Which electrons take the wave’s energy, and when the wave holds on

The electrons that take a fading wave’s energy move at almost the wave speed. The wave speeds up those slightly slower than itself and slows those slightly faster. In a Maxwellian there are more slow ones, so the electrons gain energy and the wave loses it. If the wave is strong enough, it captures those electrons and swaps energy back and forth with them instead.

Sandbox Physics · 2026-10-29

Ride along with the wave

Move to the frame that travels at the wave speed $v_\phi=\omega/k$ . There the wave is a stationary row of potential hills. An electron that is slightly slower than the wave drifts backward against the hills and gets pushed forward; one slightly faster gets pushed back. Both move toward the wave speed.

The net energy flow depends on how many electrons sit on each side. A Maxwellian falls off with speed, so at $v_\phi$ there are more slower electrons than faster ones. Pushing the slower group up costs the wave more energy than it gains from slowing the faster group. The sign of $f_0^{\prime}(v_\phi)$ is the sign of the energy flow, which is exactly the slope term from chapter 4.

Worked example

Worked example: follow the energy by velocity

Launch a weak wave with $k=0.5$ and $\alpha=0.01$ . The dielectric root gives $v_\phi=1.4157/0.5\approx2.83$ and a decay rate of $0.1534$ ; the Lab measures $-0.1530$ . The lower chart shows the kinetic energy gained at each velocity since the start. Near $\pm2.83$ , the two traveling halves of the standing wave, it dips just below the wave speed and peaks just above it: slow electrons have been pushed past the wave. In the project benchmark, cells within $0.5$ of the wave speed carry between $106\%$ and $108\%$ of the net gain from $t=20$ to $80$ . The electrons farther away end up with slightly less energy than at the start, so the resonant band can hold more than the total.

No electron far from the wave speed is involved. That is why a collisionless wave can damp at all: a small group of resonant electrons absorbs the energy, and the phase mixing of chapter 3 hides the rest of the response.

v_\phi=\frac{\omega_r}{k}=\frac{1.4157}{0.5}\approx2.83,\qquad \gamma_{\mathrm{root}}=-0.1534,\qquad \gamma_{\mathrm{fit}}=-0.1530

Derivation

A strong wave traps the resonant electrons

The standing wave in the box is the sum of two traveling waves, each with half the field amplitude, $E_w=E_0/2$ . Near the bottom of one of its potential wells, an electron oscillates like a mass on a spring with the bounce frequency $\omega_b=\sqrt{kE_w}$ . Electrons within about $2\sqrt{E_w/k}$ of the wave speed are trapped.

Linear theory holds only if the wave damps before trapped electrons complete a bounce, $|\gamma|\gg\omega_b$ . For $\alpha=0.01$ , $E_w=0.01$ , $\omega_b\approx0.071$ and $|\gamma|/\omega_b\approx2.2$ : still linear. For $\alpha=0.3$ , $E_w=0.3$ , $\omega_b\approx0.39$ , the bounce time is $16.2$ and $|\gamma|/\omega_b\approx0.4$ . The trapped electrons turn in rotating eddies in the phase-space picture, the field stops falling exponentially, and the early fitted rate is $-0.30$ instead of the linear $-0.153$ .

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

Reverse the slope and the wave grows

Add a small group of fast electrons, a bump centred at $v=4.5$ . Between the Maxwellian core and the bump the slope is positive. A $k=0.3$ wave has $v_\phi\approx3.34$ , on that slope, and the root is $1.0012+0.1981i$ : growth. The Lab measures $0.1967$ . Two opposing beams at $\pm3$ give a non-oscillating root with growth rate $0.2845$ , and the Lab measures $0.2835$ .

Growth stops when trapping catches up with it. At the field peak of the bump run, $\omega_b/\gamma\approx1.4$ ; for the beams it is about $1.2$ . Both are of order one, which is the usual rule of thumb for when an instability saturates by trapping. The energy chart shows where the energy came from: the bump region loses kinetic energy as the wave grows.

Predict, change, explain

Predict whether the gold trapping band in the scene gets wider or narrower when you double the wave strength.

Switch between the weak-wave and strong-wave presets, then try the bump and the beams. Drag a test electron into the gold band.

Explain why a test electron inside the band circles while one outside it slides past, using the bounce time shown in the readout.

The three phase-space Labs solve a one-dimensional electron model in plasma units. Each Lab exports its parameters, model version and field history.

Try explaining it yourself

1. By what factor does the trapping width change when the wave field doubles?

Read an explained answer

It scales as the square root of the field, so it grows by $\sqrt2\approx1.41$ . The bounce time shrinks by the same factor.

2. For the bump case, why does a $k=0.3$ wave grow while a $k=0.5$ wave in the same plasma would damp?

Read an explained answer

The $k=0.5$ wave travels at about $2.8$ , where the Maxwellian core still falls off and the slope is negative. Only waves whose speed lands on the rising side of the bump can draw energy from it.

3. If linear theory says the wave should keep damping, but the field levels off, what should you check first?

Read an explained answer

Compare the damping rate with the bounce frequency at that field. If they are comparable, trapping has stopped the damping; if the damping rate is much larger, look for a grid recurrence as in chapter 3.

Ride along with the wave

Worked example: follow the energy by velocity

A strong wave traps the resonant electrons

Reverse the slope and the wave grows

Predict, change, explain

Try explaining it yourself

1. By what factor does the trapping width change when the wave field doubles?

2. For the bump case, why does a k=0.3k=0.3k=0.3 wave grow while a k=0.5k=0.5k=0.5 wave in the same plasma would damp?

3. If linear theory says the wave should keep damping, but the field levels off, what should you check first?

Sources to read alongside

2. For the bump case, why does a $k=0.3$ wave grow while a $k=0.5$ wave in the same plasma would damp?