Stripe formation instability in crossing traffic flows

At the intersection of two unidirectional traffic flows a stripe formation instability is known to occur. In this paper we consider coupled time evolution equations for the densities of the two flows in their intersection area. We show analytically how the instability arises from the randomness of the traffic entering the area. The Green function of the linearized equations is shown to form a Gaussian wave packet whose oscillations correspond to the stripes. Explicit formulas are obtained for various characteristic quantities in terms of the traffic density and comparison is made with the much simpler calculation on a torus and with numerical solution of the evolution equations.