---
title: "Acoustic Radiation Force: Trapping Particles in a Standing Wave Interactive Lab | Sandbox Physics"
description: "How does a standing sound wave push particles into lines, and when does the stirred liquid win instead?"
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language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/acoustic-radiation-trap/"
last_modified: "2026-10-30"
---

FR33 · Acoustofluidics

# Acoustic Radiation Force: Trapping Particles in a Standing Wave

**How does a standing sound wave push particles into lines, and when does the stirred liquid win instead?** A standing sound wave pushes small particles toward the planes where its time-averaged energy is lowest for them. Polystyrene beads and most cells are denser and stiffer than water, so they gather at pressure nodes; softer, lighter particles gather at antinodes. The sound also stirs the liquid: at 2 MHz, polystyrene beads below about 1.4 micrometres follow those flow rolls instead of lining up.

A 2 MHz standing wave across a 377 µm water channel herds microbeads into the pressure node. Drag a bead out and watch it return.

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

Model assumptions

Two-dimensional cross-section of a water-filled channel 377 µm wide and 157 µm high, driven at a half-wave resonance across its width. Beads feel the Gor’kov radiation force of an ideal standing wave plus Stokes drag; boundary-driven Rayleigh streaming between parallel plates can be switched on. Particles do not interact. Wall corrections to drag, heating, side-wall streaming, gravity, Brownian motion and the full three-dimensional resonance are omitted. Water and polystyrene follow Barnkob et al. (2012); pyrex and the cell follow Settnes and Bruus (2012); the soft particle is hypothetical. All motion is a model simulation, not measured data.

Radiation force and potential across the channel

Probe return after release

Radiation versus streaming by particle size

More numbers and how they are checked

- **Drive frequency**: $0$
- **Pressure amplitude**: $0$
- **Streaming velocity scale u₀**: $0$
- **Largest radiation drift speed**: $0$
- **Centre-plane streaming speed**: $0$
- **Monopole coefficient f₁**: $0$
- **Dipole coefficient f₂**: $0$
- **Viscous change of the contrast factor**: $0$
- **Viscous boundary layer δ**: $0$
- **Trap stiffness at the stable plane**: $0$
- **Particle inertial time**: $0$
- **Probe release position**: $0$
- **Stable plane the probe returns to**: $0$
- **Largest gap between simulated and exact return**: $0$

White dots are minus the slope of the Gor’kov potential, taken by central differences from the field samples; the line is the closed-form force. The exact return solves the drift equation in closed form and does not use the integrator. On the centre plane, streaming adds a drift with the same sinusoidal shape, so an exact curve exists there too.

Physics tutorial

## A standing wave is a landscape for particles

**Background** A piezo transducer under a microchannel drives the water at about 2 MHz. At this frequency half a wavelength of sound just fits across the 377 µm channel, so a standing wave forms with a pressure node in the middle.

**Why it matters** The sound oscillates two million times a second, yet beads drift steadily over fractions of a second. Something averaged over many cycles must push them, and something else must stir the liquid around them.

Start with the essentials

- **Focus question**: How does a standing sound wave push particles into lines, and when does the stirred liquid win instead?
- **One-sentence intuition**: The time-averaged pressure and velocity of the wave define a potential for each particle. Its slope is the radiation force, and its sign depends on whether the particle is denser and stiffer than water. The force grows with the particle volume, while the streaming drag on a particle does not depend on its size, so small particles follow the streaming rolls instead.

### Core mathematical model

#### Gor’kov potential

$$
U=\frac{4\pi a^3}{3}\left[f_1\frac{\kappa_0}{2}\langle p^2\rangle-f_2\frac{3\rho_0}{4}\langle v^2\rangle\right]
$$

For a sphere much smaller than the wavelength. The brackets are averages over one acoustic cycle of the squared pressure and velocity.

#### Radiation force in a half-wave field

$$
F=-\frac{\partial U}{\partial y}=4\pi\Phi k a^3E_{\mathrm{ac}}\sin(2ky)
$$

Here $y$ is measured from the left wall, where the pressure is largest, and $k$ is the wavenumber.

#### Acoustic contrast factor

$$
\Phi=\frac{f_1}{3}+\frac{f_2}{2},\qquad f_1=1-\frac{\kappa_p}{\kappa_0},\qquad f_2=\frac{2(\rho_p-\rho_0)}{2\rho_p+\rho_0}
$$

A positive contrast factor sends particles to pressure nodes; a negative one sends them to pressure antinodes. Polystyrene in water gives about 0.17. The dipole term is shown without viscosity; the Lab adds the thin viscous layer around each particle, which matters for small dense beads.

#### Drift speed and return time

$$
u_{\mathrm{rad}}=\frac{F}{6\pi\eta a}=\frac{2\Phi k a^2E_{\mathrm{ac}}}{3\eta}\sin(2ky),\qquad \tau=\frac{3\eta}{4\Phi k^2a^2E_{\mathrm{ac}}}
$$

Inertia relaxes in under a microsecond, so the bead moves at the speed where Stokes drag balances the radiation force. Near the node the return is exponential with time constant τ.

#### Where streaming takes over

$$
2a_c=2\delta\sqrt{\frac{3\psi}{\Phi}},\qquad \delta=\sqrt{\frac{2\eta}{\rho_0\omega}},\qquad \psi=\frac{3}{16}
$$

Here $\delta$ is the viscous boundary-layer thickness and $\psi$ the centre-plane streaming coefficient between parallel plates. For polystyrene at 2 MHz the critical diameter is about 1.4 µm.

### Common difficulties

#### Pushed toward the quietest point

**Typical misconception** Sound pushes particles away from where it is loudest, so every particle ends at a pressure node.

**Better mental model** Where a particle ends depends on how it differs from the liquid. A particle softer and lighter than water has a negative contrast factor and gathers at pressure antinodes, the walls of this channel.

### Run the experiment

1. 01

   #### Watch the beads line up

   Start with Beads line up at the node.

   **What to observe:** Within about a second of model time the scattered beads gather into a vertical line at the centre, where the pressure colour stays faint.

2. 02

   #### Pull one bead out

   Drag the orange probe toward a side wall and let go.

   **What to observe:** The probe returns along the exact curve in the return chart: slow near the wall and the node, fastest a quarter of the way across.

3. 03

   #### Shrink the beads

   Choose Tiny beads ride the rolls, or slide the diameter below 1.4 µm.

   **What to observe:** The beads circle in four rolls instead of lining up, because streaming now moves them faster than the radiation force.

Physical conclusion

Radiation force sorts particles by size and material; streaming mixes them back. Acoustic trapping works in the size range where the first wins.

## 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.

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5. [Lab **Drum Modes & Strike Position** Listen for a drum’s overtones](https://sandboxphysics.com/labs/drum-modes/)
6. [Article **How acoustic radiation force pushes beads to the nodes of a standing wave**](https://sandboxphysics.com/stories/acoustics-ultrasound/acoustic-radiation-force/)
7. [Article **How acoustic streaming in a standing wave stirs the liquid**](https://sandboxphysics.com/stories/acoustics-ultrasound/acoustic-streaming/)
8. [Article **Reading the 2012 paper by Barnkob and co-workers that separated radiation force from streaming**](https://sandboxphysics.com/stories/acoustics-ultrasound/reading-barnkob-2012/)

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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/)

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