  We study the evolution of massless scalar waves propagating on spherically
symmetric spacetimes with a non-zero cosmological constant. Considering test
fields on both Schwarzschild-de Sitter and Reissner-Nordstrom-de Sitter
backgrounds, we demonstrate the existence of exponentially decaying tails at
late times. Interestingly the l=0 mode asymptotes to a non-zero value,
contrasting the asymptotically flat situation. We also compare these results,
for l=0, with a numerical integration of the Einstein-Scalar field equations,
finding good agreement between the two. Finally, the significance of these
results to the study of the Cauchy horizon stability in black hole-de Sitter
spacetimes is discussed.
