An application in musical acoustics:
the case of gongs and cymbals
The typical sound produced by cymbal and gong-like instruments results from the geometrical non-linearity. Controlled experiments, where a cymbal, or a gong, is harmonically forced with a slowly increasing amplitude of forcing, have shown the generic chractaeristics for the transition to chaos via two successive bifurcations. Below is shown an experimental result obtained with a gong.
One can hear the sound produced during these kind of experiments, for two different cymbals with two exciting frequencies:
Zyldjian Cymbal, excited at 283 Hz.
[MP3]
K-ride Cymbal, excited at 438 Hz.
[MP3]
Numerical simulations
Numerical simulations are conducted in order to reproduce the observed experimental scenario as well as to give insight into the wave turbulence regime. Using a numerical procedure developped by Stefan Bilbao, that combines finite difference in space and an energy-conserving scheme in time, we recover the bifurcation scenario with the apparition, for certain excitation frequencies, of the quasiperiodic regime before the chaotic one. below is shown the spectrogram of a numerical simulation for a rectangular perfect plate with free edges, excited at 645 Hz.
Below at the WAV format, you can hear the sound of a numerical simulation (the velocity of one point), for the same plate but now excited at 289 Hz. (click on the spectrogram to get the sound).
Related articles
- C. Touzé et A. Chaigne : Lyapunov exponents from experimental time series. Application to cymbal vibrations, Acta Acustica, vol 86, No 3, pp. 557-567, 2000.
- A. Chaigne, C. Touzé et O. Thomas: Nonlinear vibrations and chaos in gongs and cymbals, Acoustical Science and Technology, Acoust. Soc. of Japan, vol. 26, No. 5, pp 403-409, 2005.
[AST/ASJ | pdf]
- 28. C. Touzé, S. Bilbao, L. Longo-Mucciante, O. Cadot et A. Boudaoud : Vibrations chaotiques de plaques minces:
application aux instruments de type cymbale, 10ème Congrès Français d'Acoustique (CFA 10), Lyon, 12-16 Avril 2010.
[ pdf ]
