Conditions for periodic vibrations in a symmetric n-string

Claude Gauthier 1
  • 1 Université de Moncton


A symmetric N-string is a network of N ≥ 2 sections of string tied together at one common mobile extremity. In their equilibrium position, the sections of string form N angles of 2π/N at their junction point. Considering the initial and boundary value problem for small-amplitude oscillations perpendicular to the plane of the N-string at rest, we obtain conditions under which the solution will be periodic with an integral period.

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  • [1] Ali Mehmeti F., Nonlinear Wave in Networks, Mathematical Research 80, Akademie-Verlag, Berlin, 1994

  • [2] Berkolaiko G., Keating J.P., Two-point spectral correlation for star graphs, J. Phys. A, 1999, 32, 7827–7841

  • [3] Cattaneo C., Fontana L., D’Alembert formula on finite one-dimensional networks, J. Math. Anal. Appl., 2003, 284, 403–424

  • [4] Dáger R., Zuazua E., Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris Ser.I Math., 2001, 332, 621–626

  • [5] Dáger R., Zuazua E., Controllability of tree-shaped networks of vibrating strings, C. R. Acad. Sci. Paris Ser.I Math., 2001, 332, 1087–1092

  • [6] Gaudet S., Gauthier C., A numerical model for the 3-D non-linear vibrations of an N-string, J. Sound Vibration, 2003, 263, 269–284

  • [7] Gaudet S., Gauthier C., LeBlanc V.G., On the vibration of an N-string, J. Sound Vibration, 2000, 238, 147–169

  • [8] Gaudet S., Gauthier C., Léger L., Walker C., The vibration of a real 3-string: the timbre of the tritare, J. Sound Vibration, 2005, 281, 219–234

  • [9] Gauthier C., The amplification of non-linear travelling waves through a tree of 3-strings, Nuovo Cimento Soc. Ital. Fis. B, 2004, 119, 361–369

  • [10] Gnutzmann S., Smilansky U., Quantum graphs: applications to quantum chaos and universal spectral statistics, Adv. Phys., 2006, 55, 527–625

  • [11] Lagnese J.E., Leugering G., Schmidt E.J.P.G., Modeling analysis and control of dynamic elastic multi-link structures, Birkhäuser, Boston, 1994

  • [12] Sagan B.E., The symmetric group, The Wadsworth & Brooks/Cole Mathematic Series, Pacific Grove, California, 1991

  • [13] Sullivan D., The wave equation and periodicity, Appl. Math. Notes, 1984, 9, 1–12


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