---
title: "Kinetic plasma from phase-space stripes to wave echoes"
description: "Follow electrons in position and velocity: why plasma waves fade without collisions, which electrons take the energy, and how two pulses make an echo."
canonical_url: "https://sandboxphysics.com/stories/kinetic-plasma/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/kinetic-plasma/"
last_modified: "2026-10-29"
---

[Kinetic plasma · A course with experiments](https://sandboxphysics.com/stories/kinetic-plasma/)

FROM ONE RIPPLE TO AN ECHO

# Kinetic plasma from phase-space stripes to wave echoes

A ripple in a hot plasma fades even when no electron ever collides. Its information is not lost: it folds into finer and finer stripes in velocity, and a second pulse can bring it back as an echo. Start with a ringing slab of electrons and end by reading the 1967 paper that predicted the echo.

Sandbox Physics · 2026-10-29

Foundations · Chapters 1–3

## Where a fading wave goes

Build the plasma frequency from a displaced slab, write the Vlasov equation, and watch a ripple fold into velocity stripes that a coarse grid cannot hold.

[Start with a ringing slab →](https://sandboxphysics.com/stories/kinetic-plasma/plasma-oscillations-and-units/)

Damping, echoes, collisions · Chapters 4–8

## Who takes the energy, and how it comes back

Calculate the Landau damping rate, follow the energy to resonant electrons, make an echo with two pulses, and see how rare collisions erase it. Finish with the 1967 paper.

[Start with the damping rate →](https://sandboxphysics.com/stories/kinetic-plasma/landau-damping-rate/)

## Eight chapters, with examples and experiments

1. [01 L1 Why electrons in a plasma ring like a bell What sets the natural frequency of electrons in a plasma? Read and try →](https://sandboxphysics.com/stories/kinetic-plasma/plasma-oscillations-and-units/)
2. [02 L2 Follow every electron at once in phase space How can one equation track billions of electrons without following each one? Read and try →](https://sandboxphysics.com/stories/kinetic-plasma/phase-space-and-the-vlasov-equation/)
3. [03 L2 Where a fading ripple hides, and how a grid brings it back If the density ripple vanishes, where did the information go, and can it come back? Read and try →](https://sandboxphysics.com/stories/kinetic-plasma/phase-mixing-and-recurrence/)
4. [04 L3 Calculating how fast a plasma wave fades How fast does a small plasma wave fade, and what sets the rate? Read and try →](https://sandboxphysics.com/stories/kinetic-plasma/landau-damping-rate/)
5. [05 L3 Which electrons take the wave’s energy, and when the wave holds on When a plasma wave fades, which electrons take the energy, and when does the wave catch them instead? Read and try →](https://sandboxphysics.com/stories/kinetic-plasma/resonant-electrons-and-trapping/)
6. [06 L3 Two pulses fade, then a third signal appears How can a plasma produce a signal after both waves that caused it have disappeared? Read and try →](https://sandboxphysics.com/stories/kinetic-plasma/plasma-wave-echoes/)
7. [07 L3 How rare collisions erase a plasma’s memory If a collisionless plasma remembers everything, what makes the memory fade? Read and try →](https://sandboxphysics.com/stories/kinetic-plasma/weak-collisions-and-irreversibility/)
8. [08 L4 Reading the 1967 plasma wave echo paper What exactly did the 1967 echo paper predict, and which parts can a one-dimensional simulation check? Read and try →](https://sandboxphysics.com/stories/kinetic-plasma/reading-the-1967-echo-paper/)

## Keep the experiments beside the chapters

All three Labs run the same electron model with different starting conditions. The ionosphere Lab from the waves collection shows the plasma frequency at work in the sky.

[Watch a ripple fold into velocity stripes ↗](https://sandboxphysics.com/labs/phase-space-filaments/) [Follow the energy to resonant electrons ↗](https://sandboxphysics.com/labs/landau-damping-trapping/) [Make an echo with two pulses ↗](https://sandboxphysics.com/labs/plasma-wave-echo/) [See the plasma frequency reflect radio waves ↗](https://sandboxphysics.com/labs/ionospheric-radio-sounding/)