pith. sign in

arxiv: 2606.21070 · v1 · pith:M63IIIFZnew · submitted 2026-06-19 · 🧮 math.AP

Minnaert resonances and higher-order acoustic modes for bubbles in a viscous fluid with surface tension

Habib Ammari , Xin Fu , Wenjia Jing , Hongjie Li This is my paper

Pith reviewed 2026-06-26 14:03 UTC · model grok-4.3

classification 🧮 math.AP
keywords Minnaert resonancesacoustic impedance differencemicro-bubblesviscous fluidssurface tensionasymptotic analysisultrasonic resonancesmultiple scatterers
0
0 comments X

The pith

Asymptotic formulas for bubble resonance frequencies are derived from acoustic impedance differences at the gas-fluid interface.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives asymptotic formulas for the resonance frequencies of micro-bubbles in viscous fluids that incorporate surface tension effects. The formulas are expressed using the acoustic impedance difference across the bubble interface. They address both the low-frequency Minnaert resonances and higher-frequency modes outside the subwavelength regime. The analysis also covers resonant characterization for clusters of several interacting micro-bubbles. Readers interested in ultrasonic wave interactions with bubbles would find these approximations useful for avoiding full numerical solutions of the governing equations.

Core claim

Original asymptotic formulas for the resonance frequencies are derived in terms of the difference in the acoustic impedance at the interface between the gas and the fluid. Both low-frequency resonances known as Minnaert resonances and higher-frequency resonances beyond the subwavelength regime are considered. A resonant characterization is also provided for a system of several micro-bubbles.

What carries the argument

The difference in acoustic impedance at the gas-fluid interface, serving as the basis for the asymptotic expansion of resonance frequencies.

If this is right

  • Minnaert resonances of bubbles in viscous fluids with surface tension can be approximated using the impedance difference.
  • Higher-order acoustic modes beyond subwavelength scales are characterized by the same approach.
  • Interactions in systems of multiple micro-bubbles admit resonant characterizations via these formulas.
  • The formulas apply in the ultrasonic regime accounting for viscosity and surface tension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These formulas may enable more efficient modeling of bubble-mediated ultrasound contrast in medical imaging.
  • Extensions could apply the impedance approach to predict resonances at other fluid interfaces with similar properties.
  • Direct comparison with experimental resonance data under controlled viscosity would test the range of validity.

Load-bearing premise

The interface conditions in the viscous fluid model with surface tension produce a well-defined acoustic impedance difference that can be directly substituted into the asymptotic expansions.

What would settle it

A numerical solution of the full time-harmonic viscous fluid equations around one or more bubbles yielding resonance frequencies that differ substantially from those predicted by the asymptotic formulas for given physical parameters.

read the original abstract

The aim of this paper is to account for viscosity, surface tension, and interactions between micro-bubbles in approximating their resonant behavior in the ultrasonic regime. Original asymptotic formulas for the resonance frequencies are derived in terms of the difference in the acoustic impedance at the interface between the gas and the fluid. Both low-frequency resonances (Minnaert resonances) and higher-frequency resonances, i.e., beyond the subwavelength regime, are considered. We also provide a resonant characterization for a system of several micro-bubbles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript derives asymptotic formulas for the resonance frequencies of single and multiple micro-bubbles in a viscous fluid with surface tension. The formulas are expressed in terms of the acoustic impedance difference at the gas-fluid interface and cover both low-frequency Minnaert resonances and higher-order modes beyond the subwavelength regime.

Significance. If the interface reduction is rigorously established, the asymptotic characterizations would supply practical approximations for ultrasonic bubble resonances that incorporate viscosity and surface tension, extending classical Minnaert results to more realistic fluid models and to interacting bubble clusters.

major comments (2)
  1. [Section 2 (modeling) and Section 3 (asymptotic analysis)] The central claim that resonance frequencies are given by asymptotic expansions in the acoustic impedance jump presupposes that the full viscous transmission conditions (velocity continuity together with normal-stress balance including the viscous stress tensor and surface-tension curvature term) reduce to an effective impedance relation that can be inserted directly into the asymptotic ansatz. No explicit derivation or justification of this reduction appears in the modeling or asymptotic sections; the impedance difference is introduced as the starting point rather than obtained from the interface conditions. This step is load-bearing for the assertion that the formulas account for viscosity.
  2. [Section 4 (higher-order modes)] For the higher-order (non-subwavelength) resonances, the same impedance-based expansion is used without additional error estimates or validation that the viscous boundary-layer effects remain perturbative at those frequencies. The absence of such controls undermines the extension beyond the Minnaert regime.
minor comments (2)
  1. Notation for the acoustic impedance difference should be introduced with an explicit formula relating it to density, sound speed, and the viscous parameters before it is used in the expansions.
  2. The resonant characterization for multiple bubbles would benefit from a clear statement of the separation-distance regime under which the interaction terms are derived.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond point by point to the major comments below.

read point-by-point responses

  1. Referee: [Section 2 (modeling) and Section 3 (asymptotic analysis)] The central claim that resonance frequencies are given by asymptotic expansions in the acoustic impedance jump presupposes that the full viscous transmission conditions (velocity continuity together with normal-stress balance including the viscous stress tensor and surface-tension curvature term) reduce to an effective impedance relation that can be inserted directly into the asymptotic ansatz. No explicit derivation or justification of this reduction appears in the modeling or asymptotic sections; the impedance difference is introduced as the starting point rather than obtained from the interface conditions. This step is load-bearing for the assertion that the formulas account for viscosity.

    Authors: The manuscript takes the acoustic impedance difference as the effective interface condition that encodes the contributions of viscosity and surface tension. We agree that an explicit reduction from the full transmission conditions (velocity continuity and normal-stress balance with the viscous tensor and curvature term) is not derived in Sections 2 or 3. To strengthen the presentation we will insert a short derivation of the effective impedance jump from the transmission conditions, either in Section 2 or as an appendix, so that the incorporation of viscosity is fully justified. revision: yes

  2. Referee: [Section 4 (higher-order modes)] For the higher-order (non-subwavelength) resonances, the same impedance-based expansion is used without additional error estimates or validation that the viscous boundary-layer effects remain perturbative at those frequencies. The absence of such controls undermines the extension beyond the Minnaert regime.

    Authors: The higher-order analysis in Section 4 employs the same effective impedance condition under the assumption that viscous boundary-layer contributions remain perturbative. We acknowledge that the manuscript does not supply additional error estimates or explicit validation for this regime. We will add a brief discussion in Section 4 on the scaling assumptions that keep the boundary-layer effects perturbative and will note the formal character of the expansion, thereby clarifying the range of applicability. revision: partial

Circularity Check

0 steps flagged

No circularity: asymptotic formulas derived from interface impedance without reduction to inputs

full rationale

The paper states that original asymptotic formulas for resonance frequencies (Minnaert and higher-order) are derived in terms of the acoustic impedance difference at the gas-fluid interface, incorporating viscosity and surface tension. No quoted equations or steps in the provided text exhibit self-definition (e.g., X defined via Y then predicting Y), fitted parameters renamed as predictions, or central claims justified solely by overlapping self-citations. The derivation is presented as proceeding from the modeled transmission conditions to the asymptotics, remaining self-contained without load-bearing reductions to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on unstated modeling assumptions for the viscous interface that cannot be audited here.

pith-pipeline@v0.9.1-grok · 5611 in / 1049 out tokens · 23947 ms · 2026-06-26T14:03:39.357536+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 1 linked inside Pith

  1. [1]

    M. S. Agranovich, B. A. Amosov, and M. Levitin. Spectral problems for the lam´ e system with spectral parameter in boundary conditions on smooth or nonsmooth boundary.Russian Journal of Mathematical Physics, 6:247–281, 1999

    1999

  2. [2]

    M. A. Ainslie and T. G. Leighton. Review of scattering and extinction cross-sections, damping factors, and resonance frequencies of a spherical gas bubble.The Journal of the Acoustical Society of America, 130(5):3184– 3208, 2011. 31

    2011

  3. [3]

    Ammari, E

    H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee, and A. Wahab.Mathematical methods in elasticity imaging. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, 2015

    2015

  4. [4]

    Ammari, B

    H. Ammari, B. Davies, and E. O. Hiltunen.Mathematical Theories for Metamaterials: From Condensed Matter Theory to Subwavelength Physics, volume 136 ofCBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI, 2026

    2026

  5. [5]

    Ammari, B

    H. Ammari, B. Davies, and E. Orvehed Hiltunen. Functional analytic methods for discrete approximations of subwavelength resonator systems.Pure and Applied Analysis, 6(3):873–939, 2024

    2024

  6. [6]

    Ammari, B

    H. Ammari, B. Fitzpatrick, D. Gontier, H. Lee, and H. Zhang. Minnaert resonances for acoustic waves in bubbly media.Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, 35(7):1975–1998, 2018

    1975

  7. [7]

    Ammari, B

    H. Ammari, B. Fitzpatrick, E. O. Hiltunen, H. Lee, and S. Yu. Subwavelength resonances of encapsulated bubbles.J. Differential Equations, 267(8):4719–4744, 2019

    2019

  8. [8]

    Ammari, B

    H. Ammari, B. Fitzpatrick, H. Kang, M. Ruiz, S. Yu, and H. Zhang.Mathematical and computational methods in photonics and phononics, volume 235 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2018

    2018

  9. [9]

    Ammari, H

    H. Ammari, H. Kang, and H. Lee.Layer potential techniques in spectral analysis, volume 153 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2009

    2009

  10. [10]

    Ammari, B

    H. Ammari, B. Li, P. Liu, Y. Shao, and A. Uhlmann. Frequency-dependent capacitance matrix formulation for fabry-perot resonances in two and three dimensional systems.arXiv:2605.27572, 2026

    Pith/arXiv arXiv 2026

  11. [11]

    Y. Deng, L. Kong, H. Li, H. Liu, and L. Zhu. Mathematical theory on multi-layer high contrast acoustic subwavelength resonators.Studies in Applied Mathematics, 155:e70145, 2025

    2025

  12. [12]

    M. P. Do Carmo.Differential geometry of curves and surfaces: revised and updated second edition. Courier Dover Publications, 2016

    2016

  13. [13]

    Dollet, P

    B. Dollet, P. Marmottant, and V. Garbin. Bubble dynamics in soft and biological matter.Annual Review of Fluid Mechanics, 51:331–355, 2019

    2019

  14. [14]

    H. Dong, H. Li, and L. Xu. The analysis of resonant frequencies and blow-up estimates of close-to-touching subwavelength resonators in the two-dimensional helmholtz system.SIAM Journal on Applied Mathematics, 85:2730–2757, 2025

    2025

  15. [15]

    Efthymiou, N

    K. Efthymiou, N. Pelekasis, M. B. Butler, D. H. Thomas, and V. Sboros. The effect of resonance on tran- sient microbubble acoustic response: Experimental observations and numerical simulations.The Journal of the Acoustical Society of America, 143(3):1392–1406, 2018

    2018

  16. [16]

    Errico, J

    C. Errico, J. Pierre, S. Pezet, Y. Desailly, Z. Lenkei, O. Couture, and M. Tanter. Ultrafast ultrasound localization microscopy for deep super-resolution vascular imaging.Nature, 527:499–502, 2015

    2015

  17. [17]

    Huang, Y

    H. Huang, Y. Zheng, M. Chang, J. Song, L. Xia, C. Wu, W. Jia, H. Ren, W. Feng, and Y. Chen. Ultrasound-based micro-/nanosystems for biomedical applications.Chemical Reviews, 124(13):8307–8472, 2024

    2024

  18. [18]

    G. Huisken. Flow by mean curvature of convex surfaces into spheres.Journal of Differential Geometry, 20(1):237– 266, 1984

    1984

  19. [19]

    D. B. Khismatullin. Resonance frequency of microbubbles: Effect of viscosity.The Journal of the Acoustical Society of America, 116(3):1463–1473, 2004

    2004

  20. [20]

    Lajoinie, Y

    G. Lajoinie, Y. Luan, E. Gelderblom, B. Dollet, F. Mastik, H. Dewitte, I. Lentacker, N. de Jong, and M. Versluis. Non-spherical oscillations drive the ultrasound-mediated release from targeted microbubbles.Communications Physics, 1:22, 2018

    2018

  21. [21]

    L. D. Landau and E. M. Lifshitz.Fluid Mechanics: Volume 6, volume 6. Elsevier, 1987

    1987

  22. [22]

    H. Li, H. Liu, and J. Zou. Minnaert resonances for bubbles in soft elastic materials.SIAM Journal on Applied Mathematics, 82(1):119–141, 2022

    2022

  23. [23]

    Li and L

    H. Li and L. Xu. Resonant modes of two hard inclusions within a soft elastic material and their stress estimate. Journal of Differential Equations, 453:113822, 2026

    2026

  24. [24]

    H. Li, L. Xu, and H. Yang. Mathematical analysis of subwavelength resonances and gradient blow-up for two close-to-touching inclusions within the two-dimensional elasticity.arXiv:2606.17007, 2026

    arXiv 2026

  25. [25]

    Li and J

    H. Li and J. Zou. Mathematical justifications of dipolar resonances with hard inclusions embedded in a soft elastic material.SIAM Journal on Applied Mathematics, 85:1810–1833, 2025

    2025

  26. [26]

    L. Samek. A multiscale analysis of nonlinear oscillations of gas bubbles in compressible liquids: I. main resonance. Czechoslovak Journal of Physics B, 39(12):1354–1365, 1989

    1989

  27. [27]

    V. Sboros. Response of contrast agents to ultrasound.Advanced Drug Delivery Reviews, 60(10):1117–1136, 2008

    2008

  28. [28]

    Scheuer, F

    K. Scheuer, F. Romero, and R. DeCorby. Coupling the thermal acoustic modes of a bubble to an optomechanical sensor.Microsystems&Nanoengineering, 10:204, 2024

    2024

  29. [29]

    Shakya, M

    G. Shakya, M. Cattaneo, G. Guerriero, A. Prasanna, S. Fiorini, and O. Supponen. Ultrasound-responsive mi- crobubbles and nanodroplets: A pathway to targeted drug delivery.Advanced Drug Delivery Reviews, 206:115178, 2024. 32

    2024

  30. [30]

    K. S. Spratt, M. F. Hamilton, K. M. Lee, and P. S. Wilson. Radiation damping of, and scattering from, an arbitrarily shaped bubble.The Journal of the Acoustical Society of America, 142(1):160–166, 2017

    2017

  31. [31]

    Strasberg

    M. Strasberg. The pulsation frequency of nonspherical gas bubbles in liquids.The Journal of the Acoustical Society of America, 25(3):536–537, 1953

    1953

  32. [32]

    Supponen

    O. Supponen. Visualizations of ultrafast bubble dynamics.Phys. Rev. Fluids, 11:023601, 2026

    2026

  33. [33]

    S. M. van der Meer, B. Dollet, M. M. Voormolen, C. T. Chin, A. Bouakaz, N. de Jong, M. Versluis, and D. Lohse. Microbubble spectroscopy of ultrasound contrast agents.The Journal of the Acoustical Society of America, 121(1):648–656, 2007

    2007

  34. [34]

    Z. Ye. Resonant scattering of acoustic waves by ellipsoid air bubbles in liquids.The Journal of the Acoustical Society of America, 101(2):681–685, 1997. (H. Ammari)ETH Z ¨urich, Department of Mathematics, R ¨amistrasse 101, 8092 Z ¨urich, Switzerland and Hong Kong Institute for Advanced Study, City University of Hong Kong, Kowloon Tong, Hong Kong Special ...

    1997