Modeling of the resonance of an acoustic wave in a torus. (English. French summary) Zbl 1282.35288

Summary: 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.


35Q35 PDEs in connection with fluid mechanics
76Q05 Hydro- and aero-acoustics
35B10 Periodic solutions to PDEs
35B34 Resonance in context of PDEs
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