An explicit second-order finite-difference scheme has been used to solve the elastic-wave equation in the time domain. Solutions are presented for the perfect wedge, the lossless penetrable wedge, and the plane parallel waveguide that have been proposed as benchmarks by the Acoustical Society of America. Good agreement with reference solutions is obtained if the media is discretized at 20 gridpoints per wavelength. There is a major discrepancy (up to 20 dB) in reference-source level because the reference solutions are normalized to the source strength at 1 m in the model, but the finite-difference solutions are normalized to the source strength at 1 m in a homogeneous medium. The finite-difference method requires computational times between 10 and 20 h on a super minicomputer without an array processor. The method has the advantage of providing phase information and, when run for a pulse source, of providing insight into the evolution of the wave field and energy partitioning. More complex models, including velocity gradients and strong lateral heterogeneities, can be solved with no additional computational effort. The method has also been formulated to include shear wave effects.