L3 · Chapter 7

How a sound-driven bubble moves from linear resonance to nonlinear oscillation

Sandbox Physics · 2026-10-29

A bubble a few micrometres across is a surprisingly good oscillator: the gas inside acts as a spring and the water dragged along around it acts as a mass. Push it gently with sound and it resonates as linear theory says. Push it hard and it expands more than it shrinks, its resonance moves to lower frequency, and it sends out frequencies that were not in the drive. This chapter is about one spherical bubble, not about clouds of cavitation.

A bubble is a spring plus some water

Assume the bubble stays spherical with radius $R(t)$ . When the wall moves, the surrounding water has to flow radially with it; that water’s kinetic energy sits close to the wall and plays the role of the oscillator’s mass. Compressing the gas inside raises its pressure, which is the spring. Surface tension adds an extra pressure and the water’s viscosity adds damping.

Put together, these give the Rayleigh–Plesset equation. The left side is the inertia of the water; the right side holds the gas pressure, the far-field water pressure including the drive, surface tension and viscous stress. The gas follows a polytropic law with exponent $k=1$ for isothermal and $k=1.4$ for adiabatic air.

\rho_L\left(R\ddot R+\frac32\dot R^2\right)=p_{G0}\left(\frac{R_0}{R}\right)^{3k}-\big(p_\infty(t)-p_V\big)-\frac{2S}{R}-\frac{4\mu_E\dot R}{R}

Brennen Eq. 2.27.

p_{G0}=\bar p_\infty-p_V+2S/R_0

is the gas pressure at rest,

p_\infty(t)=\bar p_\infty-p_a\sin\omega t

, and

\mu_E

is an effective viscosity.

Bubble radius over two drive periods, schematic shape. Dashed: linear theory, a sine of amplitude

0.27R_0

. Solid: the lopsided swing of a strongly driven bubble. In the experiment’s default drive the simulated radius runs from

0.46R_0

1.81R_0

Source: C. E. Brennen · R. Barnkob

Small amplitudes: natural frequency and damping

Write $R=R_0(1+\varepsilon)$ and keep only first-order terms in $\varepsilon$ : the equation becomes a driven, damped harmonic oscillator. The natural frequency comes from the gas stiffness and surface tension together; without surface tension it falls back to the Minnaert frequency.

The damping has two parts: the viscosity of the water, and the energy the bubble radiates away as sound. In the linear equation the second acts like an “acoustic viscosity” $\mu_A$ . Brennen’s online Eq. 4.14 prints it with $R_E^2$ , which has the wrong units; units and the radiated power both call for $R_0^3$ . The experiment uses that form and checks it independently against the power of the radiated wave of Eq. 3.10; the ratio is 1.006.

Brennen notes that for most bubbles of practical interest thermal damping is the largest damping. The experiment leaves thermal damping out, so its resonance peaks are higher and sharper than a real bubble’s.

\omega_N^2=\frac{1}{\rho_LR_0^2}\left[3k(\bar p_\infty-p_V)+2(3k-1)\frac{S}{R_0}\right]

Brennen Eq. 4.8.

\ddot\varepsilon+\frac{4\nu_E}{R_0^2}\dot\varepsilon+\omega_N^2\varepsilon=\frac{p_a\sin\omega t}{\rho_LR_0^2},\qquad \mu_E=\mu_L+\frac{\rho_L\omega^2R_0^3}{4c_L}

The linearized form of Brennen Sections 4.2 and 4.4, with

\nu_E=\mu_E/\rho_L

. The

R_0^3

in the radiation term is the unit-corrected form.

Source: C. E. Brennen · R. Barnkob

Strong drive: lopsided swings, a leaning peak and new tones

The gas pressure goes as $R^{-3k}$ : it stiffens sharply under compression and softens under expansion. A strongly driven bubble therefore expands a lot and compresses little, and its radius curve is lopsided.

On average the spring has gone soft, so the resonance moves down as the drive grows and the peak leans towards low frequency. The radius curve is no longer a pure sine, and through Eq. 3.10 the wall acceleration becomes the radiated pressure, so the emitted sound contains whole multiples of the drive frequency (harmonics). Driven near twice the natural frequency, the response repeats only every two periods and the sound contains half the drive frequency (a subharmonic). These are the kinds of behaviour Brennen summarizes from the literature in Section 4.5.

All of this is the behaviour of one isolated, spherical bubble. Real cavitation usually involves many bubbles together, which interact through the sound field and the flow, deform, break up, and take up or release gas. A single bubble’s radius curve is not a cavitation cloud, and the experiment does not predict cavitation thresholds, erosion or any clinical effect. Clouds are a research field of their own: Maeda and Maxwell, for example, estimate the coherent dynamics of a whole cloud from a model and use it to adjust the pulse repetition frequency of pulsed ultrasound in a feedback loop. This course read only their abstract.

p_a(r,t)=\frac{\rho_L}{4\pi r}\frac{d^2V}{dt^2}=\frac{\rho_L}{r}\left(R^2\ddot R+2R\dot R^2\right)

Brennen Eq. 3.10, with

V

the bubble volume; the right side is evaluated at the retarded time

t-r/c_L

Source: C. E. Brennen · K. Maeda and A. D. Maxwell

Worked example: a 3 µm air bubble

Use Brennen’s data for water at 300 K, $\rho_L=996.3\,\mathrm{kg/m^3}$ , $S=0.0717\,\mathrm{N/m}$ , $\mu_L=0.863\,\mathrm{mPa\,s}$ , assume $\bar p_\infty-p_V=100\,\mathrm{kPa}$ and take $k=1.4$ . The radiation term uses the sound speed $c_L=1497\,\mathrm{m/s}$ from Table II of Barnkob et al. Surface tension puts the gas 47.8 kPa above the outside pressure: $p_{G0}=147.8\,\mathrm{kPa}$ .

From Eq. 4.8, $f_N=1.272\,\mathrm{MHz}$ ; without surface tension it would be 1.089 MHz, 14 % lower. At the natural frequency $\mu_A=0.287\,\mathrm{mPa\,s}$ , about a third of the water viscosity, and the quality factor is $Q=\omega_NR_0^2/(4\nu_E)\approx15.6$ .

Gentle drive, 1.25 MHz at 2 kPa: linear theory gives $|\varepsilon|=0.0488$ . Numerical integration puts the radius between $1-0.0490$ and $1+0.0502$ times $R_0$ , already slightly lopsided. Cut the drive to 0.5 kPa and sweep 28 frequencies, and simulation and linear theory agree within 1.06 % everywhere.

Strong drive, 1.1 MHz ( $0.865\,\omega_N$ ) at 40 kPa: linear theory predicts an amplitude of $0.271R_0$ , but the simulated radius swings between $0.460R_0$ and $1.81R_0$ . In a frequency sweep the largest expansion comes at about $0.83\,\omega_N$ , while the linear curve peaks at the natural frequency.

For scale: an empty cavity of the same size collapsing under 100 kPa takes Rayleigh’s time $0.915R_0\sqrt{\rho_L/\Delta p}=0.274\,\mu\mathrm{s}$ , about a third of the 0.786 µs period at 1.272 MHz. The experiment’s integrator reproduces the coefficient 0.9146814.

Source: C. E. Brennen · R. Barnkob

Try it in the Lab

Walk from linear resonance into nonlinearity

Choose “Gentle drive: linear resonance” and see the yellow simulated radius sit almost on the dashed teal linear theory.
Return to the default “Strong drive: the peak leans left”, find the peak in the frequency sweep and compare it with the linear curve.
Choose “Half the resonance rings an octave up” and then “Twice the resonance: half-frequency sound”, and find the $2f$ line and the $f/2$ line in the spectrum.
Drag the bubble wall to set the rest radius to 10 µm and read how the ratio of drive frequency to natural frequency changes.

Exercises with explained answers

What is the natural frequency of a 10 µm air bubble ( $k=1.4$ )? Does surface tension matter more or less than at 3 µm?
Show the answer
Eq. 4.8 gives $f_N=0.344\,\mathrm{MHz}$ ; without surface tension it is 0.327 MHz, so surface tension raises it by only 5 %, against 17 % at 3 µm. The surface-tension term scales with $1/R_0$ , so it matters more for smaller bubbles.
Why does a strongly driven bubble expand more than it compresses?
Show the answer
The gas pressure goes as $(R_0/R)^{3k}$ . Under compression it grows much faster than linearly, like a spring that stiffens the more you squeeze it; under expansion the restoring force weakens. The same drive therefore carries the wall further outwards.
Someone calls the experiment’s radius curve “the dynamics of a cavitation cloud”. What is wrong with that?
Show the answer
The experiment has one bubble that stays spherical in an unbounded liquid. It has no interaction between bubbles, no shape instability or break-up, no gas diffusion and no thermal effects, which are central to cavitation in clouds. It shows how one bubble oscillates under a given drive; a cloud adds bubble–bubble interaction, which this model leaves out.

Sources and what we recomputed

R. Barnkob, P. Augustsson, T. Laurell and H. Bruus, Acoustic radiation- and streaming-induced microparticle velocities determined by micro-PIV in an ultrasound symmetry plane, arXiv:1208.6534v1; Physical Review E 86, 056307 (2012)
Version 1, 31 August 2012, read in full. Eq. 1 (standing-wave field), Eq. 2 (radiation force), Eqs. 5–7 (streaming scale and Rayleigh streaming), Eqs. 9–10 (geometry and thermoviscous corrections), Eqs. 12–14 (velocity amplitude, velocity ratio, critical diameter), Eqs. 16–18 and Table I (wall drag), Table II (material data), Tables IV–V and Figs. 8–9 (measured energy densities, streaming coefficients and velocity ratios). We recompute the theory values; the measured data are quoted, not reproduced.
C. E. Brennen, Cavitation and Bubble Dynamics, Oxford University Press (1995), online edition
Chapter 2: Eq. 2.27 (Rayleigh–Plesset equation), Eq. 2.29 (energy integral), Eq. 2.40 (Rayleigh collapse time). Chapter 3, Section 3.8: Eq. 3.10 (radiated pressure). Chapter 4: Eqs. 4.1–4.8 (linear response and natural frequency), Section 4.4 and Eqs. 4.13–4.14 (damping; the printed radius squared in Eq. 4.14 is corrected to a cube by units), Section 4.5 (nonlinear effects, text only), Section 4.2 (water properties at 300 K). Figures after Lauterborn and Flynn were viewed only; no values were read from them.
K. Maeda and A. D. Maxwell, Controlling the dynamics of cloud cavitation bubbles through acoustic feedback, arXiv:2003.05044v1; Physical Review Applied 15, 034033 (2021)
Abstract only (version 1, 11 March 2020): model-based estimation of coherent cloud dynamics used to set the pulse repetition frequency of pulsed ultrasound in a feedback loop. Cited as an example of cloud-cavitation research; no equation or number is taken from it.