FR33 · Bubbles in sound

Driven Bubble Oscillation: Linear Resonance and Nonlinear Radius

When does a gas bubble in a sound field stop behaving like a simple resonator? A gas bubble in water rings like a mass on a spring: the gas is the spring, the water around it the mass. A 3-micrometre air bubble resonates near 1.3 MHz. Pushed gently it follows linear resonance theory. Pushed hard it grows more than it shrinks, its resonance moves to lower frequency, and it sends out harmonics and, near twice its natural frequency, half-frequency sound.

A 3 µm bubble is driven near resonance. Compare its radius with linear theory, sweep the frequency, and hear harmonics and half-frequency sound appear.

Drive frequency over natural frequency $0.865$

Largest radius over rest radius $1.81$

Smallest radius over rest radius $0.46$

Peak sound pressure sent out, 1 mm away $5.09\,\mathrm{kPa}$

Choose the bubble and the sound that drives it

Bubble radius at rest

3

R_0\;(\mu\mathrm{m})

Drive frequency

1.1

f\;(\mathrm{MHz})

Drive pressure amplitude

40

p_a\;(\mathrm{kPa})

Drag the rim of the bubble to change its size; the dashed teal ring is what linear theory predicts.

Gas behaviour

Polytropic exponent of the gas

Model, units and sources

$\mu_E=\mu_L+\frac{\rho\,\omega^2R_0^3}{4c}$

Radiation loss is added as an extra viscosity, evaluated at the drive frequency and the rest radius (Brennen Section 4.4; the online text prints the radius squared, but only the cube has the units of viscosity). Water near 300 K: density 996.3 kg/m³, surface tension 0.0717 N/m, viscosity 0.863 mPa·s, sound speed 1497 m/s; mean pressure minus vapour pressure 100 kPa. Each run settles for at least 40 drive periods before 12 periods are recorded; the sweep uses 56 frequencies.

Brennen, Cavitation and Bubble Dynamics (1995), Eqs. 2.27, 3.10, 4.3–4.8, 4.14 · Barnkob et al. (2012), Table II

Why a driven bubble stops being linear

Model assumptions

One spherical gas bubble in unbounded water, driven by a uniform sinusoidal pressure. The radius follows the Rayleigh–Plesset equation (Brennen 1995, Eq. 2.27) with a polytropic gas, surface tension, liquid viscosity and acoustic radiation loss treated as an extra viscosity (Brennen Section 4.4). Thermal damping, gas diffusion, liquid compressibility beyond that radiation term, shape instability and neighbouring bubbles are left out, so resonance peaks are taller than in real water. This is one bubble, not a cavitation cloud. The radius, sweep and emitted sound are model output, not measurements.

Radius over four drive periods

Frequency sweep at this drive pressure

Spectrum of the sound the bubble sends out

More numbers and how they are checked

Natural frequency, Brennen Eq. 4.8: $0$
Linear peak frequency, Brennen Eq. 4.6: $0$
Linear swing at this drive, Brennen Eq. 4.3: $0$
Liquid viscosity: $0$
Radiation loss as viscosity: $0$
Fastest wall speed over sound speed: $0$
Second harmonic of the emitted sound: $0$
Strongest line below the drive frequency: $0$
Drive periods simulated before recording: $0$

Yellow is always the simulated Rayleigh–Plesset bubble; dashed teal is always linear theory. At a 0.5 kPa drive the simulated swing matches linear theory within about 1 % across the sweep, and the integrator reproduces Rayleigh’s empty-cavity collapse time 0.915 and the energy integral of Brennen Eq. 2.29. Red rings in the sweep mark drives where the wall passes a tenth of the sound speed.

Physics tutorial

A bubble is a spring that does not stay linear

BackgroundA small gas bubble in water behaves like a mass on a spring. The gas inside is the spring; the water that must move when the wall moves is the mass. Sound pushing on the water drives this oscillator.

Why it mattersBubbles ring at a natural frequency set by their size, which is why they scatter ultrasound so strongly. But the gas spring stiffens when squeezed and softens when stretched, so a hard push no longer gives a symmetric swing.

Start with the essentials

Focus question: When does a gas bubble in a sound field stop behaving like a simple resonator?
One-sentence intuition: For small swings the bubble follows linear resonance theory: one peak near the natural frequency, height set by damping. As the drive grows the swing becomes lopsided, the peak moves to lower frequency, the bubble sends out harmonics, and near twice the natural frequency it can answer at half the drive frequency.

Core mathematical model

Rayleigh–Plesset equation

\rho\left(R\ddot R+\tfrac32\dot R^2\right)=p_{G0}\left(\frac{R_0}{R}\right)^{3k}-\left(p_\infty(t)-p_V\right)-\frac{2S}{R}-\frac{4\mu_E\dot R}{R}

Newton's law for the liquid around a spherical wall (Brennen Eq. 2.27). The gas pressure follows a polytropic law with exponent $k$ ; $S$ is surface tension and $\mu_E$ the liquid viscosity plus a radiation term.

Natural frequency

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

Brennen Eq. 4.8. A 3 µm air bubble in water has $f_N\approx1.27\,\mathrm{MHz}$ ; the frequency falls roughly as $1/R_0$ for larger bubbles.

Linear response

\frac{R-R_0}{R_0}=\mathrm{Re}\left\{\varphi e^{i\omega t}\right\},\qquad \varphi=\frac{\tilde p}{\rho R_0^2\left(\omega^2-\omega_N^2-i\omega\,4\nu_E/R_0^2\right)}

Brennen Eq. 4.3 for a drive $p_\infty-\bar p_\infty=\mathrm{Re}\{\tilde p e^{i\omega t}\}$ . It is valid only while $|\varphi|\ll1$ ; the dashed curves in the Lab show it.

Sound sent out by the bubble

p_a(r,t)=\frac{\rho}{4\pi r}\,\ddot V\!\left(t-\frac{r}{c}\right)=\frac{\rho}{r}\left(R^2\ddot R+2R\dot R^2\right)

Brennen Eq. 3.10: far from the bubble the emitted pressure follows the volume acceleration. Fast changes of radius, such as a sharp collapse, make loud short pulses.

Common difficulties

A radius curve is not a cavitation cloud

Typical misconceptionWhen the radius swings wildly, the Lab is showing cavitation as seen in a tank or a ship propeller.

Better mental modelThe Lab follows one spherical bubble in unbounded water. Real cavitation involves many bubbles that interact, split, lose their spherical shape and exchange gas and heat with the water. None of that is in this model.

Run the experiment

01
Start gentle
Choose Gentle drive: linear resonance.
What to observe: The yellow radius curve sits on the dashed linear theory, and the simulated sweep traces the same single peak.
02
Push hard
Choose Strong drive: the peak leans left, then raise the drive pressure.
What to observe: The bubble grows much more than it shrinks, the simulated peak moves below the natural frequency, and the spectrum fills with harmonics.
03
Drive at twice the natural frequency
Choose Twice the resonance: half-frequency sound.
What to observe: Big and small swings alternate, the pattern repeats every two drive periods, and a red line appears at half the drive frequency.

Driven Bubble Oscillation: Linear Resonance and Nonlinear Radius

A bubble is a spring that does not stay linear

Core mathematical model

Rayleigh–Plesset equation

Natural frequency

Linear response

Sound sent out by the bubble

Common difficulties

A radius curve is not a cavitation cloud

Run the experiment

Start gentle

Push hard

Drive at twice the natural frequency

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