5 · One Number Decides: the 1959 Experiment

Drivers who match the speed of the car ahead after a reaction time: one number, C, decides whether the line is calm, wobbly or crashes

viewof hermanC = Inputs.range([0.1, 1.8], {value: 0.75, step: 0.01, label: "C = λT (reaction strength × reaction time)"})

// Herman, Montroll, Potts and Rothery (1959): linear car-following with a delay, Euler steps of 0.01 s.
herman = {
  const N = 8, T = 1.0, dt = 0.01, tEnd = 50, lag = Math.round(T / dt), steps = Math.round(tEnd / dt);
  const lam = hermanC / T;
  const u = Array.from({length: N + 1}, () => new Float64Array(steps + 1));   // speed change of each car (m/s)
  for (let k = 1; k <= steps; k++) {
    const t = k * dt;
    u[0][k] = u[0][k - 1] + (t < 2 ? -1 : t < 4 ? 1 : 0) * dt;              // leader brakes, then recovers
    const kd = k - 1 - lag;
    for (let n = 1; n <= N; n++) {
      const a = kd >= 0 ? lam * (u[n - 1][kd] - u[n][kd]) : 0;
      u[n][k] = u[n][k - 1] + a * dt;
    }
  }
  const rows = [];
  for (let n = 0; n <= N; n++)
    for (let k = 0; k <= steps; k += 10)
      rows.push({car: n === 0 ? "Leader" : `Car ${n}`, n, t: k * dt, du: u[n][k]});
  const peak = n => u[n].reduce((m, x) => Math.max(m, Math.abs(x)), 0);
  return {rows, ratio: peak(N) / peak(1)};
}

Plot.plot({
  width: plotW, height: 320,
  x: {label: "Time (s)"},
  y: {label: "Speed change (m/s)", domain: [-6, 6], grid: true},
  color: {type: "linear", range: accentRange, domain: [-2, 8], legend: false},
  clip: true,
  marks: [
    Plot.ruleY([0]),
    Plot.line(herman.rows, {x: "t", y: "du", z: "car", stroke: "n", strokeWidth: 1.8}),
    Plot.text(herman.rows.filter(d => d.t === 0 && d.n === 0), {x: 6, y: -2.6, text: () => "Leader brakes", fill: "gray"})
  ]
})

{
  const C = hermanC, e = 1 / Math.E, h = Math.PI / 2;
  const regime =
    C <= e   ? "**Smooth and stable.** Each driver eases back to speed without overshooting, and the disturbance fades down the line."
    : C < 0.5 ? "**Oscillating, but still stable.** Drivers overshoot a little, yet the disturbance still fades from car to car."
    : C < h   ? "**Each pair is stable, but the line is not.** Every follower settles down eventually, yet the disturbance grows as it passes from car to car. On a long road, that is a phantom jam."
    :           "**Unstable.** Even a single follower overreacts more and more. In real traffic this ends in a collision."
  return md`${regime}

In this run, the last car's largest speed change is **${herman.ratio < 100 ? herman.ratio.toFixed(2) : "more than 100"} ×** the first follower's.`;
}

What it shows

Herman, Montroll, Potts and Rothery (General Motors, 1959) showed that one number, C = (how strongly a driver reacts) × (reaction time), decides everything. Try C below 1/2, between 1/2 and \pi/2, and above. Lecture 1 tells the story; Lab 1 has you rebuild it.

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