Unveiling traffic wave of linear adaptive cruise control: A second-order macroscopic traffic flow model

Traffic waves, the spatiotemporal propagation of congestion, are a key feature of traffic flow. As Adaptive Cruise Control (ACC) systems gain widespread adoption and show promise for improving both efficiency and safety, understanding how these waves evolve under ACC becomes increasingly important. Yet most existing analyses rely on steady-state metrics (e.g., equilibrium spacing) and neglect the ACC control-law parameters, such as feedback gains, that fundamentally shape higher-order traffic dynamics. To overcome this limitation, we embed the ACC control law directly into the momentum equation while retaining mass conservation law. The result is a higher-order macroscopic model whose dynamics are governed by a second-order partial differential equation equivalent to the linear ACC feedback law. Analyzing the flux Jacobian confirms that the system is strictly hyperbolic, thereby preserving anisotropy and ensuring physical consistency. The derivation also shows that traffic wave evolution depends on both the initial state and the ACC control parameters. We analyze wave-propagation characteristics, linear degeneracy, admissible discontinuities, and their connection to ACC string stability, with the corresponding derivations. Numerical experiments confirm that the second-order model yields markedly lower vehicle-pair speed deviations along wave paths than a first-order model subject to the same non-steady disturbances, underscoring both the necessity of a second-order treatment and the soundness of the proposed framework.