The behaviour of induced discontinuities behind a first order discontinuity wave for a quasi-linear hyperbolic system

  • Alessandra Borrelli
  • Maria Cristina Patria
Original Papers


In the present paper we study the propagation into a constant state of the induced discontinuities associated with a first order discontinuity wave for a quasi-linear hyperbolic system. Making use of the theory of singular surfaces and the ray-theory, we derive and solve completely the equations which the induced discontinuity vector\(\mathop \omega \limits^*\) must obey along the rays associated with the wave front. So we determine the evolution law of\(\mathop \omega \limits^*\) and find that it depends non-linearly on the first order discontinuities and on the geometrical features of the wave front; thus the behaviour of the induced discontinuities is known once the evolution law of the first order discontinuity wave is obtained explicitly.


Mathematical Method Wave Front Geometrical Feature Hyperbolic System Constant State 
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In questa nota studiamo la propagazione, in uno stato costante, delle discontinuità indotte associate a un'onda di discontinuità del primo ordine per un sistema iperbolico quasi-lineare. Adottando un'opportuna combinazione della teoria delle superfici singolari e delle teoria dei raggi, determiniamo in maniera completa il comportamento del vettore delle discontinuità indotte\(\mathop \omega \limits^*\) lungo i raggi associati al fronte d'onda. Troviamo che la legge di evoluzione di\(\mathop \omega \limits^*\) dipende non linearmente dalle discontinuità del primo ordine e dalle caratteristiche geometriche del fronte d'onda. L'andamento di\(\mathop \omega \limits^*\) è perciò noto una volta nota esplicitamente la legge di evoluzione delle discontinuità del primo ordine.


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  1. [1]
    E. E. Inan,The propagation of weak discontinuities in quasi-linear hyperbolic systems in the presence of a strong discontinuity. Part I: Fundamental theory. Bull. Tech. Univ. Istanbul34, 48–66 (1981).Google Scholar
  2. [2]
    P. J. Chen,The behavior of induced discontinuities behind curved shocks in isotropic linear elastic material. J. Elast.15, 43–53 (1985).Google Scholar
  3. [3]
    P. J. Chen, On induced discontinuities behind shock waves in isotropic linear viscoelastic materials. Nuovo Cimento81 B, 113–127 (1984).Google Scholar
  4. [4]
    P. J. Chen,Evolutionary behavior of induced discontinuities behind one dimensional shock waves in non-linear elastic materials. J. Elast.15, 257–269 (1985).Google Scholar
  5. [5]
    G. Boillat,La propagation des ondes. Gauthier-Villars, Paris 1965.Google Scholar
  6. [6]
    G. A. Nariboli,Wave propagation in anisotropic elasticity. J. Math. Anal. & Appl.16, 108–122 (1966).Google Scholar
  7. [7]
    G. A. Nariboli,On some aspects of wave propagation. J. Math. Phys. Sc.2, 294–310 (1968).Google Scholar
  8. [8]
    T. Y. Thomas,Extended compatibility conditions for the study of surfaces of discontinuity in continuum mechanics. J. Math. Mech.6, 311–322 (1957).Google Scholar
  9. [9]
    T. Y. Thomas,Plastic flow and fracture in solids. Academic Press, New York 1961.Google Scholar
  10. [10]
    S. Giambò and A. Palumbo,Sur l'équation de transport pour une onde simple en relativité restreinte. C.R.A.S.291 A, 665–668 (1980).Google Scholar
  11. [11]
    S. Giambó,Sur l'équation de transport pour une onde multiple en relativité restreinte. C.R.A.S.291 A, 275–278 (1980).Google Scholar
  12. [12]
    A. Palumbo,Sulle equazioni di trasporto delle discontinuità. Atti Ac. Scienze, Lettere ed Arti di Palermo, to appear.Google Scholar
  13. [13]
    A. Jeffrey,Quasilinear hyperbolic systems and waves. Pitman Publ., London 1976.Google Scholar
  14. [14]
    T. Y. Thomas,Concepts from tensor analysis and differential geometry. Academic Press, New York 1961.Google Scholar
  15. [15]
    C. Truesdell and R. A. Toupin,The classical field theories. In: Handb. d. Phys. (ed. S. Flügge), Band III/1, Springer-Verlag, Berlin 1960.Google Scholar

Copyright information

© Birkhäuser Verlag 1987

Authors and Affiliations

  • Alessandra Borrelli
    • 1
  • Maria Cristina Patria
    • 1
  1. 1.Dipartimento di MatematicaUniversitá di FerraraFerraraItaly

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