The Minnaert bubble : a new approach
Résumé
We propose a new ab initio introduction to the well known Minnaert pulsating bubble, beginning with a detailled discussion of the radial movements of an air bubble in water. In unbounded water, the {air – water} system has a continuum of eigenmodes, some of them correspond to regular Fabry-Pérot resonances. A singular resonance, the lowest one, is shown to coincide with that of Minnaert. In bounded water, the eigenmodes' spectrum is discrete, with a finite fundamental frequency. A spectacular quasi-locking of the latter occurs if it happens to exceed the Minnaert frequency, which provides an unforseen one-bubble alternative version of the famous “hot chocolate effect”. In the (low) frequency domain in which sound propagation inside the bubble reduces to a simple inflation/deflation, the –light– air bubble can be “dressed” by the outer water pressure forces, and is turned into the –heavy– Minnaert bubble. Thanks to this unexpected renormalization process, we demonstrate that the Minnaert bubble definitely behaves like a true harmonic oscillator of the mass-spring type, but with a damping term and a forcing term in apparent disagreement with those commonly admitted in literature. At last, in order to tell the water's motion associated with water compressibility (i.e. the sound) from the simple incompressible accompaniment of the bubble's breathing, we introduce a new picture analogous to the electromagnetic radiative picture in Coulomb gauge, which naturally leads us to split the water displacement in an instantaneous and a retarded parts. The Minnaert renormalized mass of the dressed bubble is then recovered.
Domaines
Matière Molle [cond-mat.soft]
Origine : Fichiers produits par l'(les) auteur(s)
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