TRN-25

Traffic shockwave speed

w = (q2 − q1) / (k2 − k1). Rankine–Hugoniot on the fundamental diagram.

Reading speed

TrafficLighthill–Whitham

Governing equation

w=\dfrac{q_2-q_1}{k_2-k_1}

$k_2 \in \left[5, 180\right]$

where

q_1: Upstream flow (veh/h)
k_1: Upstream density (veh/km)
q_2: Downstream flow (veh/h)
k_2: Downstream density (veh/km)
w: Shock speed (km/h)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

w=\dfrac{q_2-q_1}{k_2-k_1}

$k_2 \in \left[5, 180\right]$

$k_2 = 5$ $w = 26.67$

Along the curve

②

Domain of the variable — blue

k_2_{\min} = 5,\quad k_2_{\max} = 180

$k_2 \in \left[5, 180\right]$

$k_2 = 5$ $k_2 = 5$

Along the curve

③

Response curve — green

w=\dfrac{q_2-q_1}{k_2-k_1}

$k_2 \in \left[5, 180\right]$

$k_2 = 5$ $w = 26.67$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}w}{\mathrm{d}k_2}

$k_2 \in \left[5, 180\right]$

$k_2 = 6.101$ $w^{\prime} = 2.084$

Along the curve

Lecture brief

Historical brief

Highway and traffic theory crystallized with Greenshields’ 1935 speed–density line, then AASHTO stopping sight and equivalent axles. The lab is that operational arithmetic of flow, braking and pavement load. This sheet (TRN-25 — Traffic shockwave speed) is the form associated with Lighthill–Whitham. Working symbols: $q_1$ , $k_1$ , $q_2$ , $k_2$ $\rightarrow$ $w$ . LWR conservation. A jump connecting two (k, q) states travels at the chord slope.

Purpose

Purpose: compute

w

from

q_1

k_1

q_2

k_2

in Transport via

w=\dfrac{q_2-q_1}{k_2-k_1}

w = (q2 − q1) / (k2 − k1). Rankine–Hugoniot on the fundamental diagram. Use it when a real transport question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

q_1 = 1200.000\,\mathrm{veh/h}

k_1 = 20.000\,\mathrm{veh/km}

q_2 = 800.000\,\mathrm{veh/h}

k_2 = 70.000\,\mathrm{veh/km}

, the governing relation

w=\dfrac{q_2-q_1}{k_2-k_1}

yields

w = -8.00\,\mathrm{km/h}

. Two densities, two flows, a travelling front. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Upstream flow q_11200 veh/hUpstream density k_120 veh/kmDownstream flow q_2800 veh/hDownstream density k_270 veh/km

Outputs

Shock speed w-8.00 km/h

Reading speed

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Narration of this film

Two densities, two flows, a travelling front.

LWR conservation. A jump connecting two (k, q) states travels at the chord slope.

Reading speed

Watch on YouTube