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.
@article{bwmeta1.element.doi-10_2478_s11533-008-0017-9,
author = {Claude Gauthier},
title = {Conditions for periodic vibrations in a symmetric n-string},
journal = {Open Mathematics},
volume = {6},
year = {2008},
pages = {287-300},
zbl = {1146.35057},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0017-9}
}
Claude Gauthier. Conditions for periodic vibrations in a symmetric n-string. Open Mathematics, Tome 6 (2008) pp. 287-300. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0017-9/
[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 http://dx.doi.org/10.1088/0305-4470/32/45/302 | Zbl 0946.81025
[3] Cattaneo C., Fontana L., D’Alembert formula on finite one-dimensional networks, J. Math. Anal. Appl., 2003, 284, 403–424 http://dx.doi.org/10.1016/S0022-247X(02)00392-X | Zbl 1035.35058
[4] Dáger R., Zuazua E., Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris Ser.I Math., 2001, 332, 621–626 | Zbl 0986.35018
[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 | Zbl 0990.93051
[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 http://dx.doi.org/10.1016/S0022-460X(02)01122-7
[7] Gaudet S., Gauthier C., LeBlanc V.G., On the vibration of an N-string, J. Sound Vibration, 2000, 238, 147–169 http://dx.doi.org/10.1006/jsvi.2000.3153
[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 http://dx.doi.org/10.1016/j.jsv.2004.01.036 | Zbl 1236.74104
[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 http://dx.doi.org/10.1080/00018730600908042
[11] Lagnese J.E., Leugering G., Schmidt E.J.P.G., Modeling analysis and control of dynamic elastic multi-link structures, Birkhäuser, Boston, 1994 | Zbl 0810.73004
[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 | Zbl 0547.35061