A road is divided into four sections having varying widths as shown in the figure. Section-2 (S2) represents a capacity constrained condition with respect to the traffic flow passing through Section-1 (S1). Section-3 (S3) and Section-4 (S4) do not face any such capacity constraint with respect to the flow. A flow-density relationship for unconstrained and constrained flow conditions is shown in the figure.
If Section-1 observes density D2, the option representing the correct state of density in Sections 2, 3 and 4 is:
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Work out which section forces a queue onto the congested branch, and which sections let traffic spread back out onto the free-flow branch, keeping the flow rate the same everywhere in series.
Step 1: Frame this using kinematic wave (LWR) traffic flow theory.
In the Lighthill-Whitham-Richards (LWR) theory, each traffic state (a density-flow pair) sits on the flow-density curve for the road geometry at that location. When sections of different capacity are connected in series, in a steady condition the SAME flow rate must pass through every section, by conservation of vehicles, but the density needed to carry that flow differs section to section since each has its own curve.
Step 2: Fix the system flow rate from S1's state.
S1 is unconstrained and given at density $D_2$, on the ascending (uncongested, high speed) branch of the "Unconstrained flow" curve. This sets the flow rate for the whole system, since the same horizontal flow line also touches the "Constrained flow" curve at other points.
Step 3: Determine S2's state from the bottleneck's own curve.
S2, the physically narrower section, must operate on the "Constrained flow" curve, not the unconstrained one. Since S1's flow demand exceeds what S2 can carry in free-flowing conditions, a queue forms and S2's actual operating state is forced onto the CONGESTED (descending, low speed, high density) branch of the constrained curve, at $D_4$, instead of the free-flow branch at $D_1$. This is the standard queue-forms-at-a-bottleneck behavior.
Step 4: Determine S3's state just after the bottleneck.
Once vehicles pass the pinch point of S2, they discharge at S2's maximum possible throughput, the critical (capacity) flow condition. Right after the pinch, in S3, this discharging platoon sits at the critical density $D_3$ (the peak of the curve, where flow is maximum), since it has not yet had room or time to spread out further.
Step 5: Determine S4's state further downstream.
S4 is unconstrained and far enough from the bottleneck for the platoon to relax back to free-flow conditions, moving along the unconstrained curve down to its lowest, uncongested density $D_1$, while still carrying the same overall flow rate, now at a much higher speed than at the bottleneck.
Step 6: Match with the answer choices.
This gives S2 at $D_4$, S3 at $D_3$, and S4 at $D_1$, which is option (D).
Option (A) wrongly keeps S4 at $D_2$, as if no recovery beyond S1's own state occurred. Option (B) wrongly places S2 at the critical density $D_3$ and reverses S3, S4. Option (C) swaps the roles of S3 and S4.
