Radiation of elastic waves from a point force or from a localized torque into a transversely isotropic medium has been formulated in terms of displacement potentials, and transient waveforms have been computed by numerical Fourier inversion. For isotropic sandstone, this procedure yields P- and S-wave pulses whose arrival times and magnitudes agree with theory. For a range of anisotropic rocks, arrival times of quasi-P-waves and quasi-S-waves agree with asymptotic theory. For extreme anisotropy, some quasi-S-wave pulses arrive at times which are not predicted by asymptotic theory. Magnitudes have not been compared with results of asymptotic theory, but decrease with distance appears to be in agreement. This Fourier inversion method gives near-source changes in waveform which are not obtainable from the asymptotic theory.