Nonlinear Effects in the Dynamics of Car Following

It is assumed that the velocity of a car at time t is some (nonlinear) function of the spacial headway at time t − Δ, so the equations of motion for a sequence of cars consists of a set of differential-difference equations. There is a special family of velocity-headway relations that agrees well with experimental data for steady flow, and that also gives differential equations which for Δ = 0 can be solved explicitly. Some exact solutions of these equations show that a small amplitude disturbance propagates through a series of cars in the manner described by linear theories except that the dependence of the wave velocity on the car velocity causes an accleration wave to spread as it propagates and a deceleration wave to form a stable shock. These conclusions are then shown to hold for quite general types of velocity-headway relations, and to yield a theory that in certain limiting cases gives all the results of the linear car-following theories and in other cases all the features of the nonlinear continuum theories, plus a detailed picture of the shock structure.