# Modeling of the resonance of an acoustic wave in a torus

• [1] UFR de Mathématiques et Informatique Université de Cocody Abidjan 22 BP 582 Abidjan 22 CÔTE D’IVOIRE
• Volume: 20, Issue: 2, page 377-390
• ISSN: 1259-1734

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## Abstract

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A pneumatic tyre in rotating motion with a constant angular velocity $\Omega$ is assimilated to a torus whose generating circle has a radius $R$. The contact of the tyre with the ground is schematized as an ellipse with semi-major axis $a$. When $\left(\Omega R/{C}_{0}\right)\ll 1$ and $\left(a/R\right)\ll 1$ (where ${C}_{0}$ is the velocity of the sound), we show that at the rapid time scale $R/{C}_{0}$, the air motion within a torus periodically excited on its surface generates an acoustic wave $h$. A study of this acoustic wave is conducted and shows that the mode associated to $p=0$ leads to resonance. In resonance the acoustic wave $h$ moves quadratically in time and also decreases asymptotically faster when the mean pressure in the domain is low.

## How to cite

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Adou, Jérôme, Coulibaly, Adama, and Dakouri, Narcisse. "Modeling of the resonance of an acoustic wave in a torus." Annales mathématiques Blaise Pascal 20.2 (2013): 377-390. <http://eudml.org/doc/275641>.

abstract = {A pneumatic tyre in rotating motion with a constant angular velocity $\Omega$ is assimilated to a torus whose generating circle has a radius $R$. The contact of the tyre with the ground is schematized as an ellipse with semi-major axis $a$. When $(\Omega R/C_\{0\})\ll 1$ and $(a/R)\ll 1$ (where $C_\{0\}$ is the velocity of the sound), we show that at the rapid time scale $R/C_\{0\}$, the air motion within a torus periodically excited on its surface generates an acoustic wave $h$. A study of this acoustic wave is conducted and shows that the mode associated to $p=0$ leads to resonance. In resonance the acoustic wave $h$ moves quadratically in time and also decreases asymptotically faster when the mean pressure in the domain is low.},
affiliation = {UFR de Mathématiques et Informatique Université de Cocody Abidjan 22 BP 582 Abidjan 22 CÔTE D’IVOIRE; UFR de Mathématiques et Informatique Université de Cocody Abidjan 22 BP 582 Abidjan 22 CÔTE D’IVOIRE; UFR de Mathématiques et Informatique Université de Cocody Abidjan 22 BP 582 Abidjan 22 CÔTE D’IVOIRE},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Acoustic waves; pneumatic; Resonance; Air; Torus; Numerical Modeling; acoustic waves; resonance; air; torus; numerical modeling},
language = {eng},
month = {7},
number = {2},
pages = {377-390},
publisher = {Annales mathématiques Blaise Pascal},
title = {Modeling of the resonance of an acoustic wave in a torus},
url = {http://eudml.org/doc/275641},
volume = {20},
year = {2013},
}

TY - JOUR
AU - Dakouri, Narcisse
TI - Modeling of the resonance of an acoustic wave in a torus
JO - Annales mathématiques Blaise Pascal
DA - 2013/7//
PB - Annales mathématiques Blaise Pascal
VL - 20
IS - 2
SP - 377
EP - 390
AB - A pneumatic tyre in rotating motion with a constant angular velocity $\Omega$ is assimilated to a torus whose generating circle has a radius $R$. The contact of the tyre with the ground is schematized as an ellipse with semi-major axis $a$. When $(\Omega R/C_{0})\ll 1$ and $(a/R)\ll 1$ (where $C_{0}$ is the velocity of the sound), we show that at the rapid time scale $R/C_{0}$, the air motion within a torus periodically excited on its surface generates an acoustic wave $h$. A study of this acoustic wave is conducted and shows that the mode associated to $p=0$ leads to resonance. In resonance the acoustic wave $h$ moves quadratically in time and also decreases asymptotically faster when the mean pressure in the domain is low.
LA - eng
KW - Acoustic waves; pneumatic; Resonance; Air; Torus; Numerical Modeling; acoustic waves; resonance; air; torus; numerical modeling
UR - http://eudml.org/doc/275641
ER -

## References

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1. J. Adou, Etude de la résonance du mode géostrophique dans un tore, C. R. Acad. Sci. Paris 327 (1999), 1391-1396 Zbl0981.76102
2. J. Adou, Modelling of a resonant inertial oscillation within a torus, Arch. of Applied Mech. 71 (2001), 695-702 Zbl1002.76098
3. J. Adou, Sur l’origine aérodynamique du danger de sous-gonflage des pneumatiques, Entropie 234 (2001), 54-60
4. J.-P. Guiraud, R. Zeytounian, Evolution des ondes acoustiques sur une longue période: le concept d’écoulement incompressible avec densité fonction du temps, C. R. Acad. Sci. Paris (1980), 75-77 MR618100
5. C. Gulpin, Manuel de calcul numérique appliqué, (2000), EDP sciences Zbl0958.65005

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