---
title: "Landau Damping and Trapping Interactive Lab | Sandbox Physics"
description: "When a plasma wave fades, which electrons take its energy? Electrons moving at almost the wave speed take it."
canonical_url: "https://sandboxphysics.com/labs/landau-damping-trapping/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/landau-damping-trapping/"
last_modified: "2026-10-29"
---

FR31 · Which electrons take the energy?

# Landau Damping and Trapping

**When a plasma wave fades, which electrons take its energy?** Electrons moving at almost the wave speed take it. The wave pushes slightly slower ones forward and holds slightly faster ones back. In a Maxwellian the slower ones outnumber the faster ones, so these electrons gain energy and the wave loses it. A strong wave traps them instead, and they bounce inside its troughs.

Launch a weak or strong wave and see where the kinetic energy goes. A weak wave hands it to electrons near the wave speed; a strong wave traps them in rotating eddies. A fast bump or two beams make the wave grow instead.

> 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. The box holds one wavelength. A standing wave there is two traveling waves, each with half the field amplitude; the trapping band and bounce time use that half amplitude at the largest recorded field and are estimates. The growth or damping rate printed for comparison is the leading root of the dielectric function, computed separately.

Measured early rate $\gamma_{\mathrm{fit}}$ $-0.302\,\omega_p$

Dielectric-root rate $\gamma_{\mathrm{root}}$ $-0.153\,\omega_p$

Bounce time at peak field $\tau_b$ $16.2\,\omega_p^{-1}$

Trapping half-width $\Delta v_{\mathrm{trap}}$ $1.55\,v_{\mathrm{th}}$

### Field strength over time

simulated field dielectric-root rate (reference)

### Where the kinetic energy went

energy gained energy lost wave speed

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:

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

Trapping estimates use the field of one traveling half of the standing wave at the largest recorded amplitude:

$$
\tau_b=\frac{2\pi}{\sqrt{kE_w}},\qquad \Delta v_{\mathrm{trap}}=2\sqrt{E_w/k}
$$

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)
- [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

## Landau Damping and Trapping

**Background** Launch a weak or strong wave and see where the kinetic energy goes. A weak wave hands it to electrons near the wave speed; a strong wave traps them in rotating eddies. A fast bump or two beams make the wave grow instead.

**Why it matters** Linear Landau damping says how fast a weak wave dies. A strong wave, a beam or a bump changes the answer, and those cases explain how beams heat plasmas and how waves saturate.

Start with the essentials

- **Focus question**: When a plasma wave fades, which electrons take its energy, and when does the wave catch them?
- **One-sentence intuition**: The sign of the slope of the velocity distribution at the wave speed decides whether resonant electrons absorb energy from the wave or feed it.

### Core mathematical model

#### Weak-wave rate near the Bohm–Gross frequency

$$
\gamma\approx\frac{\pi}{2}\,\frac{\omega_p^{3}}{k^{2}}\,\left.\frac{\partial f_0}{\partial v}\right|_{v=\omega/k}
$$

A negative slope at the wave speed gives damping; a positive slope, as on the rising side of a bump, gives growth. The Lab uses the full dielectric root, not this approximation.

#### Bounce time of a trapped electron

$$
\tau_b=\frac{2\pi}{\sqrt{kE_{\text{wave}}}},\qquad \Delta v_{\text{trap}}=2\sqrt{\frac{E_{\text{wave}}}{k}}
$$

An electron within the trapping half-width of the wave speed oscillates in a wave trough. Linear damping holds while the bounce time is much longer than the damping time.

#### Kinetic energy gained at each velocity

$$
\Delta K(v)=\frac{v^{2}}{2}\,L\,\Bigl[\langle f\rangle(v,t)-\langle f\rangle(v,0)\Bigr]\,\Delta v
$$

The angle brackets mean an average over the box. Summed over velocity, this equals the field energy the wave has lost.

### Common difficulties

#### Every electron is resonant

**Typical misconception** All electrons trade energy with the wave equally.

**Better mental model** Most electrons slosh back and forth with the wave and return what they take. Only those near the wave speed keep a net gain or loss.

#### Model scale

**Typical misconception** The trapping band is measured from the run.

**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. The box holds one wavelength. A standing wave there is two traveling waves, each with half the field amplitude; the trapping band and bounce time use that half amplitude at the largest recorded field and are estimates. The growth or damping rate printed for comparison is the leading root of the dielectric function, computed separately.

### Run the experiment

1. 01

   #### Find where the energy goes

   Choose A weak wave just damps and watch the energy chart.

   **What to observe:** Amber bars rise near the gold wave-speed lines: the resonant electrons gain the energy the field loses.

2. 02

   #### Trap the electrons

   Choose Electrons get trapped, then drag a test electron into the shaded band.

   **What to observe:** The distribution curls into eddies at the wave speed and the field stops decaying, then pulses with roughly the bounce time.

3. 03

   #### Turn damping into growth

   Choose A fast bump feeds the wave or Two beams roll up.

   **What to observe:** The field grows at close to the dielectric-root rate until trapping saturates it.

Physical conclusion

Damping, growth and trapping are the same resonance seen at different wave strengths and distribution slopes.

## 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 **Calculating how fast a plasma wave fades**](https://sandboxphysics.com/stories/kinetic-plasma/landau-damping-rate/)
2. [Lab **Phase-Space Filaments** Watch a ripple fold into velocity stripes](https://sandboxphysics.com/labs/phase-space-filaments/)
3. [Article **Which electrons take the wave’s energy, and when the wave holds on**](https://sandboxphysics.com/stories/kinetic-plasma/resonant-electrons-and-trapping/)

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)