We study a distinguished limit for general 2 x 2 hyperbolic systems with relaxation, which is valid in both the subcharacteristic and supercharacteristic cases. This is a weakly nonlinear limit, which leads the underlying relaxation systems into a Burgers equation with a source term; the sign of the source term depends on the characteristic interleaving condition. In the supercharacteristic case, the problem admits a periodic solution known as the roll wave, generated by a small perturbation around equilibrium constants. Such a limit is justified in the presence of artificial viscosity, using the energy method.