Abstract: The linear appearance theorem states that, for a wide class of non linear hyperbolic systems, when a source term occurs, some solutions are also solutions to a linear homogeneous system, which means that the corresponding profiles are simply translated with a constant velocity. This allows to solve some problems by combining a sequence of such profiles separated by shock waves. Several examples are reported, such as the roll waves in hydraulics, acoustics waves as solution of gas dynamics systems in a duct, or a rarefaction wave in fluids, seen as limit of kinds of saw waves towards a Riemann invariant, as for the nonlinear homogeneous case. Some new numerical schemes adapted to the source term are presented, and tested on examples.