Dispersive Effective Model in the Time-Domain for Acoustic Waves Propagating in Bubbly Media

TL;DR

This paper develops a dispersive effective medium model for time-domain acoustic waves in bubbly media, avoiding Fourier transforms and allowing general incident waves, with implications for materials science and imaging.

Contribution

It introduces a novel time-domain dispersive effective model for acoustic waves in bubbly media that does not require periodicity or band-limited incident waves.

Findings

The effective model is dispersive due to bubble resonance.

The model is an integro-differential equation with a convolution term.

Bubbles can be distributed according to any smooth function, not necessarily periodically.

Abstract

We derive the effective medium theory for the linearized time-domain acoustic waves propagating in a bubbly media. The analysis is done in the time-domain avoiding the need to use Fourier transformation. This allows considering general incident waves avoiding band limited ones as usually used in the literature. Most importantly, the outcome is as follows: 1. As the bubbles are resonating, with the unique subwavelength Minnaert resonance, the derived effective wave model is dispersive. Precisely, the effective acoustic model is an integro-differential one with a time-convolution term highlighting the resonance effect. 2. The periodicity in distributing the cluster of bubbles is not needed, contrary to the case of using traditional two-scale homogenization procedures. Precisely, given any $C^1$-smooth function $K$, we can distribute the bubbles so that locally the number of such bubbles is dictated by $K$. In addition to its dispersive character, the effective model is affected by the function $K$. Such freedom and generality is appreciable in different applied sciences including materials sciences and mathematical imaging.