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.
Highlights
The problem of resonant acoustic scattering by air filled spherical cavity in a infinite liquid medium, socalled single air bubble in water, is one of those famous problems in classical physics
Due to modal degeneracy, related to these resonances, the transcendental Eq 13 was solved in β–complex plane using a generalized Muller’s method where the JWKB estimates to Transmission Resonance (TR) were used as an initial guess either
The concept of effective potential Ueff explicitly given in Eq 21 is applied according to the analogy between acoustic waves and matter waves in quantum mechanics (Nussenzveig 1969, 1992, Guimarães and Nussenzveig 1992)
Summary
The problem of resonant acoustic scattering by air filled spherical cavity in a infinite liquid medium, socalled single air bubble in water, is one of those famous problems in classical physics. In this semiclassical regime of narrow resonance (Guimarães and Nussenzveig 1992) shown in Eq 47, the internal bubble surface transmission coefficient behaves as N b , so the internal wave total number of turns can be asymptotically estimated as 1/(N b ).
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