---
title: "Plasma Wave Echo Interactive Lab | Sandbox Physics"
description: "Why does a plasma echo appear after two pulses have both faded? The first pulse leaves stripes in the electron velocities that wind tighter as time passes."
canonical_url: "https://sandboxphysics.com/labs/plasma-wave-echo/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/plasma-wave-echo/"
last_modified: "2026-10-29"
---

FR31 · A field that comes back

# Plasma Wave Echo

**Why does a plasma echo appear after two pulses have both faded?** The first pulse leaves stripes in the electron velocities that wind tighter as time passes. The second pulse adds stripes at a different wave number, which unwind part of that winding at a different rate. At one later moment the two windings cancel, the electrons bunch again and a field reappears with no third pulse.

Kick the electrons with two short potential pulses of different wavelengths. Each response fades; later a field reappears at the difference wave number, when the velocity stripes untwist. Change the delay and the pulse pair to predict when.

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

Model assumptions

Electrons move in one dimension through a fixed, uniform ion background, in a periodic box. The program evolves the electron distribution over position and velocity and computes the field from the electron density. Time is measured in inverse plasma frequencies, speed in thermal speeds and length in Debye lengths. Magnetic fields and ion motion are left out. Both pulses are instantaneous standing-wave potentials in a box four pi Debye lengths long. The predicted echo time is the free-streaming result of Gould, O'Neil and Malmberg (1967); the self-consistent field shifts the measured peak by about one percent. Collisions use a Dougherty operator that keeps particle number, momentum and energy. On a coarse velocity grid each response returns at the recurrence time, which can look like an echo.

Predicted echo time $\tau'$ $40\,\omega_p^{-1}$

Measured echo peak $t_{\mathrm{peak}}$ $40.5\,\omega_p^{-1}$

Echo over first response $E_3/E_1$ $0.117$

Earliest grid recurrence $T_R$ $154\,\omega_p^{-1}$

### The echo-mode field

simulated field pulses predicted echo grid recurrence

### Stripes wind, then unwind

measured stripe power free-streaming prediction

Equations, approximations and sources

Electron distribution with fixed ions, in units of the plasma frequency, thermal speed and Debye length:

$$
\frac{\partial f}{\partial t}+v\frac{\partial f}{\partial x}-E\frac{\partial f}{\partial v}=\nu\frac{\partial}{\partial v}\Big[(v-u)f+T\frac{\partial f}{\partial v}\Big],\qquad \frac{\partial E}{\partial x}=1-\int f\,dv
$$

The program splits each step into streaming in position, a kick from the field and, when switched on, a weak-collision step. The comparison rate is the root of the Maxwellian dielectric function:

$$
1-\frac{1}{2k^2}Z'\!\left(\frac{\omega}{\sqrt2\,k}\right)=0
$$

Each pulse is an instantaneous potential kick. Free streaming between the pulses predicts the echo time and mode:

$$
\tau'=\frac{k_2\,\tau}{k_2-k_1},\qquad k_3=k_2-k_1
$$

A velocity grid cannot hold stripes finer than one cell, so the free-streaming field comes back at the recurrence time, where the last factor is the cell width:

$$
T_R=\frac{2\pi}{k\,\Delta v}
$$

This is a numerical model with one space and one velocity dimension. It omits magnetic fields, ion motion, and the experiment geometry of the original echo measurements. Exported records include every parameter, the model version and the full field history.

- [R. Fitzpatrick, Plasma Physics lecture notes: Landau damping and the plasma dispersion function](https://farside.ph.utexas.edu/teaching/plasma/Plasma/node109.html)
- [Gould, O’Neil and Malmberg (1967), Plasma wave echo, Phys. Rev. Lett. 19, 219: Fig. 1 and Eqs. (4)–(5)](https://doi.org/10.1103/PhysRevLett.19.219)
- [Continue through the kinetic plasma course](https://sandboxphysics.com/stories/kinetic-plasma/)

[Read the kinetic plasma course: from phase-space stripes to wave echoes](https://sandboxphysics.com/stories/kinetic-plasma/)

Physics tutorial

## Plasma Wave Echo

**Background** Kick the electrons with two short potential pulses of different wavelengths. Each response fades; later a field reappears at the difference wave number, when the velocity stripes untwist. Change the delay and the pulse pair to predict when.

**Why it matters** The echo shows that Landau damping keeps a memory in the electron velocities. Its size also measures how fast weak collisions erase fine velocity structure.

Start with the essentials

- **Focus question**: How can two pulses that both fade away make a third signal appear later?
- **One-sentence intuition**: Stripes from the two pulses wind at rates set by their wave numbers; the echo appears in the difference wave number when the combined winding returns to zero.

### Core mathematical model

#### Echo time and wave number (Gould, O’Neil and Malmberg 1967)

$$
k_3=k_2-k_1,\qquad \tau'=\frac{k_2\,\tau}{k_2-k_1}
$$

The first pulse at time zero has wave number $k_1$; the second, at $\tau$, has $k_2$. When $k_2=2k_1$, the echo comes at twice the delay.

#### Winding of the second-order stripes

$$
\kappa(t)=k_2\,(t-\tau)-k_1\,t
$$

The stripe wave number in velocity crosses zero exactly at the echo time; the stripe chart draws this line in gold.

#### Weak collisions erase the echo

$$
C[f]=\nu\,\frac{\partial}{\partial v}\Bigl[(v-u)f+T\,\frac{\partial f}{\partial v}\Bigr]
$$

This Dougherty operator keeps particle number, momentum and energy. A stripe of velocity wave number $\kappa$ decays at the rate $\nu T\kappa^{2}$, so fine stripes die first.

### Common difficulties

#### An echo is a reflection

**Typical misconception** The echo is the first wave bouncing back from a wall.

**Better mental model** The box is periodic and has no walls. The field reappears because the velocity stripes realign, and it appears at a new wave number.

#### Model scale

**Typical misconception** The pulse strengths match an experiment.

**Better mental model** Electrons move in one dimension through a fixed, uniform ion background, in a periodic box. The program evolves the electron distribution over position and velocity and computes the field from the electron density. Time is measured in inverse plasma frequencies, speed in thermal speeds and length in Debye lengths. Magnetic fields and ion motion are left out. Both pulses are instantaneous standing-wave potentials in a box four pi Debye lengths long. The predicted echo time is the free-streaming result of Gould, O'Neil and Malmberg (1967); the self-consistent field shifts the measured peak by about one percent. Collisions use a Dougherty operator that keeps particle number, momentum and energy. On a coarse velocity grid each response returns at the recurrence time, which can look like an echo.

### Run the experiment

1. 01

   #### Predict, then watch

   Choose An echo at twice the delay, predict the echo time, then compare with the readouts.

   **What to observe:** The echo-mode field rises out of the noise near the gold marker, and the stripe chart shows the gold line crossing zero there.

2. 02

   #### Change the pulse pair

   Choose An echo at three delays, then Swap the pulses: no echo.

   **What to observe:** With the smaller wave number second, the winding never returns to zero and no echo forms.

3. 03

   #### Tell physics from numerics

   Choose Weak collisions erase it, then A coarse grid fakes a return.

   **What to observe:** Collisions shrink the echo at the predicted time. The coarse grid gives a large peak near the coral recurrence line instead, which disappears on a finer grid.

Physical conclusion

An echo appears only if the second pulse has the larger wave number; its time is fixed by free streaming, its size by the self-consistent field and by collisions.

## Related experiments and articles

Kinetic plasma · Reading path

### Kinetic plasma from phase-space stripes to wave echoes

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Where a fading ripple hides, and how a grid brings it back**](https://sandboxphysics.com/stories/kinetic-plasma/phase-mixing-and-recurrence/)
2. [Lab **Phase-Space Filaments** Watch a ripple fold into velocity stripes](https://sandboxphysics.com/labs/phase-space-filaments/)
3. [Article **Two pulses fade, then a third signal appears**](https://sandboxphysics.com/stories/kinetic-plasma/plasma-wave-echoes/)
4. [Article **How rare collisions erase a plasma’s memory**](https://sandboxphysics.com/stories/kinetic-plasma/weak-collisions-and-irreversibility/)
5. [Article **Reading the 1967 plasma wave echo paper**](https://sandboxphysics.com/stories/kinetic-plasma/reading-the-1967-echo-paper/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

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