  We show that in one space dimension, a linearly degenerate hyperbolic system
of rich type admits exact traveling wave solutions if the initial data are
Riemann type outside of a space interval. In a particular case of the system
including physical models, we prove the convergence of entropy solutions to
traveling waves in the $L^1$ norm as the time goes to infinity. The traveling
waves are determined explicitly in terms of the initial data and the system. We
also obtain the stability of entropy solutions in $L^1$.
