Two pulses fade, then a third signal appears

A plasma remembers a damped wave in its velocity stripes. A second kick with a different wavelength makes part of those stripes wind backward. When they come untwisted, the electrons line up again in position and a field appears: the plasma wave echo, predicted by Gould, O’Neil and Malmberg in 1967.

Sandbox Physics · 2026-10-29

Derivation

Wind the stripes, then unwind them

A first pulse at $t=0$ with wave number $k_1$ leaves, after its field has damped, a perturbation proportional to $e^{-ik_1x+ik_1vt}$ : stripes whose velocity wave number grows as $k_1t$ . A second pulse at $t=\tau$ with wave number $k_2$ modulates the whole distribution by $e^{ik_2x-ik_2v(t-\tau)}$ , including the stripes from the first pulse.

The product contains a term with spatial wave number $k_3=k_2-k_1$ . Its velocity phase is $v[k_1t-k_2(t-\tau)]$ . The bracket shrinks after the second pulse and reaches zero at one time. At that moment the term is smooth in velocity, its velocity integral survives, and a field with wave number $k_3$ appears.

k_1t-k_2(t-\tau)=0\quad\Longrightarrow\quad \tau'=\frac{k_2\,\tau}{k_2-k_1},\qquad k_3=k_2-k_1

Worked example

Worked example: three pulse pairs

The echo Lab uses a box of length $4\pi$ , so mode $m$ has $k=0.5m$ . Pulse mode 1 then mode 2, with $\tau=20$ : $\tau^{\prime}=1.0\times20/0.5=40$ and the echo is in mode 1. The simulation’s envelope peaks at $40.46$ . For modes 2 then 3 with $\tau=15$ , $\tau^{\prime}=1.5\times15/0.5=45$ ; measured $45.04$ . For modes 1 then 3 with $\tau=20$ , $\tau^{\prime}=1.5\times20/1.0=30$ in mode 2; measured $30.53$ .

The small offsets are a fraction of one oscillation period, $2\pi/1.42\approx4.4$ time units, so they are about as precise as a peak of an oscillating field can be located. Swap the order, mode 2 then mode 1: now $k_2<k_1$ , the formula gives a time before the second pulse, and the simulation finds no echo.

The echo is second order in the pulses

The echo term is a product of the two modulations, so its amplitude is proportional to both pulse strengths. In the project benchmark, doubling the first pulse multiplies the echo by $2.00$ . Doubling the second gives $1.94$ : the second pulse’s kick is no longer small compared with the stripe spacing, and higher-order terms start to take a share.

Fig. 1 of the 1967 paper draws the echo envelope rising and then falling, with different rates on the two sides. Chapter 8 reads that figure and the equation behind it and checks the two rates against a simulation.

E_{k_3}(\tau')\;\propto\;\Phi_1\,\Phi_2

Is it an echo or a recurrence?

A coarse velocity grid can produce a revived field near the echo time, as chapter 3 explained. With 32 velocity cells the echo-mode field peaks at $38.6$ : after the grid recurrences of the two pulses at $33.5$ and $36.8$ , before the predicted $40$ , and about six times taller than the real echo. Refining to 128, 256 or 512 cells removes it and leaves the echo at the same time, which is the test that separates a physical echo from a numerical one.

The benchmark adds a second test. Widening the velocity limits from $6$ to $12$ at the same cell width leaves the echo time unchanged, which rules out the periodic velocity boundary as its cause.

An experiment to think through

Predict, change, explain

For modes 1 then 3 and a delay of 30, predict the echo time and the echo mode before you run it.
Set the delay to 30 and choose modes 1 then 3. Turn on “show only the echo mode” to see the stripes untwist.
Use the stripe spectrogram to explain the echo: the gold line for the echo-producing stripes reaches zero at the echo time.

Make an echo with two pulses Watch a ripple fold into velocity stripes

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. For modes 1 then 3 with $\tau=30$ , when and in which mode does the echo appear?

Read an explained answer

$k_1=0.5$ , $k_2=1.5$ , so $\tau^{\prime}=1.5\times30/1.0=45$ and $k_3=1.0$ , which is mode 2.

2. Why is there no echo when the second pulse has the longer wavelength?

Read an explained answer

With $k_2<k_1$ the bracket $k_1t-k_2(t-\tau)$ keeps growing after the second pulse. The stripes never unwind, so no time after $\tau$ removes their velocity dependence.

3. You double the delay between the pulses. What happens to the echo time and to its distance from the second pulse?

Read an explained answer

Both double, because $\tau^{\prime}$ and $\tau^{\prime}-\tau=k_1\tau/(k_2-k_1)$ are proportional to $\tau$ . The echo also arrives after more Landau-damping periods, which is why it stands out from both pulses.

Sources to read alongside

R. W. Gould, T. M. O’Neil and J. H. Malmberg, Plasma Wave Echo, Physical Review Letters 19, 219 (1967)
Read in full from the Caltech repository copy. Figure 1 sketches the two pulses and the echo; Equations (4) and (5) give the echo time and envelope. Our reproduction checks the echo time, the echo mode and the envelope rates in a one-dimensional simulation.
J. H. Malmberg, C. B. Wharton, R. W. Gould and T. M. O’Neil, Plasma Wave Echo Experiment, Physical Review Letters 20, 95 (1968)
The laboratory observation of spatial plasma wave echoes. Only the bibliographic record and abstract were checked; none of its data are used here.

Wind the stripes, then unwind them

Worked example: three pulse pairs

The echo is second order in the pulses

Is it an echo or a recurrence?

Predict, change, explain

Try explaining it yourself

1. For modes 1 then 3 with τ=30\tau=30τ=30, when and in which mode does the echo appear?

2. Why is there no echo when the second pulse has the longer wavelength?

3. You double the delay between the pulses. What happens to the echo time and to its distance from the second pulse?

Sources to read alongside

1. For modes 1 then 3 with $\tau=30$ , when and in which mode does the echo appear?