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Uniform Space and Time Behavior for Acoustic Resonators

Long Li, Mourad Sini

TL;DR

The paper analyzes time-domain acoustic waves in the presence of a small, high-contrast Minnaert bubble and derives a uniform-in-space, time-domain asymptotic expansion of the total field. By a scaled Lippmann-Schwinger formulation and projection onto Neumann-Poincaré eigenspaces, the authors isolate a resonant contribution driven by a time-modulated Dirac source at the bubble center, whose modulation obeys a 1D second-order ODE with complex poles. The leading resonant term is governed by the Minnaert frequency $\omega_M$, while the life-time scales like ${\mathcal O}(\varepsilon^{-1})$ and the period by $\omega_M$, with an exponentially decaying remainder. This work advances understanding of Minnaert resonances in the time domain and provides a framework for bubble-based imaging and sensing in heterogeneous media.

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 $ε^{-1}$, where $ε$ 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.

Uniform Space and Time Behavior for Acoustic Resonators

TL;DR

The paper analyzes time-domain acoustic waves in the presence of a small, high-contrast Minnaert bubble and derives a uniform-in-space, time-domain asymptotic expansion of the total field. By a scaled Lippmann-Schwinger formulation and projection onto Neumann-Poincaré eigenspaces, the authors isolate a resonant contribution driven by a time-modulated Dirac source at the bubble center, whose modulation obeys a 1D second-order ODE with complex poles. The leading resonant term is governed by the Minnaert frequency , while the life-time scales like and the period by , with an exponentially decaying remainder. This work advances understanding of Minnaert resonances in the time domain and provides a framework for bubble-based imaging and sensing in heterogeneous media.

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 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 , where 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.

Paper Structure

This paper contains 14 sections, 11 theorems, 194 equations.

Table of Contents

  1. Introduction and Statement of the Main Results
  2. Proof of Theorem \ref{['th:1']}
  3. Outline of the proof of Theorem \ref{['th:1']}
  4. Functional spaces and auxiliary operators
  5. Scaled Lippmann-Schwinger equation
  6. A priori and pullback estimates
  7. Asymptotic expansion of the wave fields
  8. Proofs of Lemmas \ref{['le:0']}, \ref{['le:1']}, \ref{['le:3']} and \ref{['le:9']}
  9. Proof of Lemma \ref{['le:0']}
  10. Proof of Lemma \ref{['le:1']}
  11. Proof of Lemmas \ref{['le:3']} and \ref{['le:9']}
  12. Solutions of the system of ordinary differential equations
  13. Further auxiliary estimates
  14. Proof of inequality (\ref{['eq:32']})

Key Result

Theorem 1.1

Let $\varepsilon, T > 0$. Suppose that $f\in {\color{HW}W_1^{3,2}}(\mathbb R_+; L_{\alpha}^2(\mathbb R^3)) \cap H_0^{16}\left(\mathbb R_+; L_{\alpha}^2(\mathbb R^3)\right)$ with $\alpha > 3/2$. Then, we have the asymptotic expansion where $z^{\pm}_M(\varepsilon):= \pm \omega_M - i\varepsilon {\color{HW}\mathrm{cap}(\Omega)} \omega^2_M/({8\pi c_0})$ and $\textrm{Res}(x,t)$ satisfies as $\varepsi

Theorems & Definitions (27)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 17 more