Sound propagation in a monodisperse bubble cloud: from the crystal to the glass

We present a theoretical study of the propagation of a monochromatic pressure wave in an unbounded monodisperse bubbly liquid. We begin with the case of a regular bubble array--a bubble crystal--for which we derive a dispersion relation. In order to interpret the different branches of this relation, we introduce a formalism, the radiative picture, which is the adaptation to acoustics of the standard splitting of the electric field in an electrostatic and a radiative part in Coulomb gauge. In the case of an irregular or completely random array--a bubble glass--and at wavelengths large compared to the size of the bubble array spatial inhomogeneities, the difference between order and disorder is not felt by the pressure wave: a dispersion relation still holds, coinciding with that of a bubble crystal with the same bubble size and air volume fraction at the centre of its first Brillouin zone. This relation is discussed and compared to that obtained by Foldy in the framework of his multiscattering approach.

The bubble cloud as an N-degree of freedom harmonic oscillator

The dynamics of a two-dimensional N-bubble static cloud is investigated and shown to be well described by an N-degree of freedom harmonic oscillator model, at least at low enough frequencies. Eigenmodes and eigenfrequencies are calculated and compared with experimental results obtained with an assembly of bubbles caught up under a net in a water tank. Accordance is found to be excellent in the frequency range of validity of the model, the limits of which are discussed. An interpretation of the low-frequency branch of Foldy's dispersion relation in bubbly liquids is suggested in terms of "bubble waves" in a quasi-incompressible medium.