Abstract: For a wide class of 2×2 nonlinear hyperbolic systems involving the transport equation and a source term, the waves are also solutions to a linear homogeneous wave equation with constant coefficients. (Linear Appearance Theorem). The nonlinearities remain in the shock waves, where the entropy reduces the total energy. We consider here the balance between the energy brought by a source term and the part lost as entropy in the shock waves, and the way to take that in account in a numerical scheme.