Uniform Space and Time Behavior for Acoustic Resonators

TL;DR

This paper investigates the time-domain behavior of acoustic waves interacting with a subwavelength Minnaert bubble, deriving a point-approximation expansion that captures primary and resonant wave components, and analyzing their temporal evolution and decay.

Contribution

It provides a novel time-domain analysis of Minnaert resonance effects, including a point-approximation expansion and lifetime estimation of the resonant wave.

Findings

Resonant wave persists for a time proportional to the inverse of the bubble radius.

The resonant wave's lifetime is inversely related to the imaginary part of the Minnaert resonance.

The period of the wave is determined by the real part of the resonance.

Abstract

We deal with the time-domain acoustic wave propagation in the presence of a subwavelength resonator given by a Minneart bubble. This bubble is small scaled and enjoys high contrasting mass density and bulk modulus. It is well known that, under certain regimes between these scales, such a bubble generates a single low-frequency (or subwavelength) resonance called Minnaert resonance. In this paper, we study the wave propagation governed by Minnaert resonance effects in time domain. We derive the point-approximation expansion of the wave field. The dominant part is a sum of two terms. 1. The first one, which we call the primary wave, is the wave field generated in the absence of the bubble. 2. The second one, which we call the resonant wave, is generated by the interaction between the bubble and the background. It is related to a Dirac-source, in space, that is modulated, in time, with a coefficient which is a solution of a $1$D Cauchy problem, for a second order differential equation, having as propagation and attenuation parameters the real and the imaginary parts, respectively, of the Minnaert resonance. We show that the evolution of the resonant wave remains valid for a large time of the order $\epsilon^{-1}$, where $\epsilon$ is the radius of the bubble, after which it collapses by exponentially decaying. Precisely, we confirm that such resonant wave have life-time inversely proportional to the imaginary part of the related subwavelength resonances, which is in our case given by the Minnaert one. In addition, the real part of this resonance fixes the period of the wave.