33.5. Shock Waves

The flow of traffic along a stream can be considered similar to a fluid flow.

This concept suggests that just like fluids, traffic can flow in a continuous manner with various characteristics such as speed, density, and flow rate. When vehicles move steadily at a constant speed and density, we refer to this condition as 'state A'.

Imagine a river where water flows smoothly without obstacles. If all the water is flowing uniformly, it resembles a stream of traffic where all vehicles are moving at the same speed.

Let this be denoted as state A. Suddenly due to some obstructions in the stream (like an accident or traffic block) the steady state characteristics change and they acquire another state of flow, say state B.

In real-world scenarios, an obstruction—such as an accident—can disrupt the flow of traffic. This leads to a new condition, 'state B', where the speed, density, and flow rates change significantly.

Think about a highway. If all vehicles are traveling smoothly (state A), but then a car accident occurs ahead, suddenly all cars must slow down or stop (state B). This abrupt change reflects how shock waves in traffic form.

The sudden change in the characteristics of the stream leads to the formation of a shock wave.

When the traffic conditions suddenly change due to an obstruction, a 'shock wave' is created. This shock wave marks the boundary between the vehicles moving at different speeds—some may still be in state A, while others have already moved into state B.

Picture a line of dominoes. If you push the first one over, it starts a chain reaction—some dominoes fall while others remain standing momentarily. Similarly, a shock wave moves through traffic, affecting vehicles in succession.

There will be a cascading effect of the vehicles in the upstream direction.

As the shock wave moves backward, vehicles upstream of the obstruction will experience changes in flow and speed due to the congestion caused by the obstruction. This is a cascading effect where each vehicle's behavior influences the ones in front of it.

Imagine a crowd at a concert. If someone falls, the person behind them has to stop, which causes a chain reaction leading to people several meters back having to stop as well. This mirrors how traffic responds to disruptions.

The speed of the shock wave (represented as ω) can be calculated using the formula: ω = (qA - qB) / (kA - kB).

The shock wave speed is determined by the difference in flow rates and density between states A and B. This mathematical formula allows for the quantification of how quickly the shock wave travels through the traffic stream.

Think of a water faucet. When you turn the water on suddenly, the change in pressure causes water to shoot out quickly. Similarly, when traffic conditions change, the speed of that change can be measured, like how quick the water flows from the faucet.

There are possibilities for other types of shock waves such as forward moving shock waves and stationary shock waves.

Shock waves can also move forward or remain stationary depending on the traffic conditions. Forward-moving shock waves occur when a denser, faster stream encounters a slower one, while stationary shock waves arise when flows are equal but densities differ.

Imagine two rivers merging; if one is flowing faster, it pushes into the slower one and creates waves that travel upstream, resembling a forward-moving shock wave. If both rivers flow at the same rate but have different volumes, they will create stationary ripples instead.

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