implicit in some designs proposed by reed makers and musicians (fig. 3.2). As described above, these observed closing modes of the reed can be due to the force distribution applied for testing during the design

3.4 Reed equivalent stiffness

41

−6

1.5

x 10

right left

Opening area (m2)

1

0.5

0

−0.5

−1

−1.5 0

2

4

6 Time (s)

8

10

12

Figure 3.6.: Time evolution of the half areas of the slit (on each side of the reed opening axis) (fig. 3.3). Otherwise, the fact that such closing modes were not observed by us may be justified by the relative scarceness of finished reeds in the samples tested by us, revealing a lack of variety in the analysed reeds.

3.4. Reed equivalent stiffness When modeling the double reed as a harmonic oscillator, the quasi-static regime corresponds to an instantaneous relation between the reed displacement (z) and the force applied (F ), because time derivatives implied in inertia and damping can be neglected. In such a model, the force is distributed over the reed surface, so that it is more straightforward to measure a surface stiffness k as a ratio between the reed displacement and the pressure applied k = pz . For this measurement, a synthetic reed is inserted in the artificial mouth, being subjected to a controlled pressure from its outside. Stiffness is estimated through the instantaneous and static relation between the pressure and the displacement, so that oscillations have to be prevented in order to keep inertia and damping negligible. We use the static method as described in section 3.3.1 to prevent the oscillations. Pressure is increased progressively until the reed closes (points A to B in fig. 3.7). When the reed is almost closed, pressure can be increased with very small consequences on the reed opening (hook on the upper right end in figure 3.7, corresponding to points

42

3 2.5

Measured Fit

C

B

2 1.5 1

2

Reed area (mm −− difference from equilibrium)

Chap. 3: Geometrical and Mechanical aspects of the reed

0.5 D 0 −0.5 −5

A 0

5 10 Pressure (kPa)

15

20

Figure 3.7.: Measurement of the applied pressure as a function of the reed displacement from its rest position, and fitted linear function to the decreasing pressure branch (points C to D).

B to C): it is in fact very difficult that the measured area reaches zero because light tends to diffuse when the reed is almost closed. After the reed was kept closed for a few seconds, the same measurement is performed while decreasing the pressure (points C to D in figures 3.7 and 3.8). The reed opening area decreases at a rate similar to that found while increasing the pressure, but the curve shows an offset towards slightly smaller reed openings. In fact when the pressure is released the reed opening has a larger area than in the beginning of the experiment. A likely interpretation is that the reed blades are slightly deformed by the continued force applied to the reed, suggesting that a more complicated model should be used to explain this behavior (see sect. 3.5). Besides, the relation between the opening area and the pressure applied on the reed is mostly linear, which is quite unexpected for such a complex system. Given that the slope of the graphic does not vary much between increasing and decreasing pressures, stiffness can be calculated as this slope (neglecting the offset). In figure 3.7 it is found to be approximately 6.15 × 109 Pa · m −2 for decreasing pressures. This value can be compared with that found for the clarinet, about 0.8 × 109 Pa · m −2 , based on measurements by S. Ollivier and J.-P. Dalmont [Dalmont et al., 2003], which means that the opening area is more sensible to pressure variations in clarinet reeds than in oboe reeds. This may account to some extent to the results obtained for the comparison of oboe and clarinet non-linear characteristics (sect. 4.3.8). The measurements presented in this section were obtained for a synthetic reed. The

3.5 Viscoelastic effects

43

20

4 pressure 3 2

Pressure (kPa)

C

B

opening (mm )

reed displacement

15

10

2

5

1

0

−5 0

0

D

A

10

20

30 Time (s)

40

50

−1 60

Figure 3.8.: An example of time variations of pressure and reed displacement during a stiffness measurement. linearity of the elasticity curve observed in figure 3.7 is not perfectly maintained in natural cane reeds. In fact, the stiffness is seen to increase for small reed openings in dry reeds. Wetting the reed seems to bring the curve back to a linear elastic regime (see appendix F)

3.5. Viscoelastic effects The reed deformation suggested by the observations in the previous section can be taken into account by means of a model of viscoelasticity, a phenomenon frequently observed in material science. A simple model can be used to describe the reed deformation according to the time evolution of the force (expressed as the convolution with a kernel function, for example a decreasing exponential) [Christensen, 2003]. Such a model states that when a constant force is applied to the material, its deformation increases with time, tending to a constant value that is proportional to force. Similarly, after this force is suddenly released, the deformation is still observed, but it tends back to zero as time evolves in a way that can be modeled considering an exponential kernel as: x0 =

P kve

e −t/τ

(3.1)

The parameters of the model are then the relaxation time τ, and the viscous stiffness kve . P is the pressure step that forced the reed.

44

Chap. 3: Geometrical and Mechanical aspects of the reed

Measured

2

Reed Opening(mm )

2.7

Exponential sum fit

2.6 2.7 2.5

2.6 2.5

2.4 2.4 0 2.3 0

200

10

400 600 Time (s)

20 800

30 1000

Figure 3.9.: Measurement of the viscoelastic relaxation time and fitted sum of two exponentials (inset shows a zoom for the short-term relaxation).

3.5.1. Synthetic reed In order to estimate the parameters τ and kve , a constant pressure is applied to the reed in the artificial mouth (28 kPa), keeping it shut for 30 minutes. The pressure is then suddenly reduced to 0, so that the reed should re-open to its rest position. The reed opening is recorded on video for several minutes after relaxing the pressure. Image analysis is used to measure the reed opening area (vertical axis in figure 3.9) along the time (horizontal axis). Measurements are affected by a certain amount of noise, because the opening variations are small when compared to the absolute reed opening, but the overal trend of the reed opening can be distinctively recognized. The source of the noise is mainly the grain of the image (fluctuations in the intensity of each pixel from one image to the other). When observing the data for the full length of the measurement (about 15 minutes) the opening area is seen to increase slowly at a decelerating rate. A measurement realized several hours after the acquisition in figure 3.9 shows that the opening area stabilized at 2.83 mm 2 . This long-term evolution can be modelled using equation (3.1) with parameters τ = 850 s and kve = 150 × 109 Pa · m −2 .

However, this model fails to describe the evolution of the reed opening area in the short-term (figure 3.9, inset), because there is a quicker transition during the first few seconds after the pressure release. This relaxation period can be described by adding another function in the form of equation (3.1) with parameters τ = 2.5 s and kve = 150 × 109 Pa · m −2 . This fact is also common in the study of complex materials,

3.5 Viscoelastic effects

45

where several relaxations are observed with different time constants, corresponding to different scales in the material structure. Note that the similarity of the values found for kve in both relaxations is merely a coincidence and was not imposed in the analysis of the results. In clarinet reeds, a similar behavior is observed [Dalmont et al., 2003], with a longterm relaxation close to 900 s, and a short-term relaxation of longer duration than for double reeds (about 8 s). Both relaxation times are several orders of magnitude larger than the period of oscillation of the reed (usually smaller than 10−2 seconds), so that viscoelastic effects can be neglected for a simple model which describes the instrument behavior (for sound synthesis, for instance). In practice, viscoelastic effects tend to slightly close the reed after some time of playing, because there is an average force over the reed applied during large intervals of time. This effect is probably balanced by the musician, who slightly releases the force of the lips over the reed, but its effects are observed on experiments with the artificial mouth: when playing during long intervals, changes in frequency and intensity are observed as if the reed parameters were changing. In fact, if the reed is not stable enough and visco-elastic effects cause strong variations in the reed opening at rest, the musician periodically presses the reed by its rails in order to re-increase the opening area. As another example, during the experiments carried out in section 3.4, the closing and opening of the reed lasted for one minute in average. The short-term relaxation can contribute to the difference in opening area values before and after the reed is forced for some seconds (see figure 3.7).

3.5.2. Humidified natural cane reed The previous experiment was not repeated for a humidified cane reed, however, the relaxation of the reed can also be studied after a different time-distribution of the stress. In figure 3.10 are presented the reed-opening variations after a 3-minute measurement typical run for measuring the reed characteristics, as done in chapter 4. There are two main differences from the previous analysis with respect to the time evolution of the force applied to the reed. Firstly, the stress is not constant before the release of the pressure, and second, the time during which the stress is applied is smaller than the typical values of the slowest relaxation times found previously. This means that some ‘‘memory’’ of the state of the reed before the experiment is conserved throughout the whole run. The relaxation time can be measured on the graph, considering that the relaxation begins at 165 seconds verify in MatLab, and that the reed opening stabilises at the same value as observed around 45 seconds, the reed opening reaches 60% of its final (e −1/2 ) value at 180 seconds, yielding a relaxation time (τ) of 30 seconds. The differences of the relaxation times found in this graphic when compared to that of figure 3.9 are probably related to different material used for the reed.

46

Chap. 3: Geometrical and Mechanical aspects of the reed

4

50

3.5

40

0.4 4

0.35

20

2.5

2

0.25

10 0 0

2

100

1.5

0 200

0.2 0.15

1

0.1

0.5 0 110

0.3 2

3

opening (mm )

Pressure difference (kPa)

30

120

130

140

150 160 Time (s)

170

Pressure 0.05 Opening 0 180 190 200

Figure 3.10.: Zoom on the time evolution of the reed opening for low displacements of the reed relative to its rest position.

3.6. Partial conclusions In this chapter we analysed some important details for a model of the double reed. The internal cross-section profile was measured, showing a divergent duct throughout most of the length of the reed. These measurements will be useful in the remaining of this work. It was demonstrated that the reed opening area S(z ) varies linearly with the distance between the blades z, and during the oscillations of the reed no higher transverse modes were observed. When applying a pressure difference across a dry natural cane reed without flow, the reed opening area varies non-linearly according to the applied pressure, showing a higher stiffness when the reed is almost closed. For synthetic reeds and natural soaked reeds a mostly linear relation is observed, allowing to attribute a stiffness constant to the reed. These measurements unveiled a viscoelastic behavior of the reed. By completely releasing a pressure applied during a sufficiently long period of time and observing the opening area variations after the release, we were able to measure the relaxation times of the reed material and the amplitude of the viscoelastic effects. These are long-term effects that can be seen as a gradual shift of the opening area at equilibrium (S0 ). These results will be used in the interpretation of further experiments realised throughout the remaining of this work, and provide important information for the tuning of the model described in section 2.4.

4. The nonlinear characteristics of the reed In this chapter we try to characterise the double-reed in static regimes. The nonlinear relation between the pressure drop ∆p in the double-reed and the volume flow crossing it q is measured for slow variations of these variables. The volume flow is determined from the pressure drop in a diaphragm, a technique used in the past for similar measurements in clarinet mouthpieces [Dalmont et al., 2003]. The influence of environment and experimental conditions on this relation is verified for double-reeds, and the measurements are compared to other reed instrument exciters and to physical models.

4.1. Introduction 4.1.1. Context It is a common assumption that a self-sustained instrument model can be divided into a resonator and an exciter block [MacIntire et al., 1983]. The two blocks are connected through a set of coupling variables. In section 2.2.3 a minimal set was identified as the pressure p and volume flow q at the input of the resonator. The physics of resonators was extensively studied in the past, with increasing refinement. . In musical instruments, the resonator can be described with sufficient accuracy using a one-dimensional distribution of pressures p and volume flows q along it. In turn, the values of these variables at the bore entrance are well characterised by a linear, frequency-dependent relation (see sect. 2.3). From the reed side, a relation between p and q is also required to describe the behavior of the complete instrument. The coupled solution of the two relations determines the time evolution of variables p and q and of the complete instrument. This relation is called the non-linear characteristic of the exciter. In fact, whereas we know that in the bore a linear relation models reasonably well its behavior, in the case of the reed this is obviously not true. In fact, a negative slope in the p/q relation is required to maintain auto-oscillations of the instrument, and a non-linearity to limit the growth of the oscillations [Maganza, 1985]. It was previously shown that even for a simple model of a reed exciter (sect. 2.4) this relation is highly nonlinear. Static characteristics Due to dynamic properties of the reed (for instance, its inertia and damping, see sect. B.1) and of the flow (again, the inertia of the flow) the non-linear relation between p and q is likely to be non-instantaneous . In this chapter however, we will try to bypass this problem, and restrict the study of the reed to slow variations of the coupling variables. By slow it is understood that

48

Chap. 4: The nonlinear characteristics of the reed

the time-scales of these variations are larger than the typical delays involved in the relations between the two variables p and q in the reed. Measured in such conditions, the relation will be called the quasi-static non-linear characteristic. This restriction is imposed by the difficulty of measuring quickly varying volume flows, and the complexity of a measurement which would take into account a great range of time-scales. However, a great amount of information can be extracted from the quasistatic regime, and it can be shown that an instantaneous relation is a good approximation to the actual non-linear characteristics when the playing frequency of the instrument is low when compared to the resonance frequency of the reed [Wilson and S., 1974] [Dalmont et al., 1995]. Double-reed quasi-static characteristics The nonlinear characteristics curve measured on a double-reed can be compared to that of single-reed instruments. It can be checked whether the nonlinear behavior of the reed in low-frequency regimes can be described by a generic model such as described in section 2.4 and successfully used for single-reed instruments, or if some particularities of the flow inside the reed imply a modification of the generic model used for single-reeds, as described in chapter 2, and suggested based in theoretical considerations about the flow inside the reed [Hirschberg, 1995].

4.2. Principles of measurement and practical issues The characteristic curve requires the synchronised measurement of two quantities: the pressure drop across the reed (∆p)r and the induced volume flow q.

4.2.1. Volume flow measurements One of the main difficulties in the measurement of the reed characteristics lies in the measurement of the volume flow. There are instruments which can accurately measure the flow velocity in an isolated point (LDA, hot-wire probes) or in a region of a plane (PIV), but it can be difficult to calculate the corresponding flow by integrating the velocity field. In fact it is difficult to do a sampling of a complete cross-section of the reed because a large number of points would have to be registered. Since the flow is found to be approximately axisymmetric at the reed output (see sect. 5.6.3), the measurement along a diameter of the reed would be sufficient, but regions close to the wall are inaccessible. Another difficulty would be making sure that the profile is being measured along a diameter. On the other hand, commercial flow-meters considered for these experiments have the disadvantage of requiring a direct reading, which would have been unpractical for a complete characteristic measurement, since it requires a large number of reading in a short time interval. An indirect way of measuring the flow was then preferred to the above mentioned methods. It consists in introducing a flow resistance in series with the reed, for which the pressure can be accurately related to the flow running through it (see fig. 4.1).

4.2 Principles of measurement and practical issues

49

The diaphragm method, used successfully by S. Ollivier [Ollivier, 2002] to measure the non-linear characteristic of single reeds, is based on this principle. The resistance is simply a perforated metal disk which covers the reed output.

(∆p)s PSfrag replacements





(∆p)r







pm 1



    p   

(∆p)d

r

 

patm

2

Figure 4.1.: Use of a diaphragm to measure flow and pressure difference in the reed. Numbered rectangles correspond to the pressure probes used in the measurement. For such a resistance, and assuming laminar, viscous less flow, the pressure drop (∆p)d = pr − patm across the diaphragm can be approximated by the Bernoulli law1 :

(∆p)d = pr − patm

1 = ρ 2



q Sdiaph

2

(4.1)

where q is the flow crossing the diaphragm, Sdiaph the cross section of the hole, and ρ the density of air. In our experiment, pressure patm is the pressure downstream of the diaphragm (usually the atmospheric pressure, because the flow opens directly into free air). The volume flow q is then determined using a single pressure measurement p r .

4.2.2. Practical issues and solutions Issues The realisation of the characteristic measurement experiments encountered two main problems: Diaphragm reduces the range of (∆p)r for which the measurement is possible The addition of a resistance to the air flow circuit of the reed changes the overall nonlinear characteristic of the reed plus diaphragm system (corresponding to (∆p) s and to the dashed line in fig. 4.2). The solid line plots the flow against (∆p)r , the pressure drop needed to plot the non-linear characteristics. When the resistance is increased, the maximum value of the system’s characteristic is displaced towards high pressures 1

The flow velocity at the reed output is neglected when compared to the velocity inside the diaphragm (Sdiaph  Soutput )

50

Chap. 4: The nonlinear characteristics of the reed

[Wijnands and Hirschberg, 1995], whereas the static beating pressure2 value does not change (because when the reed closes there is no flow and the pressure drop in the diaphragm ((∆p)d ) is zero). Q (l/s) 0.3

0.2

0.1

10

20

30

40

50

60

p (kPa)

−0.1

Figure 4.2.: Comparison of the theoretical reed characteristics (solid line) with the reed plus diaphragm system characteristics (dashed) — mathematical models, based on the Bernoulli theorem [Wijnands and Hirschberg, 1995]. Therefore, if the diaphragm is too small (i. e., the resistance is too high), part of the decreasing region of the system’s characteristics becomes vertical, or even multivalued, so that there is a quick transition between two distant flow values, preventing the measurement of the characteristic curve (see figure 4.2).

Reed auto-oscillations The restriction of the measurements to low-frequency regimes is a simplification in terms of the amount of measured data because it is not necessary to measure a different curve for each frequency, and because simpler methods can be used to measure the flow in quasi-static conditions. Nevertheless, this restriction requires that the pressure and flow vary smoothly throughout the whole range of measurement. This can be difficult to achieve in practice, because of auto-oscillations, sought in a normal utilisation of the instrument, but that have to be prevented here using artificial procedures. Reed auto-oscillations arise when the reed ceases to behave as a passive resistance (a positive ∂p , which absorbs energy from the standing wave inside the reed channel) ∂q 0.8 are never observed in conical diffusers

8.5 Results of the simulation

147

A full analysis of the instrument model must be based on more than a few plots. The behavior of the instrument corresponds to its response for the whole set of physical parameters, and to likely transitions between these sets. From a musical point of view, it is the experience of using these models that can determine their suitability. From a physical point of view, ideally all possible ranges of parameters and possible transitions would have to be tested.

8.5.1. General remarks on the time-evolution curves The time variation of the simulated reed position can be compared for instance with the observations of figure 6.2. The asymmetry noticed in the observations in the fractions of periods the reed remains closed or open is also clearly visible in the simulations. As discussed in section 2.5 this is a consequence of the acoustical response of the cone. As discussed by [Gokhshtein, 1979], after the reed opens, the internal pressure decays exponentially in time during the first part of the open-state, due to the tail seen in the reflection function. This decay is stopped by the arrival of the incoming pressure wave p− which reopens the reed before the transition to the closed-state. The variation of the duration of the closed-state with the mouth pressure pm is not observed in the simulations. Figure 8.6 shows the pressure and reed-opening timeevolutions for several values of pm from the threshold of oscillation until the first register transition. The duration of the closed-state is seen to increase slightly between the threshold and slightly higher pressures, but until the register transition it does not vary significantly. In the first results shown in figure 8.4 it is not possible to observe the mechanical oscillations of the reed because these are highly damped. Figure 8.7 shows a plot of the internal pressure and reed opening for different values of the parameter α corresponding to the damping of the reed oscillator. For low values, several high frequency oscillations can bee seen in the reed opening plot. These are hardly observable in the pressure plot but they have significant perceptive consequences when hearing the sound, which becomes brighter.

148

Chap. 8: Synthesis

Adim. pressure

Adim. reed position

1

0.5

0 −1 0 1

200

400

600

0 0

200

400

600

200

400

600

200

400

600

200

400

600

0.5

0 −1 0 1

200

400

600

0 0 0.5

0 −1 0 1

200

400

600

0 0 0.5

0 −1 0 1

200

400

600

0 0 0.5

0 −1 0

200 400 time (samples)

600

0 0

200 400 time (samples)

600

Figure 8.6.: Simulation results for variable mouth pressures (pm ). pm increases from top to bottom, from a pressure just above the threshold of oscillation (here 2.6 kPa) until the transition to the second register (at 4 kPa). Adim. pressure

Adim. reed position

1

0.5

0 −1 0 1

200

400

600

0 0

200

400

600

200

400

600

0.5

0 −1 0 1

200

400

600

0 0 0.5

0 −1 0

200 400 time (samples)

600

0 0

200 400 time (samples)

600

Figure 8.7.: Simulation results for variable dampings (α)

9. Conclusion Reed instruments, and in particular double-reeds are complex systems that involve the interaction between fluid mechanics phenomena and elastic solids. Although several aspects of their behavior are quite well mimicked by rather simple models, the details of the oscillating flow inside the instrument (mainly in the upstream part) and the interaction between the moving reed and the flow are challenging problems. This is especially true in oboes and bassoons, where the geometry of the duct introduces further complications with respect to a clarinet mouthpiece, for instance. The aim of this work was to propose a model of the double-reed functioning that could be used in simulations of a complete instrument. An important goal was to propose a more refined model (than the one available in the literature) intended specifically for double reeds, or to explain why the generic model is appropriate for double-reeds as well. Since the start of the project we noticed the relative lack of experimental knowledge on double-reeds, especially when compared to single-reeds. For this reason it was important to collect some experimental data that could help determining some particular aspects of the reed and the flow.

9.1. Quasi-static model of the reed 9.1.1. The double reed as a nonlinear exciter One of the fundamental questions was to know whether the non-linearity that characterises the exciter was changed with respect to other reeds. While other reed instruments usually have a regime where the reed oscillates between non-closed states, double-reed instruments tend to play exclusively in regimes where the reed abruptly shuts during part of its cycle. A model had been proposed by Wijnands and coll. relating this tendency to an additional singular loss in the flow inside the reed [Wijnands and Hirschberg, 1995]. In terms of the non-linear characteristic curve of the reed, the maximum-low pressure (PT ) should be displaced towards higher pressures so that multiple flow values could be observed for a same pressure difference across the reed. Our measurements showed that this model does not apply to oboe or bassoon reeds (chap. 4). There is no qualitative change in the non-linear characteristic of the reed (there are no multiple flow values for a same pressure, other than hysteretic effects due to visco-elasticity of the cane reed). In quasi-static measurements for a complete oboe reed with its staple, the displacement to PT is opposite of what was predicted by Wijnands’ model. In terms of the reed

150

Chap. 9: Conclusion

model, this means that no singular losses are observed in the flow inside the reed. Moreover we have shown that this displacement can be justified by the pressure recovered along the staple due to the expansion of the flow (diffuser effect, see section 5.10). In oscillating regimes, it is not clear if the diffuser effect should be taken into account in the exciter, because the staple can be seen as a continuation of the resonator. Due to the complexity of the measurements in dynamic regimes, this ambiguity could not be completely solved. Because of this difficulty, a complete model of the instrument was built where the diffuser effect can be controlled by means of the CP (pressure recovery coefficient) parameter. This parameter does not seem very relevant when observing the pressure and reed opening time evolutions, nor their spectrum. In listening tests, the effect is audible, increasing the brightness of the sound and making the model more responsive in transitions (chap. 8).

9.1.2. Reed opening geometry With respect to the reed opening (which controls the volume flow into the reed for a given pressure difference accross the reed), different thoughts could be found in the literature concerning the variation of its shape throughout the opening range. In particular, it ´ 1989] that the reed closes from the was proposed for tenora reeds [Barjau and Agullo, corners as the two blades approach, so that both the width and the height of the opening is reduced as the blades approach. In such case, the reed area is proportional to the square of the distance between blades. This view is shared by some musicians, who seek this kind of behavior when making their reeds. Our observations show that in oboe and bassoon the reed blades touch at the same time throughout the length of the reed (sect. 3.3). Quantitative analysis of the reed opening using an algorithm developed for this purpose (appendix H) confirm that for these reeds the opening area is proportional to the distance between reeds. It is possible that the behavior is different for some reed designs, which should be checked with reeds designed specifically for this kind of behavior.

9.1.3. Elastic model The simplest approach for the mechanical behavior of the reed is to consider it as a spring, driven by the pressure difference across it. In a linear spring, the pressure difference induces a proportional deformation of the string. In the reed, the relation between pressure and deformation depends on the conditions of the reed: when it is dry, there is some non-linearity and the reed is stiffer when nearly closed. A more linear relation is achieved when the reed is humidified, a condition met when the reed is blown by the musician (sect. 3.4). Measurements of the deformation against pressure revealed a viscoelastic behavior of the reed material (confirmed through relaxation time measurements in section 3.5). Some of the long-term deformation is kept after the force is released. This should cause

9.2 Open questions

151

an offset in the reed equilibrium position through time, but no significant changes in the oscillating regime of the reed. Other measurements and observations on the reed properties were made throughout a number of experiments that allowed to gather important knowledge and data about the properties of the reed. They were listed in chapter 7 and are not recalled here.

9.1.4. Sound synthesis A synthesis model for double-reed instruments was derived from a clarinet model in chapter 8. Conical waveguides were implemented, keeping the traveling-wave formalism, and the parameters, many of them measured throughout the experimental part of this work, were adapted to double reeds. The exciter model was changed to include the quasi-static effect of the conical diffuser. The model for the conical resonator together with the change in the reed parameters already approach the timbre of the model to that of an oboe, for instance. In a subjective analysis, the pressure recovery does not greatly affect the timbre, but the behavior in the transitories seems to be improved. The application of the simulation to sound-synthesis requires an extensive work over the control and parameter mapping of the model. For a similar model which was implemented for other wind instruments in the framework of the project WINDSET1 (a partnership between IRCAM and Arturia) this work was effectuated in order to be used for musical synthesis.

9.2. Open questions Although the quasi-static behavior of the double-reed is at this point well characterised, there are many questions that remain unanswered which may affect the behavior of the double-reed when used in the instrument. One of the main issues is the step towards the dynamic model. The model used in chapter 8 for the reed is mostly a static model. Although dynamic effects of the reed mechanics are taken into account through the model of the harmonic oscillator, unsteady effects of the flow should be considered also, since they would contribute to and potentially modify the reed dynamics. An analysis of magnitudes shown in section 6.2 suggests that both the dynamics of the flow and the flow induced by the reed are important and should have consequences on the behavior of the instrument (although no consequences of the induced flow were noticed at the reed output, as mentioned in section 6.4.3). These two aspects should thus be studied in more detail. Another unsolved issue is the separation between the exciter and the resonator. There is clearly a region where the acoustic wave equations are valid. Inside the reed however, the variable boundary conditions and the velocity magnitudes call for the use of the complete equations of fluid mechanics. In our model, for the exciter we used an approximation of these equations which is valid in incompressible, stationary 1

see

http://www.arturia.com/en/brass/samples.php for sound examples

152

Chap. 9: Conclusion

conditions. Even in the case where the exciter model is extended to include dynamic effects of the flow, it will still be necessary to define a boundary for the transition to the acoustics region.

9.3. Perspectives The broad approach to the double-reed presented in this document can be seen as an advantage, because it allows to relate several distinct features of the reed properties, and to the empirical knowledge involved in its fabrication and utilisation. But it can also be seen as a weakness: in fact in many of the measurements, the variability of the results do not allow us to be more categorical. Naturally, one of the open paths for future research is to investigate the reasons of the variability observed for instance in the characteristic measurements. This can be achieved with a finer control of the experimental conditions (in particular the use of artificial lips and teeth) but also by a better measurement of the parameters of the reed during the experiment. Another kind of variability is related to the astonishingly wide range of timbres that can be associated with double reeds. With similar working principles and geometries (although different dimensions), the timbre of a double-reed can be mellow, like in the oboe, flute-like like in the guanzi, strident, like in the bombarde... Is this variability related to the reed geometry? Its mechanical properties induced by different fabrication traditions? Is it due to the way they are played (between the lips against free from lips)? These questions have to be studied through a collaboration between the musicians, who know what to change in a reed to achieve a particular result in the sound, and the scientific community which can provide the means of testing the physical consequences of these changes and provide models to explain them. A collaborative project is being planned with musicians to list the techniques used in reed making, evaluating the agreement on a particular technique between several musicians, and trying to propose experiments and models for the effects of these techniques. In order to better compare the simulations to experiments it is necessary to be able to control and measure the evolution of the parameters applied in experiments. This is the aim of the artificial mouth (appendix G), where in principle it is possible to control each action of the mouth independently in a stable way (i. e.,without being limited by muscle fatigue, breath supply, etc.), and measurement access is simpler than in the mouth of a musician. However, the relation between the parameters in the mouth and those in the simulation can be complicated, and despite the accuracy of the control for stationary conditions it is still not easy to impose precise time evolutions of the controls. These issues will be addressed by the ANR project CONSONNES2 One other aspect that undeniably calls for more attention is the dynamical description of the reed. Measurements of the reed opening and flow profiles during oscillations unveil a great number of interesting and puzzling phenomena, for which only the first 2

CONSONNES : CONtrˆole de SONs instrumentaux Naturels Et. Synthe´ tiques

9.3 Perspectives

153

questions were formulated. . . An interesting approach would be to continue the experimental measurements of the dynamic flow, in particular for lower frequencies and in bounded (because coupled to the bore) conditions, as well as a computational simulation of the fluid-structure coupling in the reed. A simple unidimensional model of the fluid structure coupling, and extended to include unstationnary flows can be tested through similar simulations as those shown in chapter 8.

154

Chap. 9: Conclusion

Appendices

A. Summary of double reed instruments Oboes and bassoons are only two examples of double-reed instruments. Several others exist, and the differences in the way they are played, their resonators, or the range of notes they are aimed at. Reed dimensions are also widely variable. These differences account to the large varieties of sounds produced by double-reed instruments. For instance, the bore shape influences the balance between odd and even harmonics in the spectrum. By reducing and damping the motion of the reed, the contact between the lips and the reed blades has a softening effect on the timbre. The difficult question to answer is thus if double-reeds have a characteristic and recognizable sound. One of the interests of the physical model is that if it is sufficiently detailed and precise, it should be able to describe and reproduce these variations in timbre, by inputing it with the correct parameters. Although this was not fully investigated in this work, it is clear that some of these effects arise by adjusting such parameters as reed damping and mass (for the lip contact) or bore tapering angle (for the bore shape). In the following sections we list some of the double instruments that can be found both in modern, ancient and traditional music, which are eventually a source of information about the behavior of the double-reed.

A.1. Classification of double reed instruments One important characteristics of double-reed instruments that influence their timbre are the bore shape which can be cylindrical, conical or in some cases smoothly progressing from cylindrical to conical (renaissance shawms) or still stepped conical. The bell shape also slightly influences the timbre and mostly the sound volume produced by the instrument. Another important characteristic is the interaction between the player’s lips and the reed, either in contact (in some cases a pirouette is added so that the lips have a fixed position on the reed) or free, either inside a bag (in bagpipes) inside a protective wind-cap or completely inside the mouth. Table A.1 lists some examples of modern, ancient and traditional double-reed instruments, and the main characteristics referred above. The note range for which some of these instruments are designed to is also listed, because it can be an interesting information when comparing two different instruments. Some of these however constitute a family (for example shawms, crumhorns) so that they can be found in several sizes and musical ranges. Traditional double-reed instruments are spread over almost all countries in Europe (Tenora, Bombarde and many more), most of Asia (Guan, Zurna and many more) and

158

Chap. A: Organology

Name

Bore

Blow

Bell

Range

Oboe

Conical

Lips

Slightly flaring

B3b − G6

Bassoon

Conical

Lips

Slightly flaring

B1b − E5

Contrabassoon

Conical

Lips

Flaring

B0b − C4

English Horn

Conical

Lips

Bulb

E 3 − C6

Heckelphone

Conical

Lips

Bulb

A 2 − G5

Oboe d’amore

Conical

Lips

Bulb

G 3 − C5

Crumhorn

Straight

Free (windcap)

No bell

(var)

Shawm

Conical

Pirouette

Flaring

(var)

Dulcian

Conical

Lips

Rauschpfife

Conical

Free (windcap)

No bell

Racket

Straight

Lips

No bell

Bagpipe (chanter)

Conical

Free (bag)

Bombarde

Conical

Lips

Flaring

Tenora

Conical

Pirouette

Flaring

Guan

Straight

Lips

No bell

Zurna

Stepped cone

Free (mouth)

Flaring

#

Obs

#

C 1 − G4

Table A.1.: Examples of double-reed instruments and their main characteristics

A.2 Variability of reed shapes

159

Northern Africa. Bagpipes exist mostly in Europe, and their chanters can have both conical or cylindrical shapes. Usually, to a conical chanter is associated a double-reed and to a cylindrical one a single-reed (which is different from a clarinet or saxophone reed because it is not striking). More examples of double-reed instruments can be found in literature such as [Midgley, 1976] or in the Wikipedia1 article for double-reeds.

A.2. Variability of reed shapes

Although a double-reed instrument is, as the name indicates, constituted by two blades, the actual shape of the reed can vary substantially from one reed to another. Looking into traditional instruments, it is possible to find reeds whose structure is considerably different (for example the duduk, which has a duck-beak like shape). Double-reeds can also be made out of different plants, which may give them different elastic properties for instance (for example, the reed used in the shehnai in northern India, made out of pala-grass and in the Pi nai or in sralai, a palm leaf is used instead of the reed) . In most of the cases however, the basic profile of the reed remains more or less similar: the entrance has an oblong cross-section, which progressively evolves into a circular cross-section towards the reed output. In some cases, the downstream reed consists of a metal conical tube (the staple), in others this tube is not part of the reed but part of the instrument, where the reed is inserted. For the most common double-reeds in the occident, even if the shape remains basically constant, the dimensions are significantly variable. In general, the lower the range of the instrument, the larger is the reed, but as can be seen in table A.2, each double-reed is not a scaled version of another reed. Table A.2 lists some typical dimensions (see fig. A.1) in some of the most common double-reeds which can be found in scientific articles (bassoon, oboe, english horn, contrabassoon, chanter and tenora). A description of the dimensions used in this table follows:

1

http://en.wikipedia.org

160

Chap. A: Organology

WI

Reed opening width, measured inside the reed opening

HI

Reed opening height at rest, measured inside the reed opening

WO

Reed tip width, measured externally

HO

Reed tip height at rest, measured externally

LR

Reed length — length of a blade from the staple (or the narrowest part of the reed duct, measured from the inside) until the tip

LSC

Scraped length — length of the part of the reed tip where the bark is removed

LST

Staple length (or the length from the narrowest part of the reed duct, measured from the inside until the butt)

NDS

Narrowest diameter of the staple or reed duct (this can correspond to two dimensions because the staple entrance is often oval)

XDS

Diameter at the staple exit, or the butt

LT

Total reed length

Bn

Hb

Ca

Cbn

Chnt

WI

13.3

7.0

7.6

16.4

16

HI

0.9 – 1.2

0.8 – 1.0

1.2

1.7

1.6

WO

14.4

7.2

7.8

17.6

HO

1.6 – 2.1

1.0 – 1.2

1.6

2.4

LR

33

26

LSC

26 – 27.8

10

LST

23

47

NDS

3.7

2 × 2.5

XDS

5.3

4.8

LT

56

73

10

49 10

Tn

20

35

18

24

20

2.0 × 3.5

4.5

1.6

5.0

5.7

3

73

36

Table A.2.: Dimensions of the most common double-reeds in the occident (see figure A.1 for the meaning of the dimension abbreviations) As a final remark, table A.2 should not be taken as a geometrical characterisation of these double reeds. Variations between reed geometries go beyond those hereby listed. A side view can vary from reed to reed, and perhaps more importantly, the profiles of

A.2 Variability of reed shapes

161

WO

WO

WI

WI

HO

HI

HI

LSC LSC LR LR

LT

LST

LST

NDS XDS

Figure A.1.: Reed dimensions

162

the thicknesses of the blades can be greatly variable.

Chap. A: Organology

B. Mechanical modes of vibration of the double-reed B.1. Mechanical response of the double-reed In the section 2.4, an elementary model was proposed for the behavior of the reed when used as an exciter of an instrument. This model is intrinsically quasi-static, that is, it is valid in regimes where its variables change slowly with time. In oscillatory regimes, other effects influence the motion of the reed and the evolution of the fluid flow. Among these we identified the inertia and damping of the reed (sect. 2.5), which can be taken into account by modelling the reed tip as a harmonic oscillator. However, this model is still a simplification of the mechanical behavior of the reed (even without considering its interaction with the flow), because the reed cannot be reduced to its tip. The blades vibrate along their surface until the region where they are attached to the staple, and for this reason several modes of vibration are expected to be observed in the reed. The analysis of these modes of vibration is important, on one hand because each mode of vibration can have a different frequency of resonance, making a single harmonic oscillator insufficient to describe the vibration of the reed tip, and on the other hand because the motion of the reed inner surface is in contact with the fluid, inducing some flow.

B.2. Modes of vibration of the reed Equation (2.31) correctly describes a linear oscillator with one degree of freedom. In general, a system with more than one degree of freedom shows several resonance peaks in a plot of its mechanical response. As long as the linear hypothesis remains valid, these systems can be described as a set of linear oscillators with different resonance frequencies (ωr in equation (2.31)) and damping coefficients (α in the same equation). Modal analysis studies the decomposition of an oscillating solid into this set of harmonic oscillators, which are usually referred to as modes . This decomposition can also be used to study the mechanical behavior of the reed as a 3-dimensional system with boundary conditions, such as the fixed ends on the butt side, the free ends on the tip and the contact conditions between the two blades on the reed’s rails. Modal analysis was previously applied by Pinard ([Pinard et al., 2003]) to clarinet single-reeds. In practice, modal analysis consists in decomposing the reed motion into its eigenmodes of vibration, each of which has its own resonance frequency ωr and damping α.

164

Chap. B: Preliminary modal analysis

The whole system is a linear combination of these modes of vibration, and they can be seen as independent harmonic oscillators. The excitation of these modes depends not only on the frequency of the driving force, but also on the distribution of the force over the reed. In reed instruments the excitation is particularly different from the excitation in percussion instruments because the force is usually distributed along the surface of the reed whereas in percussions the excitation is usually very localized.

B.3. An oscillator driven by an external force When driven by an external sinusoidal force of frequency ω/(2π ), the harmonic oscillator seen in section 2.4.3 will respond with an oscillation with the same angular frequency ω, and with an amplitude that depends on the relationship between the driving frequency ω and the resonance frequency ωr and quality factor Q = ωr /(2α ) according to the formula of a Lorentz peak centered at the resonance frequency: X (ω) F (ω)

=

ω2 r

1/m − ω2 − 2ıαω

(B.1)

If the velocity (V ) of the oscillator is measured instead of the position (X ), a similar formula can be obtained by differentiation of eq. (B.1), V (ω) = ıωX (ω): V (ω) F (ω)

=

ıω/m ω2 r

− ω2 − 2ıαω

(B.2)

As a linear combination of simple harmonic oscillators, the response of a more complex system with several modes of vibration as described in section B.2 is the sum of the responses of each individual mode. Linear superposition of the modes is assumed because the flows induced by the acoustic flow and reed motion are small enough to induce a non-linear coupling between the modes. An analysis of the spectrum will thus give an indication of the parameters ω r and α of each mode of vibration of the reed. In general, because several modes of vibration exist in the structure, it may be difficult to measure these parameters in each individual peak.

B.4. Frequency analysis of reed vibration B.4.1. Experimental approaches In vibromechanics, a usual method for imposing a vibration of the object to be studied is to rigidly connect a shaker to the object, so that it vibrates synchronously to the exciting device. In the case of the double reed, in normal playing conditions, it is driven by a force that is distributed over the reed blades, rather than applied punctually. For this reason, we preferred to use air vibrations as the excitation to the reed, provided by a loudspeaker placed in the vicinity of the reed. Two different methods were tried out to excite the reed vibration:

B.4 Frequency analysis of reed vibration

165

External excitation The reed is supported by the butt or the staple and an acoustics source (a loudspeaker) is placed next to the reed, but not too close, so that the wavefronts are approximately plane in the vicinity of the reed. Internal excitation A small loudspeaker is inserted in a small chamber with an opening where the base of the reed can be inserted, the same way it is inserted in a resonator. The method of external excitations has the disadvantage of requiring more power, because the wave field spreads throughout the room rather than concentrating over the reed. A compromise is obtained by approaching the reed to the loudspeaker, but not too much, otherwise the pressure distribution is no longer homogeneous at high frequencies. Internal excitations require a weaker sound source but it is much more difficult to maintain a constant amplitude excitation throughout the spectrum because of the influence of the acoustic modes of the loudspeaker chamber coupled to the reed. Moreover, since the propagation of the waves is longitudinal with respect to the reed, the frequency at which the pressure excitation is no longer homogeneous is lower than with external excitations (provided that the reed is placed parallel to the loudspeaker in the external case). The excitation signal was a sweep (a sinusoidal wave of varying frequency) running from 0 to 48000 Hz, 10 seconds long. In both cases, the actual amplitude of the pressure oscillations was measured close to the reed before the actual measurement of the amplitude of vibration of the reed. For external excitations it would be possible to measure pressure simultaneously with the vibration measurements, but for internal excitation it was not possible to maintain the microphone inside the reed without disturbing the reed oscillations. We then used the measured pressure variations to correct the amplitude of the excitation signal so that it would be approximately flat at mid-range frequencies (1000-4000 Hz). A second measurement of the pressure variation with the new excitation signal was used during the analysis to compute the reed response as a ratio of the measured reed vibration to the pressure forcing it. The vibration of the reed is measured using a laser vibrometer (Ometron VH300+). This device measures the velocity of the structure at the point of incidence of the laser.

B.4.2. Data analysis Analysis of the results consists simply in calculating the FFT of the reed vibration signal and dividing it by the FFT of the reference (excitation) signal sent to the loudspeaker.

B.4.3. Comparison of excitation methods Figure B.1 shows the response of one reed measured using the two different methods described in section B.4.1.

166

Chap. B: Preliminary modal analysis

250

Amplitude (dB)

Internal External 200 150 100 50 0

0.5

1

1.5

2

2.5 3 Frequency (Hz)

3.5

4

4.5

5 4

x 10

Figure B.1.: Response of reed 744 using two different excitation methods (external and internal) as described in section B.4.1. The results are quite different from one method to the other. The fact that the two measurements were taken at two different dates doesn’t seem sufficient to explain these differences. We suppose that the resonances of the loudspeaker cavity and reed can be responsible for the differences observed, because it is difficult to guarantee the reeds are placed in the same position for every measurement so that the cavity geometry is the same. In fact, measures of the actual pressure intensity inside the reed should be taken simultaneously to the measurements of reed vibrations. This has not been accomplished because this technique would require small microphones and perforation of the reed. Due to the problems identified in the internal excitation method, we chose to work with external excitation in the following sections, unless stated.

B.5. Admittance spectra of bassoon reeds In the following paragraphs we present some admittance curves of double-reeds. Most of the reeds used for these measurements are bassoon reeds made from natural cane. In one particular case we present a measurement for a synthetic bassoon reed, and a final measurement concerns a same oboe reed measured with two different excitation methods. These results require further investigation in order to investigate the origin of some discrepancies observed for different methods of excitation and orientations of the reed.

B.5.1. Comparison of two different cane reeds Figure B.2 plots the frequency response of two different cane reeds. Reed labeled 744 is tailored to be used in concert whereas reed 766 is not finalized, so that its blades are thicker at the tip. Both reeds have strong resonances between 2400 and 2700 Hertz, which we will call

B.5 Admittance spectra of bassoon reeds

167

50

Amplitude (dB)

744 766 0 −50 −100 −150 0

1000

2000

3000

4000 5000 6000 Frequency (Hz)

7000

8000

9000

10000

Figure B.2.: Comparison of two different cane reeds.

the mechanical frequency of resonance of the reed. These frequencies are similar to those of clarinet reeds, which are situated in the 2100-2400 Hz region (for example, [Pinard et al., 2003]). These values can also be compared to the ones found for chanter reeds by S. Carral [Carral and Campbell, 2005]: for a reed with a similar geometry but smaller dimensions (10 mm wide tip, 18 mm long for the chanter against 14 mm wide and 28 mm long for the bassoon), the author found a resonance frequency of about 4800 Hertz for dry reeds. The larger dimensions of bassoon reeds probably account in a great part for the lower resonance frequencies when compared to the chanter reeds. The two measured bassoon reeds show similar response spectra below 5000 Hz, although the reed that was not fully scraped sees its peaks displaced towards higher frequencies, and a higher frequency of resonance. This is probably due to the increased stiffness of the unscraped reed. However, reed makers usually scrape the tip of the reeds to increase their frequency [Kopp, 2003].

B.5.2. Effect of humidity For natural cane reeds, instrumentists notice a great dependence of the reed quality with the damping of the reed. These usually have to be soaked for a few minutes before being played, and blown a few times with moist hot air before they sound in perfect conditions. Figure B.3 plots the dependence of a reed spectrum on the time the reed is soaked before the measurement. The first resonance frequency is substantially lowered after 5 minutes of soaking ( from 1900 Hz to 1640 Hz) which is probably due to the mass of water captured by the reed. The quality of the resonances is slightly increased (peaks are narrower). Simultaneously, results of section 3.4 and appendix F suggest that the reed stiffness k increases when the reed is soaked (an increase in stiffness alone implies a higher resonance frequency).

168

Chap. B: Preliminary modal analysis

Amplitude (dB)

100 Dry 5 min soaking 30 min soaking

50

0

−50 0

500

1000

1500

2000 2500 3000 Frequency (Hz)

3500

4000

4500

5000

Figure B.3.: Comparison between different levels of humidification.

B.5.3. Comparison between plastic and cane reeds In some of our experiments it is more practical to use synthetic reeds (made out of plastic materials) whose properties are more stable because they depend less on environment conditions and on the effort applied during the experiment.

Amplitude (dB)

50 Plastic 744 766

0 −50 −100 −150 0

1000

2000

3000

4000 5000 6000 Frequency (Hz)

7000

8000

9000

10000

Figure B.4.: Comparison between plastic and cane reeds. However, as can be seen in figure B.4, the dynamical properties of the plastic reed are quite different from the cane reeds: its resonant frequency is lower (around 1900 Hz) and less damped, since the peak is narrower. Dynamical measurements show how plastic reeds are less damped when compared to cane reeds, although the lowest resonance peak stands close to the resonance frequency of a real cane reed.

B.5.4. Oboe reeds The same method can be used for oboe reeds, although due to their dimensions, their vibration is fainter than in the case of bassoon reeds. The setup remains basically

B.6 Discussion

169

the same, but two different reed positions were tried out: with the rails facing the loudspeaker and with the reed opening facing the loudspeaker. Oboe reed mechanical response 50

Amplitude (dB)

debout allonge 0 −50 −100 −150 0

1000

2000

3000

4000 5000 6000 Frequency (Hz)

7000

8000

9000

10000

Figure B.5.: Mechanical response of an oboe reed. The two measurements present different peak configuration below 5000 Hz, but there is one peak common to both experimental setups at 3492 Hz. In the first setup, the resonance peak is partially hidden under a higher peak around 4200 Hz, which however does not exist in the other setup. This is probably due to a vibration mode of the structure supporting the reed. This resonant frequency is substantially larger than that of the bassoon, as could be expected based on the smaller dimensions of the oboe reed. However, when compared to the chanter reed, the oboe reed has a smaller tip but a bigger vibrating length which can explain the lower resonant frequency of the oboe reed.

B.6. Discussion The methodology presented in this chapter is similar to other experimental setups used to measure spectral characteristics of clarinet reed vibrations using external excitations [Pinard et al., 2003] and chanter double reeds using internal excitations [Carral and Campbell, 2005]. With our setups we tried to compare results using both methods for similar reeds, however practical difficulties prevented a full recording of the exciting signal along with the response signal in the case of internal excitations, which may be an explication of the remarkable differences between the two methods. These differences were observed in several different reeds (although figure B.1 only depicts this situation for one of the reeds). The fact that the two kinds of setups were performed in different days, and that for instance it is possible that the laser vibrometer was pointed at a slightly different point on the reed, might also be part of the explanation for the differences in the results, although it cannot explain differences in the measured resonance frequency. Further measurements, performed in more controlled conditions will be required to analyse the origin of these differences.

170

Chap. B: Preliminary modal analysis

Nevertheless, and considering that external excitation is simpler than its counterpart, this method proves useful in extracting important information about the dynamical properties of the solid parts of the reed. For instance, the frequency of resonance of the reed can be used to calculate the effective mass of the reed. It is important to notice that the resonance frequency measured in this section, are the mechanical resonance frequencies, which are usually very different from the frequency of oscillation of the reed when blown alone (tuned squeak, see sect. 6.4.3) Other than the resonant frequency, the width of each resonance peak is gives us the quality factor (Q = ∆frf , where ∆f is the width of the peak at -3 dB). Given the resonant frequency, the quality factor and the further knowledge of the reed k stiffness (from section 3.4), it is possible to determine the effective mass (m = (2πf ) )2 and damping (r = m ωQr ) of the modes of vibration of the reed that are relevant to its description.

B.6.1. Importance of the dynamic aspects of the reed Given that the forcing of the reed is usually determined by the pressure variations inside the instrument (which have direct consequences in pr (t )), it is the left-hand side that will determine the importance of the time-derivatives in the right-hand side of equation (2.31). This can be better understood by doing a frequency decomposition of equation (2.31):

∆P (ω) ms

= ωr2 ∆X (ω) + αıω∆X (ω) − ω2 X (ω)

(B.3)

In equation (B.3) the capitalized variables represent the Fourier transforms of ∆p(t ) and ∆x (t ). The Fourier transform of a derivatives of x (t ) correspond to multiplying the frequency-domain function X (ω) by ıω. Written in this form, equation (B.3) allows to see that the forcing ∆p(t ) is distributed by the terms in the right-hand side. This way, it is the balance between ω r , α and ω that will determine the relevance of each of the stiffness, damping and inertia terms. For low frequencies (ω  ωr ) the inertia term can be neglected, and if ω  ωαr , damping can also be neglected. We thus see that for quasi-static regimes, defined with the former conditions on ω, equation (2.22) provides a good mechanical description of the reed.

B.6.2. Importance of higher-order modes of vibration of the reed If the pressure under the reed can in fact be considered homogeneous, the fundamental mode of vibration of the reed can be excited preferentially over the higher modes. Higher harmonics of the bore pressure can excite higher modes of the reed, but depending on the played note, the bore harmonic might or might not fall over a reed resonance. Pinard [Pinard et al., 2003] gives indications on the first two flexural (longitudinal) and torsional (transverse) modes of vibration of the clarinet reed. Due to the contact between the sides of the double reeds, the flexural modes are not expected to be observed

B.6 Discussion

171

in oboes and bassoons because flexural modes have maximum vibration regions (antinodes) along the sides. Thompson [Thompson, 1979] studied the influence of the reed resonance frequency on the tone produced by clarinets. It is probable that higher resonances of the reed might have similar reinforcement effects on some of the partials produced by the instrument. Section 6.3.5 presents some observations of the reed movement, with particular attention to high-frequency oscillations during the period that the reed remains open. These oscillations can be associated to several causes, and among them, the mechanical eigenmodes of the reed. To these oscillations on the reed position are associated simultaneous oscillations of the pressure and volume flow. With respect to higher flexural modes, these are not very important in terms the fluid-structure coupling, because they have several anti-nodes that vibrate in opposite directions. The total flow ‘‘pumped’’ by the reed (average flow) is less important for a same vibration amplitude, although it is possible that local flows are induced underneath the reed.

172

Chap. B: Preliminary modal analysis

C. The momentum equation in fluids C.1. Differential form The differential form of the conservation of momentum equation is known as the NavierStokes equation. It can be formulated as:

1

∂t ui + uj ∂j ui = − ∂i p + ν∂2 ui ρ

(C.1)

In this equation, ui stand for the components of the flow velocity field vectors, ρ is µ the volume mass of the air, and ν = ρ is the cinematic viscosity (µ being the dynamic viscosity).

C.2. Integral form Equation (C.1) can be integrated in a control volume Ω, bounded by the control surface Σ:

Z

∂t ui dV + Ω

Z

Σ

uj nj ui dS = −

Z

1 Σ

ρ

pni dS +

Z

σij nj dS

(C.2)

Σ

C.3. Vena Contracta We can now apply equation (C.2) to a Borda Tube immerse in a flow. The control volume Ω will be bounded by the duct walls, its bounding surfaces will be perpendicular to the flow, and will entirely include the borda tube. The flow is considered to be inviscid, stationary and irrotational, so that the first term in the left-hand side vanishes, as well as the last term in the right-hand side. Equation (C.2) becomes:

Z  Σ

uj nj ui +

1 ρ

pni



dS = 0

(C.3)

Along the duct walls in the upstream region the pressure is homogeneous, and the same is true for the downstream region, so that in this region the second term in equation (C.3) cancels out. Since the flow is parallel to the walls, uj nj is zero along the walls of the duct, cancelling out the first term of the equation as well. The two cross-section surfaces Σ1 and Σ2 remain in the equation.

174

Chap. C: Momentum

PSfrag replacements Σ

Ω p1

p2 u2 Sj St

S1 S2 Sw

Figure C.1.: Flow into a Borda tube In the upstream surface (Σ1 ), the flow is homogeneous along the cross-section of the duct. The velocity is ux = u1 . The pressure is also homogeneous and equals p1 . In the downstream surface (Σ2 ), the flow is zero everywhere except for the region where the jet crosses Σ2 : over the surface Σj the flow velocity is u2 . As for the pressure, it equals p1 everywhere outside the Borda tube and p2 inside the tube. Here it is homogeneous, because there is the pressure does not change across a jet. Collecting all these assumptions leads us to the formula: S1 p1 + S1 ρu12 = (S1 − St )p1 + St p2 + Sj ρu22

(C.4)

Cancelling out equivalent surfaces on both sides of the equation: S1 ρu12 = St (p2 − p1 ) + Sj ρu22 or ρ

q2 S1

= St (p2 − p1 ) − Sj ρu22

(C.5)

(C.6)

using the equation of conservation of mass: u1 S1 = u 2 Sj = q

(C.7)

∆p is related to the flow by the Bernoulli formula: 1 ∆p = ρu22 2

(C.8)

Supposing that the duct is much wider than the Borda tube, S1  St , the term on the left-hand side is much smaller than those on the right-hand side, and can consequently be neglected. It is now possible to calculate the discharge coefficient C D CD =

Sj St

=

∆p ρu22

=

1 2

(C.9)

C.4 Singular losses

175

PSfrag replacements

C.4. Singular losses

A similar method to that described in the previous section can be used to calculate the pressure drop in a singular loss scenario.

Σ Ω Sj

p1 u1

S2

S1

p2 u2

Sw

Figure C.2.: Singular losses in a channel: some of the jet’s (in the downstream region) kinetic energy is dissipated through turbulence in the mixing region (in gray) In this case, the control volume Ω encloses the mixing region, where the dissipation of the kinetic region occurs. The cross-section surfaces S1 and S2 are situated respectively in the upstream and downstream region from the mixing region, such that the flow can be considered as laminar. Considering the hypothesis of inviscid and stationary flow, as in section C.3, equation (C.3) can be used in the control surface Σ. With similar considerations, it yields Sj ρu12 + S1 p1 = S2 ρu22 + S2 p2

(C.10)

If the mixing region is not very long, such that the cross-section does not vary substantially between S1 and S2 , these areas can be considered similar. The conservation of mass can be used to simplify equation (C.10), stated in the form: Sj u1 = S 2 u2 = q

(C.11)

This yields for the pressure drop across the mixing region: p2 − p 1 = ρ



q S2

2

Sj S2



1−

Sj S2



(C.12)

C.5. Potential theory for the Vena contracta In a flow that can be described by the Euler equations [Landau and Lifchitz, 1989], or in other words, without viscosity, and if in addition it is incompressible and irrotational, it can be described by a potential φ such that,

u = ∇φ

(C.13)

For bidimensional flows, this relation can be expanded into: ux =

∂φ ∂x

uy =

∂φ ∂y

(C.14)

176

Chap. C: Momentum

P A’

A a B’

B

P0 C’

C a∞

Figure C.3.: Flow in the complex spatial plane (z)

Another interesting definition is the stream function: ux =

∂ψ ∂y

uy = −

∂ψ ∂x

(C.15)

For further analysis, we will use the following complex variables: z = x + ıy

u = ux − ıuy

w = φ + ıψ

(C.16)

Using these variables, we can see that equations (C.14) and (C.15) express the CauchyRiemann conditions for the differentiability of function w.

C.5.1. Ecoulements par des orifices This section describes the flow from a reservoir entering a straight tube through a duct whose walls are at an angle with the tube walls. The Borda tube is a particular case of this flow, where = pi. The coordinate system is placed at the center of the entrance cross-section of the tube. The upstream flow is bounded by the upstream duct, whereas downstream the flow boundaries may not coincide with the tube walls because of the Vena Contracta effect. A hodograph is the representation on the complex plane of all the flow velocity values. In Landau [Landau and Lifchitz, 1989] it is supposed that the flow velocity tends to 0 upstream. In addition, after the detachment from the duct walls, the velocity is supposed to be constant and equal to u∞ . This can only be true for the external streamlines, establishing the limit between the jet and the stagnant fluid. This assumption is

C.5 Potential theory for the Vena contracta

177

|v| = v∞ C ≡ C0 B

B’



A ≡ A0

Figure C.4.: Hodograph — Set of velocities (u) found in the flow

justified by the fact the stagnant fluid has a pressure equal to P0 all the way from the tube entrance because otherwise there would be some fluid moving between two points in the stagnant region. In the jet, however, this is not true because the flow velocity varies along the x direction. With these assumptions, the hodograph is a circular section (see fig. C.4). This hodograph can be represented in a simpler way by taking its logarithm:

ζ = − log



−

i u∞

v

∗



(C.17)

which maps the circular section into the half-strip represented in figure C.5. This will be useful for the problem solution. For the potential (w) the analysis must be done independently for the two quantities. The stream-function (ψ) is constant on each streamline (psi and φ iso-contours are orthogonal). Walking upstream from the jet, ψ must vary at a constant rate (− ∂ψ = u1 , ∂x yielding ψ = −u1 x + K). K is chosen so that the rightmost streamline has ψ = 0 (fig. C.6). ∂φ

φ can be integrated on each streamline. On the jet side we have ∂y = u1 , that is φ = u1 y + K. K is now chosen such that φ = 0 at the point where the jet detaches from the tube walls, and φ tends to −∞ . On the reservoir side, flow velocity varies roughly as 1r , r being the distance to the tube entrance. φ thus varies as ln(r ) and tends to +∞ as r → +∞. The w plan is thus crossed by an infinite horizontal strip with u1 a1 as width, where a1 is the jet width at infinity. In order to calculate the shape of the outermost streamline, we must plot a new simplified graphic which transforms the flow in the imaginary half-plane. The new

178

Chap. C: Momentum



B’

A’

B

A

C ≡ C0

−

Figure C.5.: Transformed hodograph (ζ )

C

C’

−v∞ a∞

B

A

B’

A’

Figure C.6.: Flow potential set (w

C.5 Potential theory for the Vena contracta

179

-1 A’

1 C0 ≡ C

B’

B

A

Figure C.7.: Auxiliary variable v

variable v can be related to ζ and w: w ζ

u∞ a∞

= − log v   π 2 − 1 arcsin v = i

(C.18) (C.19)

π

With these relations, we can compute the streamline shape by noticing that on the line (BC), ψ = 0 and thus the complex potential w is reduced to its real part.

 dφ  dz

BC

= u = u∞ e −iθ

(C.20)

=(u )

where θ = − arg u = − arctan i = N1 j=1 xij ) and

standard deviation (σx,i = N1

q

sumjN=1 (xij − < x >i )2 ).

The standard deviation of the signal gives an indication of the phases and regions where the flow velocity is more unstable, that is, where turbulence is stronger. In figure 6.5 some average profiles are represented for one period, along with the standard deviation, calculated as described above. Figure 6.6 plots the profile evolution (along the longitudinal axis) with the standard deviation represented as a color scale (blue for small deviations, red for large ones).

I.1.3. Image analysis and reed opening area In most cases, the reed opening oscillations was video recorded before and after the velocity measurements. Simultaneously to the recording of each picture, an impulsive signal was recorded into a WAV file synchronised with the signal captured by the microphone, in a similar manner to the procedure applied to the velocity measurements. Since the period reference marks are available from section I.1.1, the picture synchronisation signal can be used to determine the phase corresponding to each picture. This way, it would be possible to reconstruct a slow-motion movie of the reed opening movement, eventually by interpolating between images to obtain a periodic rendering of the motion. However, it seemed more useful for the current analysis to synchronise the measurements performed on each picture (see appendix H) with the velocity profile evolution. The result of this synchronisation is shown in figure 6.9. In figure 6.8 the same reed opening evolution is shown together with the average profile evolution.

I.1.4. Flow reversal The hot wire probe used in this work measures the heat loss in the wire (even if in this system the wire temperature remains constant independently of the flow). The more air runs in the vicinity of the probe the more heat is lost to it, because more fresh air is available for the probe to transfer its heat to. This implies that the probe can

I.1 Signal synchronisation

207

only measure the absolute value of the velocity and not its direction. In the case of mostly unidimesnional flow, it is still possible for the flow to reverse its sense while the measured velocity remains constant. In our measurements, flow reversal seems to be present at some points in the period (figure I.1). Far from the reed output, signal variations are smooth, but as the probe approaches the main flow, the amplitude of variation of the flow increases, and a second peak becomes visible in each period. A stronger discontinuity is observed near the minima that surround this second peak. This suggests that flow reversal takes place in these regions. Supposing that the flow is directed along the x axis, the recorded signal would be |u x |. In this case, and supposing additionally that ux (t ) and its derivative are continuous, flow reversal would be identified as a discontinuity in ∂t |ux (t )| near the time-axis (as x 6= 0 when ux (t ) = 0). long as ∂u ∂t In practice, flow reversal was identified as the two minimum points in each period of the measured signal. Nevertheless, an observation of a single signal for a specific probe position shows that it never becomes 0. This can suggest the presence of a residual component of the velocity orthogonal q to the x axis (u⊥ ). If this component is constant, the measured velocity is thus u 0 =

2 . By knowing the flow reversal points (described above), ux2 + u⊥

both the residual velocity and the main flow can be deduced. This processing provides expected results (figure I.2) for most signals, however it is problematic for some signals where more than two minima have identical values. For example, when three minima have similar values, one of them is a true minima of u x close to zero, and not a flow reversal point. One way to overcome this problem may be to use the knowledge about flow reversal times for other probe positions next to the problematic signal, however this was not tried. After some tries, this processing was abandoned because the reversal points are in some cases not identified by the automatic processing. In some cases it is also difficult to identify them manually, and a global observation of the profile evolution (such as fig. 6.6) is required. In practice, it is difficult to be sure that the apparent discontinuities correspond to flow reversals, or that the orthogonal component is constant. As a conclusion, it seems safer to analyse directly the measured signals with the knowledge of these problems, and to extract a maximum of conclusions from the observation of these signals

208

Chap. I: Signal processing

0

36.3974

72.9097

109.564

time (samples)

Figure I.1.: Plot of the flow velocity signals along a reed diameter. From bottom to top, each line corresponds to one probe position. Horizontal axis represents the time in seconds. Vertical dotted lines show period reference marks (with the same phase). Thicker plots are the signals for the reed output boundaries.

I.1 Signal synchronisation

209

30

20

velocity (m/s)

10

0

−10

−20

−30 60

80

100

120

140 160 180 time (samples)

200

220

Figure I.2.: Unfolding the velocity signal

240

260

210

Chap. I: Signal processing

J. Topography methods In several aspects it is important to know the internal geometry of the reed channel. For instance, it can be useful both for considerations on the flow inside the reed or for an acoustical characterisation of it, as a resonator of reduced dimensions (crucial in the analysis of reed oscillations without the bore — chapter 6). The inside of the reed is of difficult access, due to its reduced cross-section dimensions. Therefore, a cast of the duct provides a better way of accessing the geometry of the double-reed. Due to the neck formed at the junction between the staple and the reed, it is usually necessary to destroy the reed in order to recover the cast. Different casting materials were tried out. The preferred was a soft silicone which did not adhere to the reed and could be recovered without much dammage when opening the staple.

J.1. Hand measurements The first apporoach to measuring the reed is to mark several lines along the reed, perpendicular to the reed axis, and measuring height h and width w on these lines on a digital photograph. A first approximation of the cross-section can be obtained by approximating each reed cross-section as an ellipse with axis lengths h and w. The cross-section area is thus A = πhw.

J.2. Artisanal scanner A different approach was to use an artisanal method of topography consisting in measuring the deviation of a laser plane that incided over the reed. The cast lied on an accurately moving table (A), whose displacement can be precisely measured (B). The table and the cast are illuminated by a laser beam (C) expanded by a transparent cylinder (E) and striking the table with a small angle (20 deg.). This laser plane would trace an arc of a circumeference (due to the presence of E) over the table if the cast wasn’t present (the baseline , A on figure J.2). Images are captured from above the reed (D), the direction of observation being perpendicular to the table. Several images (fig. J.2) of the laser plane are captured for different displacements of the cast (about 1 mm). These correspond to sections of the reed parallel to each other. The photographs are then inserted into the computer in order to extract the beam deviation at each point.

212

Chap. J: Topography methods

Figure J.1.: Scanner setup

Figure J.2.: Laser line illuminating the reed cast and computer recognition of the line: A – baseline; B – line deviated by the cast

From each photograph several points on an oblique section of the reed are extracted. The complete set of photos provides a sampling of a part of the reed’s surface, however in order to reconstruct cross-sections of the reed interpolation must be done.

J.2 Artisanal scanner

213

10

20

30

40

4

50

2

60

20 15

Figure J.3.: Three-dimensional reconstitution of a reed cast

214

Chap. J: Topography methods

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