---
title: "Driven Bubble Oscillation: Linear Resonance and Nonlinear Radius Interactive Lab | Sandbox Physics"
description: "When does a gas bubble in a sound field stop behaving like a simple resonator?"
canonical_url: "https://sandboxphysics.com/labs/driven-bubble-oscillation/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/driven-bubble-oscillation/"
last_modified: "2026-10-30"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

**Background** A 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 matters** Bubbles 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 misconception** When the radius swings wildly, the Lab is showing cavitation as seen in a tank or a ship propeller.

**Better mental model** The 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

1. 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.

2. 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.

3. 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.

Physical conclusion

Linear theory gives the natural frequency and the small-signal response. A harder drive breaks the symmetry of the gas spring, shifts the resonance, and creates harmonics and subharmonics in the emitted sound.

## Related experiments and articles

Acoustics and ultrasound · Reading path

### From a pressure wave to trapped beads, echoes and a driven bubble

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a sound-driven bubble moves from linear resonance to nonlinear oscillation**](https://sandboxphysics.com/stories/acoustics-ultrasound/bubble-dynamics/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)