No Access

DIFFUSIVE REPRESENTATIONS FOR THE ANALYSIS AND SIMULATION OF FLARED ACOUSTIC PIPES WITH VISCO-THERMAL LOSSES

Laboratoire des Sciences et Technologies de la Musique et du Son, CNRS UMR 9912, Équipe Analyse/Synthése - Ircam, Centre Georges Pompidou, 1, place Igor Stravinsky, F-75004 Paris, France

Search for more papers by this author

and

D. MATIGNON

Dépt. Traitement du Signal et des Images, Télécom Paris, CNRS UMR 5141, 37-39, rue Dareau, F-75014 Paris, France

Search for more papers by this author

https://doi.org/10.1142/S0218202506001248Cited by:35 (Source: Crossref)

Abstract

Acoustic waves travelling in axisymmetric pipes with visco-thermal losses at the wall obey a Webster–Lokshin model. Their simulation may be achieved by concatenating scattering matrices of elementary transfer functions associated with nearly constant parameters (e.g. curvature). These functions are computed analytically and involve diffusive pseudo-differential operators, for which we have representation formula and input-output realizations, yielding direct numerical approximations of finite order. The method is based on some involved complex analysis.

Keywords:

AMSC: 93C20, 93D20, 35B37

References

H. Haddar, T. Hélie and D. Matignon, A Webster–Lokshin model for waves with viscothermal losses and impedance boundary conditions: Strong solutions, 6th Int. Conf. on Mathematical and Numerical Aspects of Wave Propagation Phenomena (INRIA, 2003) pp. 66–71. Google Scholar
H. Haddar and D. Matignon, Theoretical and numerical analysis of the Webster–Lokshin model, Technical report, INRIA, 2005 . Google Scholar
T. Hélie, J. Acoust. Soc. Amer. 114, 2633 (2003), DOI: 10.1121/1.1608962. Crossref, Web of Science, Google Scholar
D. Matignon, ESAIM Proc. 5, 145 (1998). Crossref, Google Scholar
M. Bruneau , Manuel d'acoustique Fondamentale ( Hermès , 1998 ) . Google Scholar
J. Dupraz , La Théorie des Distributions et ses Applications ( Cépadues , 1977 ) . Google Scholar
D. Matignon and B. d'Andréa-Novel, Spectral and time-domain consequences of an integro-differential perturbation of the wave pde, 3rd Int. Conf. on Math. and Numer. Aspects of Wave Propagation Phenomena (SIAM, 1995) pp. 769–771. Google Scholar
M. Taylor , Pseudo-Differential Operators , Princeton Mathematical Series 34 ( Princeton Univ. Press , 1981 ) . Google Scholar
O. J. Staffans, Trans. Amer. Math. Soc. 345, 527 (1994), DOI: 10.2307/2154987. Crossref, Web of Science, Google Scholar
G. Montseny, ESAIM Proc. 5, 159 (1998). Crossref, Google Scholar
D. Matignon and H. J. Zwart, Standard diffusive systems as well-posed linear systems, Syst. Control Lett., submitted . Google Scholar
D. Salamon, Trans. Amer. Math. Soc. 300, 383 (1987), DOI: 10.2307/2000351. Web of Science, Google Scholar
M. Abramowitz and I. A. Stegun , Handbook of Mathematical Functions ( Dover , 1970 ) . Google Scholar
H. Zwart, Syst. Control Lett. 52, 247 (2004), DOI: 10.1016/j.sysconle.2004.02.002. Crossref, Web of Science, Google Scholar
D. G. Duffy , Transform Methods for Solving Partial Differential Equations ( CRC Press , 1994 ) . Google Scholar
D. Héleschewitz, Analyse et simulation de systèmes différentiels fractionnaires et pseudo-différentiels linéaires sous représentation diffusive, Ph.D. thesis, Ecole Nationale Supérieure des Télécommunications, 2000 . Google Scholar
T. Hélie and D. Matignon , Signal Processing ( Elsevier , 2006 ) . Google Scholar
E. Zwicker and R. Feldtkeller , Psychoacoustique-L'oreille, Récepteur D'information, Collection Technique et Scientifique des Télécommunications ( Masson , 1981 ) . Google Scholar
T. Hélie, Physical modeling of musical instruments with dynamic systems and inversion processes, Ph.D. thesis, Université Paris-Sud, Orsay, 2002 . Google Scholar
J. O. Smith, Comput. Music J. 20, 44 (1996). Crossref, Web of Science, Google Scholar
M. Dunau, Représentations diffusives de seconde espèce: Introduction et expérimentation, Master's thesis, DEA d'Automatique, Toulouse, 2000 . Google Scholar
D. Matignon, Représentations en variables d'état de guides d'ondes avec dérivation fractionnaire, Ph.D. thesis, Université Paris-Sud, 1994, Appendix C . Google Scholar
D. P. Berners, Acoustics and signal processing techniques for physical modeling of brass instruments, Ph.D. thesis, Stanford University, 1999 . Google Scholar
R. Curtain and H. Zwart , An Introduction to Infinite-Dimensional Linear Systems Theory ( Springer-Verlag , 1995 ) . Crossref, Google Scholar
J. R. Partington , Linear Operators and Linear Systems , London Mathematical Society Student Texts 60 ( Cambridge Univ. Press , 2004 ) . Crossref, Google Scholar
H. Cartan , Elementary Theory of Analytic Functions of One or Several Complex Variables , Coll. Enseignement des Sciences ( Hermann , 1961 ) . Google Scholar