---
title: "Ultrasound Pulse-Echo Imaging: Building a Picture from Echo Times Interactive Lab | Sandbox Physics"
description: "How does an ultrasound scanner turn echo arrival times into a picture, and what limits its sharpness?"
canonical_url: "https://sandboxphysics.com/labs/ultrasound-pulse-echo/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/ultrasound-pulse-echo/"
last_modified: "2026-10-30"
---

FR33 · Ultrasound

# Ultrasound Pulse-Echo Imaging: Building a Picture from Echo Times

**How does an ultrasound scanner turn echo arrival times into a picture, and what limits its sharpness?** An ultrasound scanner sends a short pulse and records when echoes return to each element of its array. For every pixel it adds the samples that a reflector there would produce, so true echoes add up and others cancel. A shorter pulse sharpens depth; a wider array or a shorter wavelength sharpens the sideways detail.

A 5 MHz plane pulse hits three thin wires and a plastic plate. The echo times on 64 elements become an image. Drag a wire and see its blur.

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

Model assumptions

Two-dimensional water tank imaged by a linear array of point elements. One plane pulse at normal incidence, 12.8 mm wide, is sent; the active elements receive. Wires are ideal point scatterers with cylindrical spreading, and the plate is treated as a fluid layer, so shear waves are omitted. Echoes add linearly: no multiple scattering between targets, no attenuation, no element directivity and no transmit diffraction. The image is a delay-and-sum reconstruction of simulated channel data with seeded receiver noise, not measured data. Water and polystyrene follow Barnkob et al. (2012); pyrex follows Settnes and Bruus (2012).

What the elements recorded

The image made from those echoes

Sideways cut through the selected wire

More numbers and how they are checked

- **Wavelength in water**: $0$
- **Pulse envelope duration at half height**: $0$
- **Depth blur expected from the pulse**: $0$
- **Receive aperture width**: $0$
- **Depth over aperture at the selected wire**: $0$
- **First sideways minimum, simulated pulse**: $0$
- **First sideways zero, single frequency**: $0$
- **First sideways zero, paraxial formula**: $0$
- **Grating-lobe angle for this pitch**: $0$
- **Pressure transmission into the plate**: $0$
- **Reflected plus transmitted energy flux**: $0$
- **Back-face echo amplitude**: $0$
- **Back face in the image**: $0$
- **Back face, true depth**: $0$

The yellow profile comes from the simulated broadband channel data. The dashed green curve sums the elements at one frequency without any time signals, and the white markers are the paraxial first zero. A short pulse averages over many wavelengths, which fills the zeros and lowers the sidelobes. The plate coefficients are checked separately against a one-dimensional wave simulation that never uses the formulas.

Physics tutorial

## An image made of echo times

**Background** A linear array sends one short pulse into water and then listens. Each wire and each plate face sends part of the pulse back, and every element records when the echoes arrive.

**Why it matters** The array never sees the wires directly. It only has a table of pressure against time for each element. Turning that table into a picture, and knowing how sharp the picture can be, is the core of ultrasound imaging.

Start with the essentials

- **Focus question**: How does an ultrasound scanner turn echo arrival times into a picture, and what limits its sharpness?
- **One-sentence intuition**: For every pixel, the scanner works out how long sound would take to go there and come back to each element, picks those samples and adds them. Echoes from that pixel add in step, everything else partly cancels. Depth sharpness is set by the pulse length; sideways sharpness by the wavelength, the depth and the width of the receiving aperture.

### Core mathematical model

#### Delay for one pixel and one element

$$
\tau_j(x,z)=\frac{z+\sqrt{(x-x_j)^2+z^2}}{c}
$$

The plane pulse reaches depth $z$ after $z/c$; the echo then travels back to element $j$ at $x_j$. The image value is the envelope of the weighted sum of all channels at these delays.

#### Reflection and transmission at a flat face

$$
R=\frac{Z_2-Z_1}{Z_2+Z_1},\qquad T=\frac{2Z_2}{Z_2+Z_1},\qquad R^2+\frac{Z_1}{Z_2}T^2=1
$$

Pressure and normal velocity are continuous at the face, and $Z=\rho c$ is the characteristic impedance. The last relation says reflected plus transmitted energy flux equals the incident flux.

#### Depth resolution

$$
\Delta z\approx\frac{c\,\tau_{\mathrm{FWHM}}}{2}
$$

The echo envelope lasts $\tau_{\mathrm{FWHM}}$ at half height; the factor one half comes from the round trip. A wider bandwidth means a shorter pulse.

#### Sideways resolution

$$
x_{\mathrm{null}}\approx\frac{\lambda z}{L}
$$

The first zero beside a point, for a uniformly weighted aperture of width $L$ focused at depth $z$ (Jensen, Eq. 3.10). Its first sidelobe is near $2/(3\pi)$, about −13 dB, for a single frequency.

### Common difficulties

#### Depth comes from time, not from sight

**Typical misconception** The scanner measures where things are, so the plate back face appears at its true depth.

**Better mental model** The image assumes one sound speed. Sound crosses the plastic faster than water, so the back-face echo returns early and the 2 mm plate looks about 1.3 mm thick.

### Run the experiment

1. 01

   #### Follow one pulse

   Start with Three wires and a plate and watch the stage.

   **What to observe:** The flat pulse passes each wire, which sends back a circular echo. The array elements light up as the echoes arrive, and the channel data fill in curved traces.

2. 02

   #### Shorten and lengthen the pulse

   Choose A long pulse smears depth, then slide the bandwidth back up.

   **What to observe:** Each wire stretches along the depth axis when the bandwidth is small; the sideways width hardly changes.

3. 03

   #### Narrow the array

   Choose A narrow array blurs sideways, then drag the middle wire deeper.

   **What to observe:** The wire spreads sideways, and more so the deeper it sits, as the ratio of depth to aperture predicts.

Physical conclusion

A pulse-echo image is a map of round-trip times. Bandwidth sets the depth blur, aperture sets the sideways blur, and an assumed sound speed sets where things appear.

## 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 **Pressure, particle velocity and acoustic impedance of ultrasound in water**](https://sandboxphysics.com/stories/acoustics-ultrasound/pressure-velocity-impedance/)
2. [Lab **Acoustic Radiation Force: Trapping Particles in a Standing Wave** Turn energy density into pressure](https://sandboxphysics.com/labs/acoustic-radiation-trap/)
3. [Article **How much of a sound wave reflects at an interface and how much passes through**](https://sandboxphysics.com/stories/acoustics-ultrasound/reflection-at-an-interface/)
4. [Article **Why a point comes out as a blob when echo times become an image**](https://sandboxphysics.com/stories/acoustics-ultrasound/pulse-echo-imaging/)

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)