Let the disturbance fade.
Begin with 18 identical cars. Car 1 brakes for three seconds at model time 5 s. Let two model minutes pass. Does the spread of speeds grow or shrink?
FIELD NOTE 14 / TRAFFIC WAVES
A jam from almost nothing.
A brief slowdown. A line of responses. A pattern nobody intended.
0:00 model time · 8 model seconds per real second
Stopped110 km/hWhite outline = followed car
One lane, no overtaking. Cars and brake lights are schematic; the ring represents 1 km. Tap a car to follow it, or use the selector below.
Car 1 · speed and gap appear here.
The opening trial gives Car 1 a three-second brake tap at model time 5 s.
- Mean speed
- — km/h
- Speed spread · standard deviation
- — km/h
- Past the counting gate
- — cars/h
Collecting a 60 s window
Restart a controlled trial
Each trial restarts the clock and gives Car 1 the same brake tap at 5 s. All sliders work live.
MAKE THE PATTERN VISIBLE
The cars go forward.
Which way does the jam go?
Unwrap the loop into a line. Each coloured trace follows one car; time runs down the page. Red stretches show slow travel. The white trace follows your selected car.
Forward-moving cars trace down and right, then reappear at the left edge after a lap. A red congestion band leaning down and left is moving backward through the cars. Dashed horizontal lines mark brake taps. The plot retains the last three model minutes.
One car and the whole road.
Mean speedFollowed car
What the average hides.
The bars show how many cars share each speed range.
The opening trial starts with 42 evenly spaced, identical drivers. One brake tap is queued for 5 s.
Make a prediction. Change one thing. See what happens.
Model notes & references ↗FOLLOW THE SCIENCE
A wave made of responses.
A driver briefly slows. The following driver closes the gap and brakes too. Under some conditions, each response leaves a bigger disturbance for the next driver. A moving patch of congestion can survive long after the original brake tap ends.
This is a phantom traffic jam. It needs no blocked lane. Here, the same car-following rule can either smooth a disturbance or amplify it, depending on the traffic and driver settings. The stop-and-go wave emerges from those interactions. Flynn and colleagues: self-sustained traffic waves ↗
THREE CONTROLLED COMPARISONS
Give the same nudge to a different road.
Make a wave with no obstacle.
Repeat the same brake tap with 42 cars. Watch for a red band leaning left on the space–time plot. Follow the white car: is it always part of the jam?
Change how drivers recover.
Keep 42 cars and the same desired speed and time gap. Increase acceleration response from 0.6 to 1.8 m/s². What happens to the original disturbance?
These trial buttons deliberately restart. Ordinary controls keep the current positions, speeds and clock. Changing a setting before the queued brake tap cancels that automatic tap; you can apply one yourself.
The cars and the pattern take different journeys.
FOLLOW THE WHITE CAR
Into the jam. Out again.
The selected car slows as it enters a crowded region and speeds up as space opens ahead. It always travels clockwise or stops. It never reverses around the loop.
FOLLOW THE RED REGION
A disturbance passed backward.
Drivers join the queue at its upstream end and leave at its downstream end. Those boundaries can move upstream even though the cars move downstream. Different cars carry the pattern at different times.
A mean speed is only one piece of the story.
A road where every car travels at 35 km/h and a road where half are stopped and half travel at 70 km/h have the same mean. The histogram separates those situations. Standard deviation measures the spread around the mean; zero means all cars share a speed. Watch it grow in the wave-making trial and shrink in the light-traffic trial.
Mean speed: v̄ = (1/N) Σvi
Speed spread: σ = √[(1/N) Σ(vi − v̄)²]
Speed spread: σ = √[(1/N) Σ(vi − v̄)²]
Speed and flow answer different questions.
Speed tells you how quickly cars are moving. Flow counts how many pass a location in a given time. The counting gate measures crossings over the previous 60 model seconds and scales that count to cars per hour. Before 60 seconds have elapsed, it uses the shorter available window, so its early readings fluctuate more.
For uniform traffic, flow q equals density ρ times speed v. Adding cars raises density but can reduce speed. In uneven traffic, the gate’s recent flow also depends on which part of the wave has been passing it. Raising a desired-speed setting does not guarantee more cars get through.
Uniform flow: q = ρv · Density here: ρ = N / 1 km
Try your roadside ideas.
Choose the pulled-over car or speed camera. The lane remains open: drivers reduce their target speed near the marked location. Compare a short approach with a longer one, then clear the scene while the traffic keeps moving. Does the existing disturbance vanish immediately?
The pulled-over-car scenario assumes a target of 25 km/h at the marker; the camera scenario assumes 40 km/h. These are deliberate experimental inputs. The result depends on how drivers are assumed to respond, as well as the traffic density and their following behaviour.
No randomness is needed to make the first wave.
The opening trial uses identical drivers and a single imposed brake tap. Driver variation adds differences in preferences, but the collective pattern can already emerge from deterministic interactions. Perfectly uniform traffic without a disturbance stays uniform in this idealized model.
Give a change time to work around the loop.
A slider changes the drivers’ current rules, not the traffic’s past. Existing queues need time to clear. Use the live controls to see that adjustment, or restart a controlled trial to compare two settings from a reproducible beginning.
The mathematics, assumptions & references
The Intelligent Driver Model
Each car has a position x, speed v, and acceleration determined by its speed, the bumper-to-bumper gap s, and its closing speed Δv relative to the car ahead. The free-road term accelerates toward the desired speed v₀; the interaction term reduces acceleration as the required gap grows relative to the available gap.
dx/dt = v
dv/dt = a[1 − (v/v₀)⁴ − (s*/s)²]
s* = s₀ + max[0, vT + vΔv / (2√(ab))]
Δv = v − vahead
dv/dt = a[1 − (v/v₀)⁴ − (s*/s)²]
s* = s₀ + max[0, vT + vΔv / (2√(ab))]
Δv = v − vahead
T is the desired time gap, a is the acceleration parameter, b = 1.5 m/s² is a comfortable-braking parameter, and s₀ = 2 m is the standing gap. Braking can exceed b. The equation responds continuously to the current state; T is not an explicit perception or reaction delay. The wave-making setting uses a deliberately gentle 0.6 m/s² acceleration response to make the instability easy to explore. These parameters are illustrative, not fitted to a particular road or population. Martin Treiber’s IDM documentation ↗
A periodic road and reproducible comparisons
The loop is 1,000 m long, with 4.5 m cars and one lane. The first car follows the last across the coordinate seam. There is no passing, merging, lane choice, speed-dependent collision damage, or driver learning. A reset spaces the cars uniformly and solves the identical-driver steady-speed equation for the selected density, speed and time gap. Variation or a roadside scene can then disturb that initial state.
The three comparison trials have the same 80 km/h desired speed, 1.1 s time gap, zero driver variation and clear road. Light traffic uses 18 cars and a = 0.6 m/s². Wave maker uses 42 cars and the same acceleration parameter. Faster response keeps 42 cars and raises a to 1.8 m/s². Each applies a 3 s brake tap at t = 5 s. The tap imposes at least 3 m/s² deceleration on the selected car until it ends; the car cannot reverse.
Driver variation assigns fixed, deterministic offsets to each car’s desired speed and time gap, each bounded by the chosen percentage. Changing the number of cars adds them midway into the largest remaining gaps at no more than their neighbours’ speed, or removes unselected cars. This is an explicit intervention, not a simulation of a real on-ramp. It preserves the surviving cars’ positions and speeds.
Roadside response and numerical method
A roadside scene sits at x = 750 m. A smooth target-speed reduction begins at the chosen approach distance and reaches the scenario target at the marker. The target returns smoothly over the next 45 m. Clearing the scene restores normal desired speeds without deleting queues or resetting the clock.
Accelerations are evaluated from the same old state for every car. The model advances in 0.05 s ballistic steps and accounts for stopping within a step, so no car acquires a negative speed. A numerical non-overlap guard handles unusually abrupt live edits; the three default comparison trials run without invoking it. Charts sample every 0.5 s and retain three minutes. Road positions are scaled around the loop; vehicle widths and brake lights are enlarged for readability.
The gate counts actual simulated crossings. A followed car’s completed lap is measured between consecutive gate crossings; a car partway around the loop at the beginning must cross once before its first full lap can be timed. Sound is a synthetic speed cue with no claim to reproduce engine acoustics. Hidden tabs freeze model time; changing playback speed affects viewing time only.