Tunneling effects in resonant acoustic scattering of an air bubble in unbounded water

Abstract

The problem of acoustic scattering of a gaseous spherical bubble immersed within unbounded liquid surrounding is considered in this work. The theory of partial wave expansion related to this problem is revisited. A physical model based on the analogy between acoustic scattering and potential scattering in quantum mechanics is proposed to describe and interpret the acoustical natural oscillation modes of the bubble, namely, the resonances. In this context, a physical model is devised in order to describe the air water interface and the implications of the high density contrast on the various regimes of the scattering resonances. The main results are presented in terms of resonance lifetime periods and quality factors. The explicit numerical calculations are undertaken through an asymptotic analysis considering typical bubble dimensions and underwater sound wavelengths. It is shown that the resonance periods are scaled according to the Minnaert’s period, which is the short lived resonance mode, called breathing mode of the bubble. As expected, resonances with longer lifetimes lead to impressive cavity quality Q-factor ranging from 10¹⁰ to 10⁵. The present theoretical findings lead to a better understanding of the energy storage mechanism in a bubbly medium.

Key words
Acoustic Scattering; Minnaert Resonance; Semi-Classical Methods; Whispering Gallery Modes; Mie and Rayleigh Scattering