Waves in Two and Three Dimensions

  • Wilfried Kausel
Part of the Springer Handbooks book series (SHB)


This chapter deals with the generalization of the wave equation to describe wave propagation on two-dimensional surfaces and sound waves in a three-dimensional space. Again linearity is postulated, which is only justified if amplitudes are sufficiently small. It will be shown how wave equations can be derived for rectangular and circular membranes , plates and disks and how analytic results can be obtained for a three-dimensional case with relatively simple boundary conditions. This chapter will also review techniques for the calculation of resonance frequencies and for the prediction of associated modal shapes.


finite element method


  1. 3.1
    N.H. Fletcher, T.D. Rossing: The Physics of Musical Instruments, 2nd edn. (Springer, New York 1990)zbMATHGoogle Scholar
  2. 3.2
    P. Hagedorn: Mechanical oscillations. In: Mechanics of Musical Instruments, Courses and Lectures/International Centre for Mechanical Sciences, Vol. 355, ed. by A. Hirschberg, J. Kergomard, G. Weinreich (Springer, Wien 1995) pp. 7–78Google Scholar
  3. 3.3
    P.M. Morse, U. Ingard: Theoretical Acoustics (Princeton Univ. Press, New Jersey 1987)Google Scholar
  4. 3.4
    M. Heckl: Vibrations of one- and two-dimensional continuous systems. In: Handbook of Acoustics, ed. by M.J. Crocker (Wiley, New York 1998) pp. 579–596Google Scholar
  5. 3.5
    H. von Helmholtz: Die Lehre von den Tonempfindungen als Physiologische Grundlage Für die Theorie der Musik (F. Vieweg, Braunschweig 1863)zbMATHGoogle Scholar
  6. 3.6
    H. Levine, J. Schwinger: On the radiation of sound from an unflanged circular pipe, Phys. Rev. 73, 383–406 (1948)MathSciNetCrossRefGoogle Scholar
  7. 3.7
    C.P.A. Braden: Bore Optimisation and Impedance Modelling of Brass Musical Instruments, Ph.D. Thesis (Univ. Edinburgh, Edinburgh 2006)Google Scholar
  8. 3.8
    J.A. Kemp: Theoretical and Experimental Study of Wave Propagation in Brass Musical Instruments, Ph.D. Thesis (Univ. Edinburgh, Edinburgh 2002)Google Scholar
  9. 3.9
    V. Pagneux, N. Amir, J. Kergomard: A study of wave propagation in varying cross-section waveguides by modal decomposition. Part I. Theory and validation, J. Acoust. Soc. Am. (JASA) 100(4), 2034–2048 (1996)CrossRefGoogle Scholar
  10. 3.10
    A.M. Bruneau, M. Bruneau, P. Herzog, J. Kergomard: Boundary layer attenuation of higher order modes in waveguides, J. Sound Vib. 119(1), 15–27 (1987)CrossRefGoogle Scholar
  11. 3.11
    W.E. Zorumski: Generalized radiation impedances and reflection coefficients of circular and annular ducts, J. Acoust. Soc. Am. (JASA) 54(6), 1667–1673 (1973)CrossRefGoogle Scholar
  12. 3.12
    N. Amir, V. Pagneux, J. Kergomard: A study of wave propagation in varying cross-section waveguides by modal decomposition. Part II. Results, J. Acoust. Soc. Am. (JASA) 101(5), 2504–2517 (1997)CrossRefGoogle Scholar
  13. 3.13
    T. Hélie, X. Rodet: Radiation of a pulsating portion of a sphere: Application to horn radiation, Acustica 89(4), 565–577 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2018

Authors and Affiliations

  • Wilfried Kausel
    • 1
  1. 1.Dept. of Musical AcousticsUniversity of Music and Performing Arts, ViennaViennaAustria

Personalised recommendations