L1 · Chapter 1

Pressure, particle velocity and acoustic impedance of ultrasound in water

Sandbox Physics · 2026-10-29

A 5 MHz ultrasound wave crosses water at about 1500 metres per second, yet the water itself barely moves: each small parcel shifts back and forth by a few nanometres. What travels is a pair of small disturbances, one in pressure and one in velocity. This chapter writes that pair down and introduces the number the rest of the course keeps using: the acoustic impedance.

Two small disturbances that travel together

Still water has pressure $p_0$ and density $\rho_0$ . When a sound wave passes, the pressure gains a small extra part $p_1$ and each parcel of water gains a small velocity $\mathbf v_1$ . While both stay small, products of them can be dropped, and Newton’s second law and mass conservation become two linear equations.

The first says a pressure difference accelerates the water. The second says that where water is squeezed in, the pressure rises. The constant in between is the compressibility $\kappa_0$ : the less compressible the liquid, the more the pressure rises for the same squeeze.

Eliminating the velocity gives the wave equation. The sound speed depends only on density and compressibility. For water at 25 °C we use $\rho_0=997\,\mathrm{kg/m^3}$ and $c_0=1497\,\mathrm{m/s}$ , so $\kappa_0=448\,\mathrm{TPa^{-1}}$ .

\rho_0\frac{\partial \mathbf v_1}{\partial t}=-\nabla p_1,\qquad \kappa_0\frac{\partial p_1}{\partial t}=-\nabla\cdot\mathbf v_1

The left equation is the linearized equation of motion. The right one combines continuity with

p_1=c_0^2\rho_1

\nabla^2 p_1=\frac{1}{c_0^2}\frac{\partial^2 p_1}{\partial t^2},\qquad c_0=\frac{1}{\sqrt{\rho_0\kappa_0}}

Differentiate once more in time and eliminate the velocity to get the wave equation for the pressure.

Source: J. A. Jensen · M. Settnes and H. Bruus

Impedance: how hard the medium pushes back

Take a plane wave travelling along $x$ , $p_1=p_a\cos(kx-\omega t)$ . Put it into the equation of motion and the particle velocity comes out in phase with the pressure, with a fixed ratio between them.

That ratio is the characteristic impedance $Z=\rho_0c_0$ , measured in rayl ( $\mathrm{Pa\,s/m}$ ). Water has about 1.49 MRayl, polystyrene about 2.47 MRayl and Pyrex glass about 12.7 MRayl. The same pressure drives a smaller velocity in a material of higher impedance.

Impedance also decides how a wave splits where two materials meet, which is the subject of the next chapter. Ultrasound images show tissue boundaries because the impedance differs on the two sides.

v_1=\frac{p_1}{Z},\qquad Z=\rho_0c_0

This holds only for a plane wave travelling one way. A wave travelling the other way has

v_1=-p_1/Z

; once two waves add into a standing wave, pressure and velocity swing a quarter period apart.

Source: J. A. Jensen · R. Barnkob · M. Settnes and H. Bruus

How much energy the wave carries

Pressure times velocity is power per unit area. Averaged over time for a plane wave it gives the intensity $I$ . Dividing the intensity by the sound speed gives the average acoustic energy stored per unit volume.

A standing wave is different. Two opposite waves add to $p_1=p_a\cos(ky)\cos(\omega t)$ , and the energy sloshes back and forth in place with no net flow. Averaged over the whole standing wave, the energy density is $E_{\mathrm{ac}}=\kappa_0p_a^2/4$ , half that of a travelling wave of the same amplitude. Chapters 5 and 6 use this quantity to state how strong the field is.

I=\langle p_1v_1\rangle=\frac{p_a^2}{2Z},\qquad \frac{I}{c_0}=\frac{\kappa_0p_a^2}{2},\qquad E_{\mathrm{ac}}=\frac{\kappa_0p_a^2}{4}

Angle brackets average over one period. The last expression is the space-averaged energy density of a standing wave whose peak pressure is

p_a

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.

Source: R. Barnkob · J. A. Jensen

Worked example: 5 MHz, 100 kPa ultrasound in water

First the wavelength: $\lambda=c_0/f=1497/(5\times10^6)\,\mathrm{m}=0.299\,\mathrm{mm}$ . The impedance of water is $Z=997\times1497\,\mathrm{Rayl}=1.49\,\mathrm{MRayl}$ .

Take a pressure amplitude $p_a=100\,\mathrm{kPa}$ , about one atmosphere. The particle velocity amplitude is $v_a=p_a/Z=0.0670\,\mathrm{m/s}$ , only $4.5\times10^{-5}$ of the sound speed. Divide by the angular frequency for the displacement: $\xi_a=v_a/\omega=2.13\,\mathrm{nm}$ . The wave advances fifteen hundred metres each second while the water moves a few nanometres, which is why the linear equations work.

The intensity is $I=p_a^2/(2Z)=3.35\times10^3\,\mathrm{W/m^2}$ , or 0.335 W per square centimetre, and the travelling-wave energy density is $I/c_0=2.24\,\mathrm{J/m^3}$ . A standing wave with the same peak pressure holds $1.12\,\mathrm{J/m^3}$ on average. The channel experiment of Chapter 5 uses $E_{\mathrm{ac}}=20\,\mathrm{J/m^3}$ , which means a standing-wave peak pressure of about 423 kPa.

Source: R. Barnkob · J. A. Jensen

Try it in the Lab

Watch a pulse cross the tank at the speed of sound

Open the “Three wires and a plate” preset and watch the pulse leave the array, hit the wires and the plate, and come back.
The middle wire is 17 mm deep. Predict when its echo arrives: $2\times17\,\mathrm{mm}/1497\,\mathrm{m/s}$ . Then find that echo in the “What the elements recorded” chart and check the time axis.
Switch the plate to Pyrex glass and see how much brighter the plate echo gets. The next chapter explains why.

Turn energy density into pressure

In the acoustic trap, set the energy density to $5\,\mathrm{J/m^3}$ and then $80\,\mathrm{J/m^3}$ . Use $p_a=\sqrt{4E_{\mathrm{ac}}/\kappa_0}$ to find the standing-wave peak pressure in each case.
Compare how long the beads take to reach the node. When the energy density grows sixteen times, how does the return time change?

Exercises with explained answers

The same $100\,\mathrm{kPa}$ plane wave travels in water and in Pyrex glass. What is the particle velocity amplitude in each?
Show the answer
In water $v_a=10^5/(1.49\times10^6)=0.0670\,\mathrm{m/s}$ . Pyrex has $Z=2230\times5674\,\mathrm{Rayl}=12.7\,\mathrm{MRayl}$ , so the velocity is only $0.0079\,\mathrm{m/s}$ , about 8.5 times smaller.
If the pressure amplitude doubles, by what factor do the intensity and the standing-wave energy density change?
Show the answer
Both scale with $p_a^2$ , so both grow four times. The particle velocity and displacement scale with $p_a$ and only double.
Why is a $100\,\mathrm{kPa}$ wave in water still a “small-amplitude” wave? Give a dimensionless number.
Show the answer
Compare the particle velocity with the sound speed: $v_a/c_0=p_a/(\rho_0c_0^2)=4.5\times10^{-5}$ . The neglected nonlinear terms are of that relative size, so the linear equations are accurate here. Nonlinear effects grow with amplitude and with distance travelled; the models in this course leave that build-up 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.
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.