We give a direct, method of characterizing traveling wave solutions to a Riemann problem within the Colombeau space $\mathcal G$. Solutions are represented by nets of approximated solutions. The propagation of singularities is also described. Although we assume that $f$ in (1) is smooth, while the usual assumption is $f\in C^2(\mathbf R)$ (cf. [28]), our approach is simple, flexible, includes generalized function solutions and give possibilities for a unified treatment of various different approaches.