World Scientific
  • Search
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at [email protected] for any enquiries.


    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.

    AMSC: 93C20, 93D20, 35B37


    • 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 ScienceGoogle Scholar
    • D. Matignon, ESAIM Proc. 5, 145 (1998). CrossrefGoogle 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 ScienceGoogle Scholar
    • G. Montseny, ESAIM Proc. 5, 159 (1998). CrossrefGoogle 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 ScienceGoogle 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 ScienceGoogle 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 ScienceGoogle 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 ) . CrossrefGoogle Scholar
    • J. R.   Partington , Linear Operators and Linear Systems , London Mathematical Society Student Texts   60 ( Cambridge Univ. Press , 2004 ) . CrossrefGoogle Scholar
    • H.   Cartan , Elementary Theory of Analytic Functions of One or Several Complex Variables , Coll. Enseignement des Sciences ( Hermann , 1961 ) . Google Scholar
    Remember to check out the Most Cited Articles!

    View our Mathematical Modelling books
    Featuring authors Frederic Y M Wan, Gregory Baker and more!