For constant and oscillating boundary conditions, the 1-D advection-diffusion equation with constant coefficients, which describes a contaminant flow, is solved by the explicit finite difference method in a semi-infinite medium. It is shown how far the periodicity of the oscillating boundary carries on until diminishing to below appreciable levels a specified distance away, which depends on the oscillation characteristics of the source. Results are tested against an analytical solution reported for a special case. The explicit finite difference method is shown to be effective for solving the advection-diffusion equation with constant coefficients in semi-infinite media with constant and oscillating boundary conditions.