We prove the large time asymptotic stability of traveling wave solutions to the scalar solute transport equation (contaminant transport equation) with spatially periodic diffusion-adsorption coefficients in one space dimension. The time dependent solutions converge in proper norms to a translate of traveling wave solutions as time approaches infinity. In case of classical traveling waves, the convergence rate is exponential in time for a class of small initial perturbations; and for general order one perturbations, the convergence holds in supremum norm. In case of degenerate Ho lder continuous traveling waves, the convergence holds in L 1 norm. As a byproduct, uniqueness up to translation of degenerate traveling waves follows. We use maximum principle, L 1 contraction, spectral theory, and a space-time invariance property of solutions. 1997 Academic Press where c is the solute concentration, v is a constant water velocity, D is hydraulic diffusion; 0<n<1, k(x) is spatial random stationary process. The form k(x) c n is called the Freundlich isotherm, and other types of nonlinear functions are also possible, such as Langmuir and convex isotherms, see [10], [11], [13], [26] for details. The spatial function k(x) models the article no. DE963228