The Minnaert bubble: an acoustic approach

We propose an ab initio introduction to the well-known Minnaert pulsating bubble at graduate level. After a brief recall of the standard stuff, we begin with a detailed discussion of the radial movements of an air bubble in water. This discussion is managed from an acoustic point of view, and using the Lagrangian rather than the Eulerian variables. In unbounded water, the air-water system has a continuum of eigenmodes, some of them correspond to regular Fabry Perot 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 unforeseen 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 'breathing' (i.e. 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 spring-bob type, but with a damping term and a forcing term in apparent disagreement with those commonly admitted in the literature. Finally, we underline the double role played by the water. In order to tell the water motion associated with water compressibility (i.e. the sound) from the simple incompressible accompaniment of the bubble 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 part. The Minnaert renormalized mass of the dressed bubble is then automatically recovered.