FR33 · An interactive course

Acoustics and ultrasound: radiation force, streaming, echoes and bubbles

Sandbox Physics · 2026-10-29

Ultrasound moves water by only a few nanometres, yet it can hold beads in a line, stir a liquid into rolls, draw a picture of what it hits and make a bubble ring at new frequencies. This course follows one sound field in water through all four, with the formulas worked out and checked against simulations you can run.

Start with pressure and impedance Open an experiment

Half-wave resonance across a channel with hard side walls. Teal: pressure amplitude, largest at the walls and zero at the centre. Amber: velocity amplitude, zero at the walls and largest at the centre. Beads with

\Phi>0

collect on the pressure node. Original schematic.

From a pressure wave to a bead, an image and a bubble

The first chapter writes down what a sound wave in water is made of and introduces the acoustic impedance. The second finds how much of a wave bounces back at a boundary, and the third shuts the wave between two walls to make a standing wave, comparing it with the standing waves on a drum skin and a metal plate. The fourth turns echo times into an image and works out why a point comes out as a blob. The fifth and sixth return to the standing wave: one derives the force that pushes beads to its nodes, the other the slow flow that carries the smallest particles round in rolls. The seventh drives a single gas bubble and follows it from linear resonance into lopsided, nonlinear swings. The last chapter reads the 2012 measurement that separated radiation force from streaming.

You need derivatives, sines and cosines and the resonance curve of a damped oscillator. Partial derivatives, Gaussian pulses and the Rayleigh–Plesset equation are introduced where they first appear. FR33

01 · L1
Pressure, particle velocity and acoustic impedance of ultrasound in water
From the linear acoustic equations, work out how far and how fast 5 MHz ultrasound moves water, and why impedance sets the energy a beam carries.
02 · L1
How much of a sound wave reflects at an interface and how much passes through
From continuity of pressure and velocity, derive the normal-incidence reflection and transmission and check the echoes from plastic and glass by energy flux.
03 · L2
Standing waves on drum skins, on plates and in a liquid-filled channel
Add two opposite sound waves into a standing wave, find a microchannel’s resonance and nodes, and see what a node means on a drum, a plate and in a liquid.
04 · L2
Why a point comes out as a blob when echo times become an image
Turn round-trip times into an image, then use the pulse bandwidth and array aperture to find how much a point blurs and where ghost images come from.
05 · L3
How acoustic radiation force pushes beads to the nodes of a standing wave
Derive the radiation force on a small particle from the Gor’kov potential, find the stable point, and work out force, stiffness and return time for a 5 µm bead.
06 · L3
How acoustic streaming in a standing wave stirs the liquid
From the viscous wall boundary layer, write the Rayleigh streaming field in a microchannel and find the particle size where radiation and streaming tie.
07 · L3
How a sound-driven bubble moves from linear resonance to nonlinear oscillation
Describe one bubble in water with the Rayleigh–Plesset equation, find its natural frequency and damping, and watch strong drives tilt the peak and add tones.
08 · L4
Reading the 2012 paper by Barnkob and co-workers that separated radiation force from streaming
Read the 2012 particle-velocimetry paper by Barnkob and co-workers with Chapters 5 and 6 in hand: what they measured, how they fitted it, what is open.

Experiments along the way

01Trap beads with a standing waveWatch beads drift to the pressure node of a microchannel, drag one out and time its return, and find the size where streaming takes over.02Make an image from echoesSend one pulse into a water tank, record the echoes on a linear array and see how bandwidth, aperture and pitch blur three wires and a plate.03Drive one bubbleShake a micrometre bubble with sound, compare its radius with linear theory and listen for the harmonics and subharmonics it sends out.

All three experiments run models, not measurements. The particle trap uses closed-form fields from Barnkob and co-workers (2012); the imaging tank simulates channel data for point wires and fluid plates; the bubble integrates the Rayleigh–Plesset equation for one spherical bubble. Each shows its independent checks beside the model output, and none makes a clinical or engineering claim.

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.
M. Settnes and H. Bruus, Forces acting on a small particle in an acoustical field in a viscous fluid, arXiv:1110.6037v1; Physical Review E 85, 016327 (2012)
Version 1, 27 October 2011, read in full. Eq. 1 (viscous penetration depth), Eqs. 4–5 (first-order pressure and compressibility), Eqs. 48a–e (Gor’kov potential with the viscous dipole coefficient) and Table I (Pyrex and cell parameters).
J. A. Jensen, Notes for the International Summer School on Advanced Ultrasound Imaging, Technical University of Denmark (21 May 2024)
Chapter 2, Eqs. 2.45–2.51 (linear acoustic equations and wave equation). Chapter 3, Eqs. 3.8–3.11 (line aperture, first zero, first sidelobe), Eqs. 3.15–3.17 (grating lobes) and Eq. 3.24 (F-number). Chapter 5, Eqs. 5.3–5.4 (pulse and round-trip delay). The impedance reflection coefficients are not taken from these notes.
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.