The finite-difference method is a powerful technique for studying the propagation of elastic waves in boreholes. Even for the simple case of an open borehole with vertical homogeneity, the snapshot format of the method displays clearly the interaction between the borehole and the rock, and the origin and evolution of phases. We present an outline of the finite-difference method applied to the acoustic logging problem, including a boundary condition formulation for liquid-solid cylindrical interfaces which is correct to second order in the space increments. Absorbing boundaries based on the formulations of Reynolds (1978) and Clayton and Engquist (1977) were used to reduce reflections from the grid boundaries. Results for a vertically homogeneous sharp interface model are compared with the discrete-wavenumber method and excellent agreement is obtained.The technique is also demonstrated by considering sharp and continuous transitions (damaged zones) at the borehole wall and by considering the effects of washouts and horizontal fissures on acoustic logs. The latter two cases are examples of wave propagation in media with properties which vary in two dimensions. For the models considered, amplitudes of head waves and head wave multiples (leaky PL modes) are frequently enhanced by washouts. The compressional body waves are less affected by the washouts and horizontal fissures than the guided Stoneley waves which are reflected and only partially transmitted at changes in borehole radius. Amplitude changes of up to 6 dB are observed in the compressional wave due to the borehole deformation. For the Stoneley wave, borehole deformations can cause changes in amplitude of 20 dB and dramatic changes in waveform.