nach oben

Erschienen in:

2018 | OriginalPaper | Buchkapitel

19. Finite-Difference Schemes in Musical Acoustics: A Tutorial

verfasst von : Stefan Bilbao, Brian Hamilton, Reginald Harrison, Alberto Torin

Erschienen in: Springer Handbook of Systematic Musicology

Verlag: Springer Berlin Heidelberg

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

The functioning of musical instruments is well described by systems of partial differential equations. Whether one's interest is in pure musical acoustics or physical modeling of sound synthesis, numerical simulation is a necessary tool, and may be carried out by a variety of means. One approach is to make use of so-called finite-difference or finite-difference time-domain methods, whereby the numerical solution is computed as a recursion operating over a grid. This chapter is intended as a basic tutorial on the design and implementation of such methods, for a variety of simple systems. The 1-D wave equation and simple difference schemes are covered in Sect. 19.1, accompanied by an analysis of numerical dispersion and stability, as well as implementation details via vector-matrix representations. Similar treatments follow for the case of the ideal stiff bar in Sect. 19.2, the acoustic tube in Sect. 19.3, the 2-D and 3-D wave equations in Sect. 19.4, and finally the stiff plate in Sect. 19.5. Some more general nontechnical comments on more complex extensions to nonlinear systems appear in Sect. 19.6.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Wave Field Synthesis

Nächstes Kapitel Real-Time Signal Processing on Field Programmable Gate Array Hardware

19.1

R. Courant, K. Friedrichs, H. Lewy: Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100(1), 32–74 (1928)MathSciNetCrossRef

19.2

R.D. Richtmyer: Difference Methods for Initial-Value Problems (Interscience, New York 1957)MATH

19.3

G.E. Forsythe, W.R. Wasow: Finite-Difference Methods for Partial Differential Equations (Wiley, New York 1960)MATH

19.4

K.S. Yee: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag. 14(3), 302–307 (1966)CrossRef

19.5

A. Taflove: Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems, IEEE Trans. Electromagn. Compat. EMC 22(3), 191–202 (1980)CrossRef

19.6

P. Ruiz: A Technique for Simulating the Vibrations of Strings with a Digital Computer, Master’s Thesis (Univ. Illinois, Urbana 1969)

19.7

L. Hiller, P. Ruiz: Synthesizing musical sounds by solving the wave equation for vibrating objects: Part II, J. Audio Eng. Soc. 19(7), 542–550 (1971)

19.8

J. Kelly, C. Lochbaum: Speech synthesis. In: Proc. 4th Int. Congr. Acoust., Copenhagen (1962) pp. 1–4

19.9

R. Bacon, J. Bowsher: A discrete model of a struck string, Acustica 41, 21–27 (1978)

19.10

X. Boutillon: Model for piano hammers: Experimental determination and digital simulation, J. Acoust. Soc. Am. 83(2), 746–754 (1988)CrossRef

19.11

A. Chaigne, A. Askenfelt: Numerical simulations of struck strings. I. A physical model for a struck string using finite difference methods, J. Acoust. Soc. Am. 95(2), 1112–1118 (1994)CrossRef

19.12

S. van Duyne, J.O. Smith III: Physical modelling with the 2-D digital waveguide mesh. In: Proc. Int. Comput. Music Conf., Tokyo (1993) pp. 40–47

19.13

F. Fontana, D. Rocchesso: Physical modelling of membranes for percussion instruments, Acta Acust. united Acust. 84(3), 529–542 (1998)

19.14

L. Savioja, T. Rinne, T. Takala: Simulation of room acoustics with a 3-D finite-difference mesh. In: Proc. Int. Comput. Music Conf., Århus (1994) pp. 463–466

19.15

S. van Duyne, J.O. Smith III: The 3-D tetrahedral digital waveguide mesh with musical applications. In: Proc. Int. Comput. Music Conf., Hong Kong (1996) pp. 9–16

19.16

D. Botteldooren: Acoustical finite-difference time-domain simulation in a quasi-Cartesian grid, J. Acoust. Soc. Am. 95(5), 2313–2319 (1994)CrossRef

19.17

S. Bilbao: Numerical Sound Synthesis: Finite Difference Schemes and Simulation in Musical Acoustics (Wiley, Chichester 2009)CrossRef

19.18

P. Morse, U. Ingard: Theoretical Acoustics (Princeton University Press, Princeton 1968)

19.19

J.O. Smith III: Physical Audio Signal Procesing 2004)

19.20

J. Strikwerda: Finite Difference Schemes and Partial Differential Equations (Wadsworth Brooks, Pacific Grove 1989)MATH

19.21

B. Gustafsson, H.-O. Kreiss, J. Oliger: Time Dependent Problems and Difference Methods (Wiley, New York 1995)MATH

19.22

M. Ducceschi, S. Bilbao: Linear stiff string vibrations in musical acoustics: assessment and comparison of models, J. Acoust. Soc. Am. 140(4), 2445 (2016)CrossRef

19.23

N. Fletcher, T. Rossing: The Physics of Musical Instruments (Springer, New York 1998)CrossRef

19.24

A.G. Webster: Acoustical impedance, and the theory of horns and of the phonograph, Proc. Natl. Acad. Sci. U.S.A. 5(7), 275–282 (1919)CrossRef

19.25

M. Campbell, C. Greated: The Musician’s Guide to Acoustics (Oxford University Press, Oxford 1987)

19.26

A.H. Benade: On the propagation of sound waves in a cylindrical conduit, J. Acoust. Soc. Am. 44(2), 616–623 (1968)CrossRef

19.27

R. Caussé, J. Kergomard, X. Lurton: Input impedance of brass musical instruments – Comparison between experiment and numerical models, J. Acoust. Soc. Am. 75(1), 241–254 (1984)CrossRef

19.28

D.H. Keefe: Acoustical wave propagation in cylindrical ducts: Transmission line parameter approximations for isothermal and nonisothermal boundary conditions, J. Acoust. Soc. Am. 75(1), 58–62 (1984)CrossRef

19.29

J. Kergomard, R. Caussé: Measurement of acoustic impedance using a capillary: An attempt to achieve optimization, J. Acoust. Soc. Am. 79(4), 1129–1140 (1986)CrossRef

19.30

S. Bilbao, R. Harrison, J. Kergomard, B. Lombard, C. Vergez: Passive models of wave propagation in acoustic tubes, J. Acoust. Soc. Am. 138, 555–558 (2015)CrossRef

19.31

S. Bilbao, R. Harrison: Passive time-domain numerical models of viscothermal wave propagation in acoustic tubes of variable cross-section, J. Acoust. Soc. Am. 140, 728–740 (2016)CrossRef

19.32

T. Hélie, R. Mignot, D. Matignon: Waveguide modeling of lossy flared acoustic pipes: Derivation of a Kelly–Lochbaum structure for real-time simulations. In: IEEE Workshop Appl. Signal Process. Audio Acoust., New Paltz (2007) pp. 267–270

19.33

R. Mignot, T. Hélie, D. Matignon: Digital waveguide modeling for wind instruments: Building a state-space representation based on Webster–Lokshin model, IEEE Trans. Audio Speech Lang. Process. 18(4), 843–854 (2010)CrossRef

19.34

O.V. Rudenko, S.I. Soluyan: Theoretical Foundations of Nonlinear Acoustics (Consultants Bureau, New York 1977)CrossRef

19.35

S. Adachi, M. Sato: Time-domain simulation of sound production in the brass instrument, J. Acoust. Soc. Am. 97(6), 3850–3861 (1995)CrossRef

19.36

S. Adachi, M. Sato: Trumpet sound simulation using a two-dimensional lip vibration model, J. Acoust. Soc. Am. 99(2), 1200–1209 (1996)CrossRef

19.37

H. Levine, J. Schwinger: On the radiation of sound from an unflanged circular pipe, Phys. Rev. 73(4), 383–406 (1948)MathSciNetCrossRef

19.38

R. Harrison, S. Bilbao, J. Perry, T. Wishart: An environment for physical modeling of articulated brass instruments, Comput. Music J. 39(4), 80–95 (2015)CrossRef

19.39

R. Harrison-Harsley: Physical Modelling of Brass Instruments Using Finite-Difference Time-Domain Methods, PhD Thesis (Acoustics and Audio Group, University of Edinburgh, Edinburgh 2017)

19.40

J.O. Smith III: Efficient simulation of the reed-bore and bow-string mechanisms. In: Proc. Int. Comput. Music Conf., The Hague (1986) pp. 275–280

19.41

T. Hélie: Unidimensional models of acoustic propagation in axisymmetric waveguides, J. Acoust. Soc. Am. 114(5), 2633–2647 (2003)CrossRef

19.42

H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer, New York 2011)MATH

19.43

J. Conway, N.J.A. Sloane: Sphere Packings, Lattices and Groups (Springer, New York 1988)CrossRef

19.44

D. Botteldooren: Finite-difference time-domain simulation of low-frequency room acoustic problems, J. Acoust. Soc. Am. 98, 3302–3308 (1995)CrossRef

19.45

J. Botts, L. Savioja: Integrating finite difference schemes for scalar and vector wave equations. In: IEEE-ICASSP, Vancouver (2013) pp. 171–175

19.46

S. Bilbao: Modeling of complex geometries and boundary conditions in finite difference–finite volume time domain room acoustics simulation, IEEE Trans. Audio Speech Lang. Process. 21(7), 1524–1533 (2013)CrossRef

19.47

S. Bilbao, B. Hamilton, J. Botts, L. Savioja: Finite volume time domain room acoustics simulation under general impedance boundary conditions, IEEE Trans. Audio Speech Lang. Process. 24(1), 161–173 (2016)CrossRef

19.48

G.D. Smith: Numerical Solution of Partial Differential Equations: With Exercises and Worked Solutions (Oxford Univ. Press, Oxford 1965)MATH

19.49

W.F. Spotz, G.F. Carey: A high-order compact formulation for the 3-D Poisson equation, Numer. Methods Partial Differ. Equ. 12(2), 235–243 (1996)MathSciNetCrossRef

19.50

B. Hamilton, S. Bilbao, C.J. Webb: Revisiting implicit finite difference schemes for 3-D room acoustics simulations on GPU. In: DAFx (Univ. of Erlangen, Erlangen 2014)

19.51

B. Hamilton: Finite Difference and Finite Volume Methods for Wave-Based Modelling of Room Acoustics, PhD Thesis (Acoustics and Audio Group, University of Edinburgh, Edinburgh 2016)

19.52

A. Love: The small free vibrations and deformation of a thin elastic shell, Philos. Trans. R. Soc. Lond. A 179, 491–546 (1888)CrossRef

19.53

E. Reissner: The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech. 12, 69–77 (1945)MathSciNetMATH

19.54

R. Mindlin: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, J. Appl. Mech. 18, 31–38 (1951)MATH

19.55

A. Chaigne, C. Lambourg: Time-domain simulation of damped impacted plates I. Theory and experiments, J. Acoust. Soc. Am. 109(4), 1422–1432 (2001)CrossRef

19.56

K. Arcas, A. Chaigne: On the quality of plate reverberation, Appl. Acoust. 71(2), 147–156 (2010)CrossRef

19.57

E. Jansson: Acoustics for Violin and Guitar Makers (Department of Speech, Music and Hearing, Stockholm 2002)

19.58

N. Giordano: Simple model of a piano soundboard, J. Acoust. Soc. Am. 102(2), 1159–1168 (1997)MathSciNetCrossRef

19.59

A. Chaigne, C. Touzé, O. Thomas: Nonlinear vibrations and chaos in gongs and cymbals, Acoust. Sci. Technol. 26(5), 403–409 (2005)CrossRef

19.60

A.H. Nayfeh, D.T. Mook: Nonlinear Oscillations (Wiley, New York 1979)MATH

19.61

K.-J. Bathe: Finite Element Procedures (Prentice Hall, Upper Saddle River 1996)MATH

19.62

M.J. Elejabarrieta, A. Ezcurra, C. Santamaria: Vibrational behaviour of the guitar soundboard analysed by the finite element method, Acta Acust. united Acust. 87(1), 128–136 (2001)

19.63

J. Berthaut, M.N. Ichchou, L. Jezequel: Piano soundboard: Structural behavior, numerical and experimental study in the modal range, Appl. Acoust. 64(11), 1113–1136 (2003)CrossRef

19.64

S.A. Van Duyne: Digital Filter Applications to Modeling Wave Propagation in Springs, Strings, Membranes and Acoustical Space, Ph.D. Thesis (Center for Computer Research in Music and Acoustic, Stanford Univ., Stanford 2007)

19.65

C. Camier, C. Touzé, O. Thomas: Non-linear vibrations of imperfect free-edge circular plates and shells, Eur. J. Mech. A 28(3), 500–515 (2009)CrossRef

19.66

M. Ducceschi, C. Touzé, S. Bilbao, C.J. Webb: Nonlinear dynamics of rectangular plates: Investigation of modal interaction in free and forced vibrations, Acta Mechanica 225(1), 213–232 (2014)MathSciNetCrossRef

19.67

C. Lambourg, A. Chaigne, D. Matignon: Time-domain simulation of damped impacted plates. II. Numerical model and results, J. Acoust. Soc. Am. 109(4), 1433–1447 (2001)CrossRef

19.68

S. Bilbao: A digital plate reverberation algorithm, J. Audio Eng. Soc. 55(3), 135–144 (2007)MathSciNet

19.69

S. Bilbao: Percussion synthesis based on models of nonlinear shell vibration, IEEE Trans. Audio Speech Lang. Process. 18(4), 872–880 (2010)CrossRef

19.70

S. Timoshenko, S. Woinowsky-Krieger: Theory of Plates and Shells, Vol. 2 (McGraw-Hill, New York 1959)MATH

19.71

O. Thomas, S. Bilbao: Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties, J. Sound Vib. 315(3), 569–590 (2008)CrossRef

19.72

S. Bilbao, L. Savioja, J.O. Smith III: Parametrized finite difference schemes for plates: Stability, the reduction of directional dispersion and frequency warping, IEEE Trans. Audio Speech Lang. Process. 15(4), 1488–1495 (2007)CrossRef

19.73

A. Torin: Percussion Instrument Modelling In 3D: Sound Synthesis Through Time Domain Numerical Simulation, PhD Thesis (Acoustics and Audio Group, University of Edinburgh, Edinburgh 2015)

19.74

S. Bilbao, J.O. Smith III: Energy conserving finite difference schemes for nonlinear strings, Acustica 91, 299–311 (2005)

19.75

H. Conklin: Piano strings and phantom partials, J. Acoust. Soc. Am. 102, 659 (1997)CrossRef

19.76

S. Bilbao: Conservative numerical methods for nonlinear strings, J. Acoust. Soc. Am. 118(5), 3316–3327 (2005)CrossRef

19.77

J. Chabassier: Modeling and Numerical Simulation of the Piano Through Physical Modeling, Ph.D. Thesis (Ecole Polytechnique, Paris 2012)

19.78

S. Bilbao: A family of conservative finite difference schemes for the dynamical von Karman plate equations, Numer. Methods Partial Differ. Equ. 24(1), 193–216 (2008)MathSciNetCrossRef

19.79

T. Rossing, N. Fletcher: Nonlinear vibrations in plates and gongs, J. Acoust. Soc. Am. 73(1), 345–351 (1983)CrossRef

19.80

C. Vyasarayani, S. Birkett, J. McPhee: Modeling the dynamics of a vibrating string with a finite distributed unilateral constraint: Application to the sitar, J. Acoust. Soc. Am. 125(6), 3673–3682 (2010)CrossRef

19.81

S. Bilbao, A. Torin: Numerical modeling and sound synthesis for articulated string/fretboard interactions, J. Audio Eng. Soc. 63(5), 336–347 (2015)CrossRef

19.82

S. Bilbao, A. Torin, V. Chatziioannou: Numerical modeling of collisions in musical instruments, Acta Acust. united Acust. 101(1), 155–173 (2015)CrossRef

19.83

A. Hirschberg, J. Gilbert, R. Msallam, A. Wijnands: Shock waves in trombones, J. Acoust. Soc. Am. 99(3), 1754–1758 (1996)CrossRef

19.84

G. Sod: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys. 27(1), 1–31 (1978)MathSciNetCrossRef

19.85

R. Leveque: Finite Volume Methods for Hyperbolic Problems (Cambridge University Press, Cambridge 2002)CrossRef

19.86

B. Lombard, D. Matignon, Y. Le Gorrec: A fractional Burgers equation arising innonlinear acoustics: Theory and numerics. In: Proc. 9th IFAC Symp. Nonlinear Contr. Syst., Toulouse (2013)

Titel: Finite-Difference Schemes in Musical Acoustics: A Tutorial
verfasst von: Stefan Bilbao
Brian Hamilton
Reginald Harrison
Alberto Torin
Verlag: Springer Berlin Heidelberg
Buch: Springer Handbook of Systematic Musicology
Print ISBN: 978-3-662-55002-1

Electronic ISBN: 978-3-662-55004-5

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-662-55004-5_19

Premium Partner

Marktübersichten

Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.

Zur Marktübersicht