A general expression is derived for the dispersion relations and the impulse response of a radially layered borehole. The model geometry consists of a central fluid cylinder surrounded by an arbitrary number of solid annuli. A Thomson-Haskell type propagator matrix is used to relate stresses and displacements across the layers. Although the model is completely general, the geometries considered here are restricted to those of a cased hole. Layers of steel, cement, and formation surround the innermost fluid layer. Synthetic microseismograms containing all body and interface waves are calculated for a variety of model parameters.Formation body wave arrivals are relatively unaffected by the presence of a casing. They may, however, be hard to identify if cement velocities are close to or larger than those of the formation. The Stoneley and pseudo-Rayleigh wave arrivals are strongly influenced by the casing parameters. They respond to the combined effects of the steel, the cement, and the formation.