---
title: "Radio Echoes from the Ionosphere Interactive Lab | Sandbox Physics"
description: "Why does a radio echo suggest a higher layer than the height where the wave actually turns? A pulse slows as it enters the plasma. Converting its round-trip delay into a height with the vacuum speed of light therefore places the apparent reflection above the actual turning point."
canonical_url: "https://sandboxphysics.com/labs/ionospheric-radio-sounding/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/ionospheric-radio-sounding/"
last_modified: "2026-10-23"
---

B-X5 · Radio sounding

# Radio Echoes from the Ionosphere

**Why does a radio echo suggest a higher layer than the height where the wave actually turns?** A pulse slows as it enters the plasma. Converting its round-trip delay into a height with the vacuum speed of light therefore places the apparent reflection above the actual turning point.

Send a radio pulse into an electron layer and follow its return or escape. Sweep the frequency, record the echo delays and compare the virtual height with the true turning height.

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

This is an ideal cold, unmagnetized, collisionless plasma with one flat parabolic density layer and fixed ions. Paths and narrow-band group delays follow geometrical optics. Density is illustrative. Earth curvature, polarization splitting, absorption, antenna response, irregularities, multiple layers and hops, finite pulse bandwidth and full-wave cutoff scattering are omitted. The data are simulated forward predictions, with no measured amplitudes or density-profile inversion. They cannot forecast a real radio link.

02 / Measure the delay

## The echo is a clock. Is it also a ruler?

Keep the real turning point beside the height inferred from time. The plasma makes those two answers different.

Peak plasma frequency $\text{—}$

True turning height $\text{—}$

Round-trip delay $\text{—}$

Virtual height or slant range $\text{—}$

Ground landing distance $\text{—}$

### Record a vertical frequency sweep

Send 45 narrow-band test pulses and keep every return or missing echo. Click a recorded ionogram point to launch that frequency again.

### The electron layer behind the echoes

The profile is the same one shown in the 3D cutaway. Gold marks the selected ray’s true turning height.

### What the delay measures

For a vertical sounding, the vacuum light speed turns half the travel time into a virtual height. At an oblique angle that same quantity is a virtual slant range. The true turning height follows the plasma density and the conserved horizontal wave number.

$$
h'=\frac{c\tau}{2}\quad(\theta_0=0),\qquad n^2(z_t)=\sin^2\theta_0
$$

At the exact critical limit this ideal parabolic ray takes indefinitely long to turn. Near a cutoff, real finite-bandwidth pulses need a full wave calculation. This forward simulation does not infer an unknown density profile from field measurements.

### Sources and reading scope

- [NOAA NCEI · Vertical Incidence Soundings](https://www.ngdc.noaa.gov/stp/IONO/ionogram.html)

  Read the short institutional introduction about virtual height and critical-frequency traces. No archived image is used as this simulation’s data.

- [UKSSDC · Basic ionosonde theory](https://www.ukssdc.ac.uk/ionosondes/ionosonde_basics.html)

  Read the introductory density, frequency and group-delay discussion. Magnetic mode splitting is outside this model.

- [R. Fitzpatrick · Waves in unmagnetized plasmas](https://farside.ph.utexas.edu/teaching/315/Waves/node75.html)

  Read equations 9.27–9.35 for the cold-plasma dispersion and group speed. The parabolic-layer solution is derived for the stated profile.

## Read the experimental evidence

The simulation uses the assumptions stated above. Read how the historical apparatus measured the effect, what its records supported, and where the interpretation was limited.

[Radio signals and the ionosphere](https://sandboxphysics.com/stories/probe-evidence/wireless-signals-and-ionosphere/) [All six reading paths](https://sandboxphysics.com/stories/probe-evidence/)

Physics tutorial

## Radio Echoes from the Ionosphere

**Background** Send a radio pulse into an electron layer and follow its return or escape. Sweep the frequency, record the echo delays and compare the virtual height with the true turning height.

**Why it matters** An echo gives a travel time. Learning the height of an atmospheric layer takes a model of how the pulse moves.

Start with the essentials

- **Focus question**: Why does a radio echo suggest a higher layer than the height where the wave actually turns?
- **One-sentence intuition**: The group delay measures a virtual height; the turning point belongs to the density profile and the selected frequency.

### Core mathematical model

#### The plasma changes the wave speed

$$
\omega^2=c^2|\mathbf{k}|^2+\frac{N_e e^2}{\epsilon_0m_e},\qquad v_g=cn<c
$$

This cold unmagnetized plasma law gives the group speed of a narrow-band pulse. The phase speed is different.

#### Horizontal symmetry fixes the ray

$$
p=n\sin\theta=\sin\theta_0,\qquad n^2(z_t)=p^2
$$

Angles are measured from vertical. Horizontal wave number stays fixed; the vertical component reaches zero at the turn.

#### Delay gives a virtual height

$$
h'=\frac{c\tau}{2},\qquad \tau=\frac{2}{c}\int_0^{z_t}\frac{dz}{\sqrt{n^2(z)-p^2}}
$$

The height definition applies to vertical soundings. For oblique paths half the light-speed delay is a virtual slant range.

#### An oblique wave can return at a higher frequency

$$
f_m=\frac{1}{2\pi}\sqrt{\frac{N_m e^2}{\epsilon_0m_e}},\qquad f_{\mathrm{threshold}}=\frac{f_m}{\cos\theta_0}
$$

This secant-law threshold assumes a flat, single layer. Its equality is an infinite-delay limit in the ideal parabolic ray model.

### Common difficulties

#### Virtual height and a mirror

**Typical misconception** The ionogram shows where a physical reflecting surface sits.

**Better mental model** The electron density changes continuously. A pulse bends and slows along its path; its travel time alone does not give the true turning height.

#### A cutoff and a working radio link

**Typical misconception** A returning path guarantees a usable signal at its landing point.

**Better mental model** This model calculates paths and delays. Absorption, antenna patterns, noise and a real receiver determine signal strength and practical availability.

### Run the experiment

1. 01

   #### Start with a vertical echo

   Select “A vertical echo” and press “Launch pulse”. Compare the true turning height with the virtual height.

   **What to observe:** The pulse returns to the same sounder. The virtual height is larger because propagation slows inside the layer.

2. 02

   #### Record an ionogram

   Press “Acquire vertical sweep”. Increase peak density and repeat. Export the CSV to inspect each simulated delay.

   **What to observe:** The return limit moves upward with the square root of peak density. The trace bends to long delays near the limit.

3. 03

   #### Change only the direction

   Compare the two 9 MHz presets. Open the turning-point view after the oblique launch.

   **What to observe:** The vertical ray penetrates; the oblique ray turns and reaches a distant ground point. Its delay is labeled as slant range.

Physical conclusion

A frequency sweep measures echo time. Distinguishing virtual height from true height requires the plasma propagation model.

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

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