This is more a conceptual question that requires a physical answer rather than a mathematical one. The question is
Explain why a density wave moves forward for light traffic. Consider both cases in which the traffic is getting heavier down the road and lighter.
I'm not totally sure what the density wave is and how it moves in traffic so fundamentally I am stuck on this problem. Any hints?
Start by imagining some average density of cars on the road, which changes slowly. You are asked to think about cases where it increases or decreases, but you might start by assuming it is constant. A density wave will be a local increase in density above the background level. The idea is that you expect increased density to slow the cars down. You are expected to think about what that means for the location and height of the maximum. The cars that start out in the peak of the wave are presumably moving slower than would be expected in the baseline case, so they don't proceed down the highway as fast. This might move the peak of the wave forward along the highway, but at a lower speed than baseline. The cars in front of them continue at baseline speed, so the density gets lower, allowing the cars to accelerate. This might damp the wave out. The cars behind the peak continue at their background speed, arriving at the wave. This might increase the amplitude and move the wave backward relative to the traffic flow.